Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the in...

the aim of this article is to extend the study of the lorentz transformation of order (m,n) from m=1 and n>=1 to all m,n>=1, obtaining algebraic structures called a bi-gyrogroup and a bi-gyrovector space. a bi-gyrogroup is a gyrogroup each gyration of which is a pair of a left gyration and a right gyration. a bi-gyrovector space is constructed from a bi-gyrocommutative bi-gyrogroup that admits ...

A regular way to define an additive coproduct (or coaddition) on the q-deformed differential complexes is proposed for quantum groups and quantum spaces related to the Hecke-type R-matrices. Several examples of braided coadditive differential bialgebras (Hopf algebras) are presented.

‎The Lorentz transformation of order $(m=1,n)$‎, ‎$ninNb$‎, ‎is the well-known ‎Lorentz transformation of special relativity theory‎. ‎It is a transformation of time-space coordinates of the ‎pseudo-Euclidean space $Rb^{m=1,n}$ of one time dimension and ‎$n$ space dimensions ($n=3$ in physical applications)‎. ‎A Lorentz transformation without rotations is called a {it boost}‎. ‎Commonly‎, ‎the ...

The gyroscopic force acting on a ferromagnetic soliton reflects the tendency of spins to precess. Its components increase linearly with (generalized) velocities soliton. proportionality coefficients form tensor, generalization Thiele's gyrovector. Originally introduced as an auxiliary quantity, tensor turns out be mathematical object fundamental importance long history going back Lagrange. We r...

Magnetic Hopfions are localized magnetic solitons with a non-zero 3D topological charge (Hopf index). Here we present an analytical calculation of the Hopfion gyrovector and show that it does not vanish even in infinite sample. The method is based on concept emergent field. particular case simplest non-trivial toroidal Hopf index |QH|=1 cylindrical dot considered dependencies components sizes c...

in this paper, we consider a generalization of the real normed spaces and give some examples.

Defects introduced in ferromagnetic nanodisks may deeply affect the structure and dynamics of stable vortex-like magnetization. Here, analytical techniques are used for studying, among other dynamical aspects, how a small cylindrical cavity modify the oscillatory modes of the vortex. For instance, we have realized that if the vortex is nucleated out from the hole its gyrotropic frequencies are ...