In the article, the sequence of processes constructed for the random process , which is de fined as a function of the Brownian motion process , are considered . The central moments of the sequences converge to the corresponding moments of the initial process . The accuracy of approximations is illustrated by examples.

The sequential probability ratio test (SPRT) is considered, when the actual probability distribution of observations is unknown and differs from the theoretical one, but belongs to its e-neighborhood in the  L1or C-metric. The least favorable distribution (that maximizes the type I error probability of the SPRT) of observations is constructed for each metric and each e fixed in advance.

The method of approximate evaluation of matrix-valued functional integrals, generated by the solution of the Dirac equation is proposed. The method is based on the expansion of the functional in a series. The terms of the series have the form of a product of linear functionals with an increasing total power. The convergence of this series for a wide class of functionals is proved.

The computing scheme of numerical solution of the first-kind integral equation with the triple Cauchy kernel in classes of bounded and unbounded functions by the method of orthogonal polynomials is obtained.

For linear autonomous algebraic-differential systems a new type of feedback controls is proposed. Their effectiveness is seen in the example of the problem of calming solutions. The distinctive feature of the proposed types of controls is the possibility of its implementation for the systems that do not possess the property of complete controllability.

Constructive conditions of unique solvability and algorithms for constructing the solutions of periodic boundary value problem for the matrix differential equation, which is a generalization of the Lyapunov and Riccati equations are obtained.

Пусть D / K гензелева дискретно-нормированная слабо разветвленная центральная K-алгебра с делением нечетного индекса. В статье установлено, что для существования циклической унитарной K/k-инволюции (K/k сепарабельное расширение степени 2) необходимо существование в поле вычетов k поля k примитивного корня р-й степени для любого простого делителя р индекса алгебры D. Из этого утверждения немедленно вытекает описание широкого класса нециклических инволюций в вышеупомянутых алгебрах.

In this article we consider the computational complexity of two problems related to bicliques (complete bipartite subgraphs) of a graph in the class of split graphs. Given a graph and an integer k, the biclique cover problem asks whether the edge set of the graph can be covered with at most k bicliques; the biclique vertex-cover problem asks whether the vertex set of the graph can be covered with at most k bicliques. These problems are known to be NP-complete even in the class of bipartite graphs. In this article, we show that the both problems remain NP-complete in the class of split graphs.

We construct an example of a serial module, whose endomorphism ring modulo by its Jacobson radical is neither von Neumann regular nor one-sided self-injective. This solves a problem posed by Alberto Facchini.

Within the concept of yield surfaces based on the principle of preserving the potential nature of elastic deformation the model of material is proposed. The active elastoplasticity process is presented by the alternation of plastic and elastic states. Deviatory cross sections of the yield surface are defined. Defining relationships are formulated. The model examples for orthotropic materialare considered.

A system of three differential equations describing the process of continuous bacteria cultivation in a chemostat is considered. For a simple food chain described by the dynamic Michaelis-Menten chemostat model a two-parameter analytical solution is obtained. An algorithm and software allowing one to find an explicit form of solutions with the given limit properties have been constructed with the usage of the CAS Mathematica capabilities. Examples, in which it is possible to model the survival or extinction of one or two microorganisms and to find initial concentration ranges, provide “competitive exclusion” or coexistence of the both organisms, are given.

Non-topological solitons of high dispersive nonlinear Schrodinger equations are obtained.

In this work, we investigated the influence of laser radiation and radiation of light-emitting diode sources (LEDs) on the proliferative activity of animal cells in culture, the postembryonic development of sturgeon fishes upon short single irradiation of fertilized caviar, the activity of fish sperm, the zooplankton hatching upon irradiation of cysts depending on the parameters of the acting factor (power density, wavelength, polarization, degree of coherence). The methods of enhancement of the biological activity and the therapeutic action of optical radiation due to its modulation, the mutual action with a constant magnetic field, the combined consecutive radiation action in different spectral regions were developed. The mechanisms of photochemical and non-photochemical (non-resonant) processes lying in the basis of the observed light-induced effects were examined.

The history of laser sensing in Belarus is given. Examples of lidar developments of various designations and their applications are considered. The role of the Institute of Physics, National Academy of Sciences of Belarus in the creation of the European and former Soviet-Union lidar networks is emphasized.

The processes of lepton pair production by means of the two-photon mechanism in hadron collisions at LHC and Tevatron energies are considered. These processes could be used for luminosity calibration of LHC and for a «new physics» search at the collider, in particular, heavy leptons. Constructing the needed Monte Carlo generator we have obtained total and differential cross sections of the above mentioned processes for elastic hadron scattering in natural approximation.

We investigate the OC-convexity defined by the intersections of conic semispaces of partial convexity. A polynomial time algorithm is developed to determine the OC-convex approximation of directional convex hull of a finite set of polytopes in case of the direction set being finite.

In this article universal sets of points in the plane for compatible triangulations have been found; it is shown that the disjoint compatible matching graph, in general, is not connected; it has been proved that an arbitrary perfect matching has a disjoint compatible spanning tree.

To the 80th anniversary