In terms of the relational approach to space-time geometry and physical interactions, we show that Dirac equation for a free fermion in momentum representation can be obtained starting from binary system complex relations (BSCR) between elements two abstract sets. With derivation performed, 4-dimensional pseudo-Euclidean space is not needed priori but naturally emerges considerations rather gen...

Quantum Euclidean spaces, as Moyal deformations of are the model examples noncompact noncommutative manifold. In this paper, we study quantum space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example semifinite spectral triple nonflat geometric structure. We develop abstract symbol calculus for pseudo-differential operators noncommuting deriv...

This article is presented new method of description information systems in abstract 4-dimensional pseudo-Euclidean information space (4-DPIES) with using special relativity (SR) methods. This purpose core postulates of existence 4-DPIES are formulated. The theorem setting existence criteria of the invariant velocity of the information transference is formulated and proved. One more theorem allo...

We provide a general framework to study invariant properties of various gradient-like and Laplace-like differential operators naturally associated geometric structures on $\mathbb{R}^n$, which encompass Euclidean, Minkowski, pseudo-Euclidean symplectic structures.

A rep-tiling ~is a self replicating, lattice tiling of R". Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, and self-replicating means that there is a non-singular linear map ¢: R"-o R" such that, for each T e J, the image 4~(T) is, in turn, tiled by Y-. This topic has recently come under investigation, not only because of its recreational appeal,...

The samples of geometric interpretation of space-time features of special relativity theory and phyllotaxis botanic phenomenon demonstrate variance of Minkovski’s theory application in mathematic modeling of natural phenomena. It is a well-known fact that in 1908, i.e. three years after A. Einstein’s theory of relativity was published, mathematician G. Minkowski made public geometric interpreta...

in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

We introduce new Bézier-like formulas with quaternionic weights for rational curves and surfaces in Euclidean 3-space. They are useful for representation of Möbius invariant geometric objects. Any Bézier curves and surfaces on 2-spheres can be converted to the quaternionic Bézier (QB) form of twice lower degree. Our focus is on bilinear QB-surfaces: we derive their implicitization formula and s...

We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of Killing–Yano tensors. Passing from the spinning spaces to the Dirac equation in curved backgrounds we point out the role of the Killing–Yano tensors in the construction of the Dirac-type operato...