Related papers: Generalized Quaternion and Rotation in 3-space E (…

We propose a generalization of spin algebra using three-index objects. There is a possibility that a triple commutation relation among three-index objects implies a kind of uncertainty relation among their expectation values.

We present a unifying framework for understanding several different versions of the Hopf fibration, and use this framework to reconcile two methods of representing rotations of 3-space by unitary matrices--the mathematician's convention…

In this note, we provide a important considerations of a familiar topic: the gradient of a vector field. The gradient of a vector field is a common quantity represented in continuum mechanics. However, even for Cartesian coordinate systems,…

Aiming at applications to the scientific visualization of three dimensional simulations with time evolution, a keyboard based control method to specify rotations in four dimensions is proposed. It is known that four dimensional rotations…

We add some comments to our old paper \cite{F-U} where the metric tensor was introduced as the gauge theory of general coordinate transformation. This formulation is more satisfactorily completed than the original one if it is required to…

Vector fields are a highly abstract physical concept that is often taught using visualizations. Although vector representations are particularly suitable for visualizing quantitative data, they are often confusing, especially when…

Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always…

We give a simple and self contained introduction to quaternions and their practical usage in dynamics. The rigid body dynamics are presented in full details. In the appendix, some more exotic relations are given that allow to write more…

In this paper, we use four-dimensional quaternionic algebra to describing space-time field equations in curvature form. The transformation relations of a quaternionic variable are established with the help of basis transformations of…

The goal of this note is to give a characterization of generalized quaternion $2$-groups by using their posets of cyclic subgroups.

In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral…

We consider a univariate beta integral composed from general modular quantum dilogarithm functions and prove its exact evaluation formula. It represents the partition function of a particular $3d$ supersymmetric field theory on the general…

A modified Gibbs's rotation matrix is derived and the connection with the Euler angles, quaternions, and Cayley$-$Klein parameters is established. As particular cases, the Rodrigues and Gibbs parameterizations of the rotation are obtained.…

In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved.…

We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.

In this paper we study ternary algebras of third-order hypermatrices. By hypermatrix we mean a complex-valued variable with three indices, which is also called a three-dimensional matrix or spatial matrix. We assume that a hypermatrix is…

In this paper, we study general rotational surfaces in the 4- dimensional pseudo-Euclidean space E4-2 and obtain a characterization of flat general rotation surfaces with pointwise 1-type Gauss map in E4-2 and give an example of such…

It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex,…

We introduce and study the algebras of generalized quaternion type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these…

A generalization of the quotient integral formula is presented and some of its properties are investigated. Also the relations between two function spaces related to the spacial homogeneous spaces are derived by using general quotient…