Том 63, № 3 (2023)

We show that the integral identity obtained by D.E. Apushkinskaya and S.I. Repin (2020) for approximate solutions of the biharmonic obstacle problem that satisfy a pointwise constraint on the second divergence is valid for arbitrary approximate solutions. Using this result, we obtain a new estimate for the deviation of approximate solutions from exact ones in the case when the approximate solutions do not satisfy the pointwise constraint on the second divergence.

A model of a financial pyramid is proposed in which each participant makes a decision about entering and exiting the pyramid based on the maximin principle using his (or her) ideas about the characteristics of other participants. If the pyramid organizers are able to carry out the whole process quickly enough (so that the payoffs to agents who participated in the pyramid and left it in time do not matter too much), then exactly those agents who overestimated the share of losers in the total mass of agents will lose.

A mathematical description of the household economic behavior with the help of the Kolmogorov–Fokker–Planck equation is studied. This equation describes the dynamics of the household distribution density with respect to two characteristics: financial state and income. The agreement between statistical data and the solution of the Kolmogorov–Fokker–Planck equation is examined using statistical Rosstat data on the economic state of households in Russia. The problem is formalized as minimizing the deviation of the solution to the Kolmogorov–Fokker–Planck equation from the statistical data by managing household consumption. The extremal problem is solved numerically, and numerical results are presented.

A plane problem of a potential fluid flow around a cylinder of arbitrary section over an uneven bottom with a flow velocity at infinity directed parallel to the bottom is considered. The circulation of the velocity field is determined from Goldshtik’s postulate: the maximum velocity on the contour of the cylinder must be minimal. Two numerical schemes of the boundary element method are developed for solving this problem. The first scheme determines the flow on a bounded but arbitrarily defined bottom surface. The second scheme determines the flow around a contour in a half plane. The comparison of calculations using these numerical schemes with the exact solutions shows the convergence of the method as the grid elements increase. The pressure on the cylinder surface and on the bottom obtained using numerical calculations by the 
–
 model is compared with experimental data. Streamlines are also compared taking into account the separation region.

We consider direct and inverse problems of three-dimensional quasi-static elastography underlying a cancer diagnosis method. They are based on a model of a tissue exposed to surface compression with deformations obeying linear elasticity laws. The arising three-dimensional displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by a variable Young’s modulus and a constant Poisson ratio. The problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. This is the direct problem. The inverse problem is to find the Young modulus distribution from given tissue displacements. A significant increase in Young’s modulus within a certain tissue domain suggests possible malignancy. Under certain assumptions, simple formulas for solving both direct and inverse problems of three-dimensional quasi-static elastography are derived. Three-dimensional inverse test problems are solved numerically with the help of the proposed formulas. The resulting approximate solutions agree fairly well with the exact model solutions. The computations based on the formulas require only several tens of milliseconds on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.

The paper gives a description of the integral representation of the Shapley value for polynomial cooperative games. This representation obtained using the so-called Shapley functional. The relationship between the proposed version of the Shapley value and the polar forms of homogeneous polynomial games is analyzed for both a finite and an infinite number of participants. Special attention is paid to certain classes of homogeneous cooperative games generated by products of non-atomic measures. A distinctive feature of the approach proposed is the systematic use of extensions of polynomial set functions to the corresponding measures on symmetric powers of the original measurable spaces.

We propose a new algorithm of finding a stable flow in a network with several sources and sinks. It is based on the idea of preflows (applied in the 1970s for a faster solution of the classical maximal flow problem) and has time complexity  for a network with O(nm) vertices and m  edges. The obtained results are further generalized to a larger class of objects, the so-called stable quasi-flows with bounded deviations from the balanced relations in nonterminal vertices.