We consider the geometric and algebraic properties of the first-order differential equation on smooth finite-dimensional real manifolds. An affine connection without torsion is compared with a differential flow (autonomic or non-autonomic) on a manifold, with all the original trajectories being some geodesic lines of this affine connection. Using differential-algebraic characteristics of affine connectivity, we study some classes of first-order equations on smooth finite-dimensional real differentiable manifolds.

A theory of dielectric susceptibility of a nematic liquid crystal doped with isotropic nanoparticles has been developed using a simple model of a spherical nanoparticle embedded into an anisotropic dielectric medium. A simple expression is obtained for the effective polarizability of a nanoparticle in the nematic dielectric matrix. This expression is used in the theory of the macroscopic dielectric susceptibility of the nanocomposite which is developed using the anisotropic Lorentz-Lorenz equation. Finally, a shift and a splitting of the plasmon resonance of a nanoparticle are calculated determined by the polarizability anisotropy of the liquid crystal host.

Using the classical Lorentz-Hilbert-Einstein action, we derive the kinetic Vlasov-Maxwell-Einstein equation for particles in the gravitational and electromagnetic fields. The method of synchronization of intrinsic times of different particles is proposed. Based on the obtained expressions for actions (including in the post-Newtonian approximation), we analyze the connection of the cosmological lambda-term and dark energy.

We have derived an analytical expression for high-frequency electrical conductivity of a thin metal wire with a dielectric core in the case of the diffuse interaction of metal electrons with the boundaries of the conductive layer. We have analyzed the dependence of the modulus and the argument of electrical conductivity on the ratio of the radii of the wire and the dielectric core, on the effective mass along a straight line perpendicular to the axis of the wire, on the radius of the wire, and on the electric field frequency. The analysis has shown that the effective mass of charge carriers and the boundaries of the metal layer influence its electrical conductivity. The kinetic problem has been generalized to the case of an ellipsoidal Fermi surface of a metal layer, which is a natural generalization of a simpler and more frequently used model of a spherical Fermi surface in the description of transport phenomena. The paper is addressed to designers of integrated circuit elements with specified parameters.

We have studied the eigenfunctions of a characteristic equation corresponding to the kinetic equation for a Bose gas with a constant frequency of particle collisions. The solution of the homogeneous Riemann boundary value problem is used. The theory of orthogonality of eigenfunctions is generalized to quantum gases, namely, to a Bose gas. The orthogonality properties underlying the kinetic theory allow us to find an analytical solution of problems with boundary conditions.

We have calculated for the first time the conductivity of a submicron metal layer. A correction to the Wiedemann-Franz law is taken into account in the presence of specular-diffuse reflection of electrons.

The dynamics of tunneling of Bose-condensed atoms in a double-well trap is studied with allowance for the processes of elastic interatomic interaction in each well and nonlinear pair tunneling through the barrier between the wells. Solutions of the resulting system of nonlinear evolution equations describing nonstationary tunneling show that there are both periodic and aperiodic modes of evolution of the difference between the populations of the wells. The features of the time evolution of the system are determined by the initial population difference, the initial phase difference, and the nonlinearity parameter of the system. The possibility of the existence of quantum self-trapping and phase control of the system dynamics is indicated.

The paper discusses the history of the discovery of the Rayleigh-Jeans law of thermal radiation. This discovery was an important step in the prehistory of the discovery of the universal Kirchhoff function. The discovery of this function by M. Planck in December 1900 led to the gradual development of quantum theory and to the appearance of quantum mechanics in 1925-1926, which was in fact the beginning of a new physical era, i.e. quantum physics.

Entering of incorrect information in the user-machine dialog may be intentional or unintentional. In any case, it is necessary to ‘train’ the system, which serves the machine, to understand and to recognize errors with the information input and to correct them. Thus, the aim of the work is to develop an algorithm for protection against violations of information input and correction of the final result. For this purpose, existing and traditional algorithms are studied. Their advantages and disadvantages in relation to the task are revealed. In particular, a similar situation with the detection of the input error, its recognition and correction is often observed in the most primitive situations - in calculators translating one number system (NS) to another. This primitiveness, first of all, indicates the fundamental nature of the problem to be solved. Therefore, in the presented work we use the already existing algorithms to discuss and develop the ways to correct the major errors that are found in Online Calculators translating one number system (NS) to another.