Abstract. Let K be a field. In this paper, we found the necessary and sufficient conditions for K-homomorphisms of noncommutative finite-dimensional central division K-algebras with henselian valuations being K-isomorphisms. A similar result is obtained for the case of algebras with involutions. A category interpretation of these results is given.

In this paper, we investigate the stability with respect to initial data in the uniform norm of an implicit difference scheme approximating a nonlinear transport equation. An iterative process is used to implement the difference scheme. The convergence of the iterative process and the stability of the difference scheme are proven in the case of initial data guaranteeing the absence of shock waves. In the case of the occurrence of shock waves, estimates of the growth of spatial derivatives at each time layer are obtained. An adaptive computational algorithm for solving the transfer equation during the formation of shock waves is built.

Herein we consider mathematical models of infection in a population consisting of two types of people: those who transmit infection to others (type 1) and those who do not participate in the spread of infection (type 2). On the basis of the percolation theory and a model of the urn test type, a critical value of the proportion of infected persons in the population is determined, after which the infection process may become explosive. The probabilities of continuous infection and the interruption of its transmission are investigated. On the basis of Feigenbaum logistic mapping for the epidemic process, it is possible to estimate the change in the value of the parameter of the number of contacts and the bifurcations arising in this case, which are modelled in accordance with the scenario of transition to deterministic chaos through the doubling of the cycle period. In modes of stochasticity there are local modes of periodicity, the identification of which, if the model is adequate to the real situation, allows predicting and controlling the epidemic process, translating it or keeping the process in a stable cyclic state.

In this paper, we study the semiclassical approximation of multiple functional integrals. The integrals are defined through the Lagrangian and the action. Of all possible trajectories, the greatest contribution to the integral is given by the classical trajectory x̅_cl for which the action S takes an extremal value. The classical trajectory is found as a solution of the multidimensional Euler – Lagrange equation. To calculate the functional integrals, the expansion of the action with respect to the classical trajectory is used, which can be interpreted as an expansion in powers of Planck’s constant. The numerical results for the semiclassical approximation of double functional integrals are given.

In this paper, we considered non-static vacuum spherically symmetric solutions of the Einstein equations and harmonicity conditions in the coordinate system with a non-zero space-time component in the metric. For the case of the weak field, a particular solution of the approximate equations was obtained, which corresponds to a nonstatic source whose boundary moves with a constant speed. For the exact Einstein’s equations we obtained a wave-type solution, determined by two implicitly specified functions, depending on the retarded argument and on the radial coordinate, respectively. The connection between these solutions and the Birkhoff theorem is discussed.

In the relativistic quark model, based on the point form of Poincaré-invariant quantum mechanics, an integral representation of the form-factor of the pseudoscalar P⁰(π⁰,η,η′) meson of  P⁰ (qq̅)→γγ decay is obtained taking into account the anomalous magnetic moments of u-, d- and s-quarks. IIn the developed formalism the values of the constituent quark masses and the parameters of the wave functions are calculated using the lepton decay  P^±(qQ̅ ) → ℓ^±ν_ℓ± constant f_P^± and the pseudoscalar density constant g_P±. It is shown that taking into account the gluonium component in η/η′-mesons and using the structure functions of light sector quarks lead to the behavior of the form factors of pseudoscalar π⁰-, η-, η′-mesons in the area of a small transferred momentum to the lepton pair, which is consistent with the modern experimental data

The results of a qualitative analysis of a semiclassical model of light generation in low-dimensional solidstate lasers, including quantum-dot microlasers and, on its basis, a numerical modelling of the regular pulsation regime that occurs under conditions of nonlinear shift and broadening of the resonant spectral gain line due to the influence of dipoledipoles interaction and absorption in quasi-resonant transitions on the dielectric susceptibility of the active medium are herein presented. The modelling of lasing was carried out for the parameters of semiconductor quantum-dot structures.

We investigate the explicit, analytical expressions for the fifth-order electromagnetic corrections in the fine structure constant α to the anomalous magnetic moment of leptons aL (L = e, μ, τ) from diagrams with insertions of the vacuum polarization operator consisting of pure lepton loops. Our approach is based on the consecutive application of dispersion relations for the polarization operator and the Mellin – Barnes transform for the propagators of massive particles. Exact analytical expressions for the corrections to aL from vacuum polarization by four identical loops are obtained. Asymptotic expansions are found in the limit of both small and large values of the lepton mass ratio  (r = m_ℓ /m_L), r≪ 1 and r→∞ The resulting expansions are compared with the corresponding expressions given in the literature.