In this paper, we express De Moivre’s formula for split quaternions and …nd roots of a split quaternion using this formula. MSC 2000 : 53B30, 15A66.

A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization traditional and complex based on quaternions. The novelty that the range membership function set quaternions with modulus less than or equal to one, which real imaginary parts can be used for four different features. discussion made intuitive interpretation quaternion-valued grades possible a...

The properties of spinors and vectors in (2 + 2) space split quaternions are studied. Quaternionic representation rotations naturally separates two SO(2,1) subgroups the full group symmetry norms quaternions, SO(2,2). One them represents symmetries three-dimensional Minkowski space-time. Then, second subgroup, generated by additional time-like coordinate from basis can be viewed as internal mod...

In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of R. This has been known for quite some time and is perhaps the simplest realization of Hamilton’s expectations about the potential of quaternions for physics. One reason for the renewed interest is the fact that the resulting substitution of matrices by quaternions speeds up considerably the numerical calculatio...

Attitude quaternion [1] is the attitude parameterization of choice for spacecraft attitude estimation for several reasons: 1) it is free of singularities, 2) the attitude matrix is quadratic in the quaternion components, and 3) the kinematics equations is bilinear and an analytic solution exists for the propagation. However, the components of the attitude quaternion are not independent of each ...

A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the root-mean-square deviation (RMSD). We present a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors. We prove that the quaternion me...

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R.

Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions with five examples. One of those examples gives ...

Dual quaternion algebra and its application to robotics have attracted considerable interest in the last two decades. quaternions great geometric appeal easily capture physical phenomena inside an algebraic framework, which is useful for both robot modeling control. Mathematical objects, such as points, lines, planes, infinite cylinders, spheres, coordinate systems, twists, wrenches, are all we...

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven surge applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues high-dimensionality, they all induce cost in terms network size or computational runtime. work examines the use quaternions, form hypercomplex...