A simple and direct, based on the equations of motion, derivation of the variational principle and effective actions for a spherical charged dust shell in general relativity is offered. This principle is based on the relativistic version of the D’Alembert principle of virtual displacements and leads to the effective actions for the shell, which describe the shell from the point of view of the exterior or interior stationary observers. Herewith, sides of the shell are considered independently, in the coordinates of the interior or exterior region of the shell. Canonical variables for a charged dust shell are built. It is shown that the conditions of isometry of the sides of the shell lead to the Hamiltonian constraint on these interior and exterior dynamical systems. Special cases of the “hollow” and “screening” shells are briefly considered, as well as a family of the concentric charged dust shells.

Discussed are some problems of two (or more) mutually coupled systems with gyroscopic degrees of freedom. First of all, we mean the motion of a small gyroscope in the non-relativistic Einstein Universe \(\mathbb {R}\times S^3 (0,R)\); the second factor denoting the Euclidean \(3\)-sphere of radius \(R\) in \(\mathbb {R}^4\). But certain problems concerning two-gyroscopic systems in Euclidean space \(\mathbb {R}^{3}\) are also mentioned. The special stress is laid on the relationship between various models of the configuration space like, e.g. , \(\mathrm {SU}(2)\times \mathrm {SU}(2)\), \(\mathrm {SO}(4,\mathbb {R})\), \(\mathrm {SO}(3,\mathbb {R})\times \mathrm {SO}(3,\mathbb {R})\) etc. They are locally diffeomorphic, but globally different. We concentrate on classical problems, nevertheless, some quantum aspects are also mentioned.

We find the subgroup of classical acceleration-enlarged Newton–Hooke Hopf algebra which acts covariantly on the twisted acceleration-enlarged Newton–Hooke space-times. The case of classical acceleration-enlarged Galilei quantum group is considered as well.

The resonant production of the fourth family slepton \(\tilde {l}_{4}\) via R-parity violating interactions of supersymmetry at the Large Hadron Collider has been investigated. We study the decay mode of \(\tilde {l}{}_{4}\) into the fourth family neutrino \(\nu _{4}\) and \(W\) boson. The signal will be a like-sign dimuon and dijet if the fourth family neutrino has Majorana nature. We discuss the constraints on the R-parity violating couplings \(\lambda \) and \(\lambda '\) of the fourth family charged slepton at the LHC with the center of mass energies of 7, 10 and 14 TeV.

Some neutrino physics aspects of neutrino-less double beta decay are discussed: this includes the possibility to test or rule out the inverted neutrino mass ordering, distinguishing neutrino mass models, or the effects of light sterile neutrinos.

Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with a nonlinear term of the form \({\rm sign}(\varphi )\) are presented. They are constructed from cubic polynomials in the scale invariant variable \(t/r\). One class of solutions describes the process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.

The PYTHIA 8 generator is used to estimate the percentage of the non-diffractive and diffractive events at the LHC energies. It is shown that a simple condition of the absence of charged hadrons in the central pseudorapidity region is sufficient to remove almost all non-diffractive events. This opens the way to investigate diffraction without waiting for the future specialized detectors.

The concept of the “wounded” hadronic constituents is formulated. Preliminary estimates indicate that it may help to understand the transverse mass dependence of the particle production in hadron–nucleus and nucleus–nucleus collisions.

We have given a simpler derivation of the Rotne–Prager tensor based on the exact fluid velocity formula for a uniform flow past a sphere in low Reynolds number regime. For two identical spheres in uniform flow, we have given the hydrodynamic interaction profiles for different distances and angles, which are formed between the flow directions and the connection lines of two spheres. The lift forces perpendicular to the flow directions are responsible for the migration phenomenon in the vorticity direction of a chiral object in shear flow.