Vol 97, No 2 (2018)

The contact problem for a thin elastic rigid plate described by the elasticity equations and a viscoelastic layer is solved. The ratio of the thicknesses of the plate and the layer is a small parameter, while the ratio of the Young’s moduli of the layer and the plate is proportional to the cube of this parameter. The asymptotic expansion of the solution is constructed. A theorem on the estimate of the error of asymptotic approximation is formulated. Such problem appears in geophysics, in modeling of the Earth crust–magma interaction.

We investigate one-dimensional (2p × 2p)-matrix Dirac operators D_X,α and D_X,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators D_X,α and D_X,β (self-adjointness, discreteness of the spectrum, etc.) are identical to the corresponding properties of some Jacobi matrices B_X,α and B_X,β with (p × p)-matrix entries. The relationship found is used to describe these properties as well as conditions of continuity and absolute continuity of the spectra of the operators D_X,α and D_X,β. Also the non-relativistic limit at the velocity of light c → ∞ is studied.

In the present paper we study the generic ranks of special matrix-valued maps defined by certain systems of parameters via Kronecker products. We introduce the notions of minimal superabundant, balanced and reducible systems. The main result of the paper is a theorem for maps with minimal superabundant systems of parameters. For such systems it associates the value of the generic rank with the balancedness. The proof of this theorem is based on a reduction by the parameters and consists of verifying the fact of reducibility.

The nonisothermal two-phase porous media flow of oil and hot water in a horizontal oil reservoir is considered. An asymptotic method for calculating the flow and solving related optimal control problems is proposed. Namely, the problem of choosing optimal control actions to maximize the oil production for a given level of financial costs and to minimize the costs for a given level of oil production is considered.

We consider a continuous-time branching random walk on ℤ^d, where the particles are born and die on a periodic set of points (sources of branching). The spectral properties of the evolution operator for the mean number of particles at an arbitrary point of ℤ^d are studied. This operator is proved to have a positive spectrum, which leads to an exponential asymptotic behavior of the mean number of particles as t → ∞.

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C_n(s₁, s_2,…, s_k) C_2n(s₁, s_2,…, s_k, n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.

The relationship between the rate of approximation of a monotone function by step functions (with an increasing number of values) and the Hausdorff dimension of the corresponding Lebesgue–Stieltjes measure is studied. An upper bound on the dimension is found in terms of the approximation rate, and it is shown that a lower bound cannot be constructed in these terms.

For an arbitrary matrix, we prove the existence of a skeleton approximation of rank r whose accuracy estimate is only r + 1 times worse than the estimate of the optimal approximation of rank r in the Frobenius norm.

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S_p. Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.

An analogue of Maslov’s canonical operator for rapidly decaying functions is defined. The construction generalizes the ∂/∂τ-canonical operator on homogeneous manifolds from distributions to smooth localized functions. The main novelty is that the wave profile must be specified explicitly.

The initiation of seismic activity on the continental shelf and its destructive effect on composite oil pipelines laid on the seabed is modeled numerically. The dynamic behavior of the medium is described by determining systems of elasticity and acoustic equations with an explicit separation of all layers. The composite is described as an orthotropic material. An algorithm is proposed that estimates the amount and type of oil pipeline destruction at a given level of seismic activity and given strength characteristics of the composite. A distinctive feature of the developed approach is that the problem is split into two stages calculated on different scales: the full-wave computation of seismic wave propagation from the earthquake source to the day surface and the computation of a composite pipeline element as an anisotropic object of complex shape. A grid-characteristic method on hexahedral and tetrahedral grids is used for the numerical computation.

A new integral representation of the solutions of multi-term matrix equations with commuting matrices is proposed. Spectral decompositions of these solutions are derived. In the special case they coincide with the decompositions for the solutions of Krein equations obtained earlier. The results are applicable to the Sylvester and Lyapunov equations for linear and some bilinear systems. The practical significance of the obtained spectral decompositions is that they allow one to characterize the contribution of individual eigen-components and their combinations into the asymptotic dynamics of perturbation energy in linear and some bilinear systems.

Page 607, the third line of the Abstract. The words “in the family” should be omitted.

Page 609, right column. Several substantial errors have been made in the translation of the main Theorem 4 from Russian into English, so this theorem is given in full.