Vol 509, No 1 (2023)

In this paper, we obtain two-way estimates of the Alexandrov’s n-width of a compact set of infinitely smooth periodic functions that are bounded embedded in the space of continuous functions on the unit circle.

For the first time, bicompact schemes are generalized to non-stationary Navier–Stokes equations for a compressible heat-conducting fluid. The proposed schemes have an approximation of the fourth order in space and the second order in time, are absolutely stable (in the frozen-coefficients sense), conservative, and efficient. One of the new schemes is tested on several two-dimensional problems. It is shown that when the mesh is refined, the scheme converges with an increased third order. A comparison is made with the WENO5-MR scheme. The superiority of the chosen bicompact scheme in resolving vortices and shock waves, as well as their interaction, is demonstrated.

The paper deals with the study of the limit distribution of the \(j\)-chromatic numbers of a random k-uniform hypergraph in the binomial model \(H(n,k,p)\). We consider the sparse case when the expected number of edges is a linear function of the number of vertices \(n\), i.e. is equal to \(cn\) for \(c > 0\) not depending on \(n\). We prove that for all large enough values of \(c\), the \(j\)-chromatic number of \(H(n,k,p)\) is concentrated in one or two consecutive numbers with probability tending to 1.

In this paper, we study weak saturation numbers of binomial random graphs. We proved stability of the weak saturation for several pattern graphs, and proved asymptotic stability for all pattern graphs.

Let A ≥ m_A > 0 be a closed positive definite symmetric operator in a Hilbert space ℌ, let \({{\hat {A}}_{F}}\) and \({{\hat {A}}_{K}}\) be its Friedrichs and Krein extensions, and let _∞ be the ideal of compact operators in ℌ. The following problem has been posed by M.S. Birman: Is the implication A^–1 ∈ G_∞ ⇒ (\({{\hat {A}}_{F}}\) )^–1 ∈ G_∞(ℌ) holds true or not? It turns out that under condition A^–1 ∈ G_∞ the spectrum of Friedrichs extension \({{\hat {A}}_{F}}\) might be of arbitrary nature. This gives a complete negative solution to the Birman problem.Let \(\hat {A}_{K}^{'}\) be the reduced Krein extension. It is shown that certain spectral properties of the operators (\({{I}_{{{{\mathfrak{M}}_{0}}}}}\) + \(\hat {A}_{K}^{'}\))^–1 and P₁(I + A)^–1 are close. For instance, these operators belong to a symmetrically normed ideal G, say are compact, only simultaneously. Moreover, it turns out that under a certain additional condition the eigenvalues of these operators have the same asymptotic.Besides we complete certain investigations by Birman and Grubb regarding the equivalence of semiboubdedness property of selfadjoint extensions of A and the corresponding boundary operators.

An inverse problem of determination of a variable coefficient in electrodynamic equations with a nonlinear conductivity is considered. It is supposed that the unknown coefficient is a smooth function of space variables and finite in \({{\mathbb{R}}^{3}}\). From a homogeneous space a plane wave going in a direction fall down on a heterogeneousness. The direction is a parameter of the problem. The module of the electrical strength vector for some diapason of directions and for moments of the time close to arriving the wave at points of a surface of a ball, inside of which the heterogeneousness is contained, is given as the information for solution of the inverse problem. It is shown that this information reduces the inverse problem to the well known X-ray tomography. Algorithms of the numerical solution of the later problem is well developed.

A geometric inverse problem of identifying an isotropic, linearly elastic inclusion in an isotropic, linearly elastic plane is considered. It is assumed that constant stresses are given at infinity, and the displacements and acting loads are known on some closed curve enclosing the inclusion. In the case when the inclusion is a quadrature domain, a method for identifying its nodal points has been developed. A numerical example is considered.

The paper considers a fundamentally new approach to solving the problem of determining the size of structural formations in ultrasonic diagnostics, based on the theoretically justified possibility of estimating the size of inhomogeneities of the studied medium by analyzing the statistical characteristics of the ultrasonic signal scattered on these inhomogeneities. This possibility is conditioned by the fact that the statistical distribution of the ultrasound image data varies from Rayleigh distribution to Reiss distribution depending on the relation between the coherence area size of the scattered signal and the beamwidth. The work aims at the development of a new method of statistical data analysis, which will effectively detect a significant coherent component in the echo signal and thereby be used as a mathematical tool to estimate the size of medium inhomogeneities in ultrasound imaging. Such approach to the analysis of ultrasound images would provide a possibility of quantitative estimation of structural formations and thereby would increase significantly the information value of ultrasound diagnostics and possibility of pathology detection at early stages of its formation that opens perspectives for treatment efficiency increase.

The paper is devoted to mathematical modeling of the melting process in a sample under the influence of a pulsed thermal load based on the solution of the two-phase Stefan problem. The free boundary is ignoring during the calculation, since the numerical model is based on the Samarsky approach. The calculation in axially symmetric geometry allowed us to show that about a quarter of the incident energy is consumed in the center of the melt region. This is five times more than estimates based on the solution of the one-dimensional heat equation give. Considering the evaporation of the substance a good correspondence between the calculated and experimental temperatures of the cooling surface and the rate of narrowing of the melt region is obtained. The results of mathematical modeling confirmed the existence of an evaporation cooling mode when tungsten is heated by an electron beam significantly above the melting threshold.