Some complex quaternionic equations in the type AX - XB = C are investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

We determine the one-instanton corrections to the universal hypermultiplet moduli space coming both from Euclidean membranes and NS-fivebranes wrapping the cycles of a (rigid) Calabi-Yau threefold. These corrections are completely encoded by a single function characterizing a generic four-dimensional quaternion-Kähler metric without isometries. We give explicit solutions for this function descr...

A new method for 3D rigid motion estimation is derived under the most general assumption that the measurements are corrupted by inhomogeneous and anisotropic, i.e., heteroscedastic noise. This is the case, for example, when the motion of a calibrated stereo-head is to be determined from image pairs. Linearization in the quaternion space transforms the problem into a multivariate, heteroscedasti...

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...

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and χ is a ring class character such that LK(E,χ, 1) 6= 0 we prove a G...

With a focus on investigations in the area of overactuated spacecraft, a new high-performance robust three-axis finite-time attitude control approach, which is organized in connection with the quaternion based estimation scheme is proposed in the present research with respect to state-of-the-art. The approach proposed here is realized based upon double closed loops to deal with the angular rate...

The field of neural networks has seen significant advances in recent years with the development deep and convolutional networks. Although many current works address real-valued models, studies reveal that hypercomplex-valued parameters can better capture, generalize, represent complexity multidimensional data. This paper explores quaternion-valued network application for a pattern recognition t...

In order to identify ship collision risk for the security of maritime transportation in a close-quarters situation, novel real-time awareness approach is proposed by developing non-linear velocity obstacle set, QSD-NLVO. More specifically, Quaternion Ship Domain model was introduced into algorithm, and conflict position reasonably defined. By replacing with domain, method can more assess safety...

Quaternion-valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a σ-bandli...

Quaternion models have been developed for both direction of arrival estimation and beamforming based on crossed-dipole arrays in the past. However, for almost all the models, especially for adaptive beamforming, the desired signal is still complex-valued and one example is the quaternion-Capon beamformer. However, since the complex-valued desired signal only has two components, while there are ...