We study a mixed problem for a one-dimensional wave equation in a curvilinear half-strip. The initial conditions have a discontinuity of the first kind at a single point. The mixed problem models the problem of a longitudinal impact on a finite elastic rod with a movable boundary. Using the method of characteristics, we obtain the solution in an explicit analytical form. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists.

The known representation of the solution of the linear stochastic differential equation of Skorohod on Poisson space with random coefficients and an initial condition contains as an unknown parameter a family of transformations of the probability space of the leading random process determined by the solution of the integral stochastic equation. In this paper, we consider cases when the solution of this integral equation can be found in explicit form. Explicit solutions are obtained in two cases in the class of linear Skorohod equations on Poisson space with random coefficients and an initial condition linearly dependent on the time of the first jump of the leading process. The first three moments of the solution of the original SDEs are estimated and a numerical example is given. The obtained formulas for calculating the moments of the solution of Skorohod SDE with the leading Poisson process can be used in constructing approximate formulas for calculating the mathematical expectations of nonlinear functionals of the solution, similar to those considered earlier for Skorohod equations with the leading Wiener process.

The article is devoted to the orthogonal regression analysis, which is associated with the representation of the regression function by Fourier series by the multidimensional-matrix (mdm) orthogonal polynomials, in opposite to the (usual) regression analysis, when the regression function is approximated by the (usual) polynomial (by the degrees of the independent mdm input variable). We will also distinguish the classical regression analysis, when the scalar or might the classical vector-matrix mathematical approaches are used, and the mdm regression analysis, when the mdm variables and the mdm mathematical approach are used. In this article, the orthogonal regression analysis is developed on the base of the orthogonal polynomials and the mdm mathematical approach, so called the mdm orthogonal polynomial regression analysis. The known results from the theory of the orthogonal mdm polynomials and Fourier series of the vector argument are generalized to the case of the mdm argument and function. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for the potential studies are obtained. The general case of the approximation of the mdm function of the mdm argument by the Fourier series is realized programmatically as the single program function and its efficiency is confirmed by the computer calculations. The properties of the estimations of regression coefficients and unknown parameters are studied and their distributions when the normal distribution of the measurement’s errors are obtained for the arbitrary covariance matrix of the errors of measurements and the arbitrary degree of the approximating polynomial. These results allow testing the hypothesis and building the hyper-rectangular confidence areas relating the orthogonal regression function. Theoretical results are confirmed by computer simulation.

A finite-difference computational algorithm for solving the equations of convective flows of incompressible fluid in two-dimensional irregular domains using generalized curvilinear coordinates is constructed. The physical domain is mapped into a computational domain (unit square) in the space of generalized coordinates. The equations of mixed convection in primitive variables are written in generalized curvilinear coordinates and approximated in the computational domain on uniform non-staggered grids. The constructed computational algorithm is based on splitting difference schemes. The obtained results are mapped onto a nonuniform difference grid in the physical domain. The results of solving boundary value problems of heat and mass transfer of incompressible fluid in domains of complex shape are presented.

The application of the laws of motion of two bodies in a medium obtained by the Belarusian scientific school to the problem of so-called super-velocity stars, which is relevant for astrophysics today, is investigated. A scenario is considered that justifies the generation of hypervelocity stars and is based on the laws of motion of binary stars in the interstellar medium, which consists of visible (baryonic) matter and dark matter. It has been proven that in different environments the center of mass of two stars (or galaxies) cannot be at rest relative to the medium and the background gravitational field additionally created by it, but moves with acceleration along a cycloid or quasi-cycloid trajectory. After a sufficient period of time, the speed of the center of mass reaches high values that characterize super-high-speed stars: speeds ≥(700–3750) km · s^–1 and more. Since the stars are “tied” to their center of mass, they, like the center of mass, begin to move at approximately the same speed along intricate trajectories-coils reminiscent of lace: we have the so-called lace effect of movement. Special cases have been noted in the motion of two bodies (stars) of comparable masses and their center of mass in the medium: 1) if the masses of the stars are equal, then their center of mass in both homogeneous and inhomogeneous media is at rest, the lace effect of motion is absent and the generation of high-speed stars does not occur; 2) if the medium is homogeneous (its density ρ = const), then in the Newtonian theory of gravity, for any masses of stars, their center of mass is at rest, the lace effect of motion is absent and the generation of super-high-speed stars does not occur. In accordance with the necessary formulas derived in the work, numerical estimates were carried out illustrating the process of generation of high-speed stars up to stars with relativistic velocities (1/2–2/3)c km · s^–1, where c = 3 · 10⁵ km · s^–1 speed of light in vacuum.

The problem of object detection in Earth remote sensing images is studied, which is important for agricultural monitoring, urban planning, early warning of natural disasters, etc. Due to the different sizes of objects, complex background, and dense distribution of small objects in remote sensing images, problems such as high percentage of missed objects and insufficient accuracy of their coordinates often arise. In this regard, an improved method for YOLOv11, ABS-YOLO, is proposed, which significantly improves the performance of object detection by integrating Averaged Convolution (AConv), Bidirectional Weighted Feature Pyramid (BiFPN), and Swin Transformer attention mechanism. Experimental results show that, compared with YOLOv11, the proposed object detection method ABS-YOLO with AConv, BiFPN, and Swin Transformer achieves 3.9 % increase in mAP50 estimations and 2.6 % increase in mAP50-95 on the NWPU VHR-10 dataset with significant improvement in precision and recall rates. This method allows achieving a balance between efficiency and accuracy of remote sensing object detection due to the proposed improvements.