Dual Quaternion Embeddings for Link Prediction

The applications of knowledge graph have received much attention in the field of artificial intelligence. The quality of knowledge graphs is, however, often influenced by missing facts. To predict the missing facts, various solid transformation based models have been proposed by mapping knowledge graphs into low dimensional spaces. However, most of the existing transformation based approaches ignore that there are multiple relations between two entities, which is common in the real world. In order to address this challenge, we propose a novel approach called DualQuatE that maps entities and relations into a dual quaternion space. Specifically, entities are represented by pure quaternions and relations are modeled based on the combination of rotation and translation from head to tail entities. After that we utilize interactions of different translations and rotations to distinguish various relations between head and tail entities. Experimental results exhibit that the performance of DualQuatE is competitive compared to the existing state-of-the-art models.

Quaternionic contact 4n + 3-manifolds and their 4n-quotients

We study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION

In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

Interpolating Industrial Robot Orientation with Hermite Spline Curve Based on Logarithmic Quaternion

Smooth orientation planning has an important influence on the working quality and service life as for industrial robot. Based on the logarithmic quaternion, a compact method to map a spline curve from Cartesian space to quaternion space is proposed, and consequently the multi-orientation smooth interpolation of quaternion is realized. Combining with the relevant example case, the detailed method and steps of multi-orientation interpolation are introduced for mapping Hermite spline curve into quaternion space, and the validity of the principle is verified by using the example case. The present multi-orientation smooth interpolation of quaternion has the characteristics of simple construction, easy implementation and intuitive understanding. The method is not only applicable to multi-orientation interpolation of quaternion with Hermite spline curve, but also can extended to the spline curves such as Bezier spline and B-spline.

Three-dimensional texture visualization approaches: applications to nickel and titanium alloys

This paper applies the three-dimensional visualization techniques explored theoretically by Callahan, Echlin, Pollock, Singh & De Graef [J. Appl. Cryst.(2017),50, 430–440] to a series of experimentally acquired texture data sets, namely a sharp cube texture in a single-crystal Ni-based superalloy, a sharp Goss texture in single-crystal Nb, a random texture in a powder metallurgy polycrystalline René 88-DT alloy and a rolled plate texture in Ti-6Al-4V. Three-dimensional visualizations are shown (and made available as movies as supplementary material) using the Rodrigues, Euler and three-dimensional stereographic projection representations. In addition, it is shown that the true symmetry of Euler space, as derived from a mapping onto quaternion space, is described by the monoclinic color space groupPccin the Opechowski and Guccione nomenclature.