Vol 65, No 10 (2025)

In this paper, a model of the evolution of the Lorenz curve, describing the distribution of income between economic agents, is proposed. It is proved that the evolution of income distribution is consistent with Lorenz majorization in the Ramsey—Bewley model. A Pigou—Dalton transfer (tax and subsidy) system, which generates a stationary income distribution chosen by the welfare state, is constructed. Numerical calculations allow us to formulate a conjecture about the stability of the Lorenz curve corresponding to the selected income distribution.

We study movable singularities of the Blasius equation in the complex plane. Numerical algorithms of their localization are given that allow to find singularities with high accuracy. All these singularities are equivalent and may be represented by one of them. We obtain an asymptotic expansion in the neighborhood of the singularity in explicit form and compute its coefficients. This power-logarithmic expansion is shown to be convergent and giving a local parametrization of the Riemann surface of the Blasius function.

The first initial boundary value problem for a second-order Petrovsky parabolic system with coefficients satisfying the double Dini condition in a bounded domain on the plane is considered. The lateral boundaries of the region are defined by continuously differentiable functions. It is established that if the right-hand parts of the boundary conditions of the first kind are continuously differentiable, and the initial function is continuous and bounded together with its first and second derivatives, so the solution to the problem belongs to the space of functions that are continuous and bounded together with their higher derivatives in the closure of the domain. The corresponding estimates have been proved. An integral representation of the solution is obtained. If the lateral boundaries of the domain admit the presence of angles, and the boundary functions have piecewise continuous derivatives, then in this case it is established that the higher derivatives of the solution are continuous everywhere in the closure of the domain, with the exception of corner points, and thus bounded.

An asymptotic analysis of the Dirichlet spectral problem in a thin-walled cylinder with a periodic family of partitions perpendicular to the generators of the cylindrical surface establishes the presence of open gaps in the waveguide spectrum, determining the size and position of the spectral segments. At different values of the relative thickness of the baffles various types of localization of the eigenfunctions of the model problem with the Floquet parameter on the periodicity cell are observed. Accordingly, the passing waves are localized on the baffles (large thicknesses) or near their edges (small thicknesses). The results were obtained using various procedures of dimensionality reduction and analysis of the boundary layer phenomenon near the junction zone of the baffles to the cylinder, which (layer) is described by the Dirichlet problem in a flat T-shaped joint of a single strip and a strip of varying thickness perpendicular to it. The localization method is determined by the presence or absence of a discrete spectrum for the latter task.

The turbulent flow of a viscous incompressible fluid in a circular tube caused by a pressure drop is investigated. It is assumed that the characteristic Reynolds number, calculated from the maximum velocity of the averaged flow and the length of the pipe, is large, and the radius of the pipe is small compared to its length. To find solutions to the Navier–Stokes equations, an asymptotic method of many scales is used, in which velocities and pressures are represented as series consisting of the sum of steady and perturbed terms, instead of the traditional decomposition of the solution into time-averaged values and their fluctuations. The paper finds a viscous self-sustaining steady flow that occurs in a pipe against the background of fast turbulent fluctuations. The connection of such a solution with Prigogine's theory of dissipative structures for open nonlinear systems of parabolic type is indicated. A solution has been found for the radial steady velocity, which describes the self-induced outflow of fluid from the core of the flow to a solid/permeable wall. As a result, solutions for the longitudinal velocity have been obtained that differ markedly from the laminar regimes. A qualitative comparison with well-known experiments and works on the direct numerical simulation (DNS) has been performed.

The problem of constructing a system of four point masses, the totality of which is equal to a predetermined solid, is solved. A family of such systems has been built, depending on six parameters. The freedom to choose parameters makes it possible to set the task of finding the set of masses that best approximates the momentums of the distribution of masses of the third order of the body. The problem in this formulation is solved in relation to the nucleus of comet 67P/Churyumov–Gerasimenko. The criterion for the quality of the coincidence of the momentums of the third-order mass distribution is the standard deviation of the momentums of the point mass system from the corresponding momentums of the comet nucleus. A system of four material points is constructed, minimizing the value of the root-mean-square error. This value turned out to be less than the similar values obtained in previous studies. It is noteworthy that the masses of the found points turned out to be different from each other and all are located inside the core, so that the three smallest of them are located in a larger fraction of the core, and the fourth, which concentrates ≈ 28.5% of the total mass of the core, is located inside a smaller fraction. This mass distribution is in good agreement with the well-known estimate of 27% of the volume of a smaller fraction of the comet’s core from the total volume.