The Newton equation describing the particle motion in constant external field force on canonical, Lie-algebraic and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of particle is generated. We also indicate, that in the case of spatial coordinates commuting in Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and positiondependent forces.1 There also exist so-called fuzzy space noncommutativity [43]. However, in this article such a type of deformation will be not under consideration.2 We consider only spatial deformations, i.e. time plays a role of parameter. 3 In the case of "position force" one can recognize well-known inverted oscillator force (see e.g. [46]).

We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

We provide the classical mechanics of many particles moving in canonically twistdeformed space-time. In particular, we consider two examples of such noncommutative systems -the set of N particles moving in gravitational field as well as the system of N interacting harmonic oscillators.1 For earlier studies see [39] and [40].

The classical and quantum model of high spin particles is analyzed within the manifestly covariant framework. The model is obtained by supplementing the standard Lagrange function for relativistic point particle by additional terms governing the dynamics of internal degrees of freedom. They are described by C(3, 1) Clifford algebra (Majorana) spinors. The covariant quantization leads to the spectrum of the particles with the masses depending on their spins. The particles (and anti-particles) appear to be orphaned as their potential anti-particle partners are of different mass.

We present a new algorithm for the approximate evaluation of the inverse square root for single-precision floating-point numbers. This is a modification of the famous fast inverse square root code. We use the same “magic constant” to compute the seed solution, but then, we apply Newton–Raphson corrections with modified coefficients. As compared to the original fast inverse square root code, the new algorithm is two-times more accurate in the case of one Newton–Raphson correction and almost seven-times more accurate in the case of two corrections. We discuss relative errors within our analytical approach and perform numerical tests of our algorithm for all numbers of the type float.