In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kähler foliations. The limiting case is characterized by the existence of quaternion-Kähler Killing spinors. We end this paper by giving some examples.

ABSTRACT : Quaternion Dirac equation has been obtained from the square root of Klein-Gordon equation in compact and consistent way. Dirac matrices are described as quaternion valued and the Dirac Hamiltonian is considered as Hermitian with real eigenvalues of energy. Dirac spinors and free particle energy solution has been obtained in terms of one component, two-component and four-component Dir...

This open access textbook presents a comprehensive treatment of the arithmetic theory quaternion orders. Essential for students and researchers.

In this article, we study the Hyers-Ulam stability of first-order linear quaternion-valued differential equations. We transfer a equation into real system. The results for equations are obtained according to equivalent relationship between vector 2-norm and quaternion module.

We propose a unitary diagonalisation of a special class of quaternion matrices, the socalled η-Hermitian matrices A = AηH , η ∈ {ı, ȷ, κ} arising in widely linear modelling. In 1915, Autonne exploited the symmetric structure of a matrix A = AT to propose its corresponding factorisation (also knownas the Takagi factorisation) in the complex domain C. Similarly, we address the factorisation of an...

The traditional methods for representing forward kinematics of manipulators have been the homogeneous matrix in line with the D-H algorithm. In this paper a new method known as quaternion algebra is described and it is used for three dimensional representation and orientation in robot kinematics. This method is compared with homogeneous transform in terms of easiness of representation, computat...

A new quad-quaternion model is herein established for an electromagnetic vector-sensor array, under which a multidimensional algebra-based direction-of-arrival (DOA) estimation algorithm, termed as quad-quaternion MUSIC (QQ-MUSIC), is proposed. This method provides DOA estimation (decoupled from polarization) by exploiting the orthogonality of the newly defined “quadquaternion” signal and noise...

We investigate the problem of quaternion beamforming based on widely linear processing. First, a quaternion model of linear symmetric array with two-component electromagnetic (EM) vector sensors is presented. Based on array's quaternion model, we propose the general expression of a quaternion semiwidely linear (QSWL) beamformer. Unlike the complex widely linear beamformer, the QSWL beamformer i...

Two-layered neural networks are well known as autoencoders (AEs) in order to reduce the dimensionality of data. AEs are successfully employed as pre-trained layers of neural networks for classification tasks. Most of the existing studies conceived real-valued AEs in real-valued neural networks. This study investigated complexand quaternion-valued AEs for complexand quaternion-valued neural netw...

The quaternion Fourier transform (QFT), which is a nontrivial generalization of the real and complex Fourier transform (FT) using quaternion algebra has been of interest to researchers for some years (see e.g. [3, 5]). It was found that many FT properties still hold but others have to be modified. Based on the (right-sided) QFT, one can extend the classical wavelet transform (WT) to quaternion ...