Vol 24, No 4 (2024)

A semi-analytical approximation of the normal derivative of the simple layer heat potential near the boundary of a two-dimensional domain with $C^{5} $ smoothness is proposed. The calculation of the integrals that arise after piecewise quadratic interpolation of the density function with respect to the variable of arc length $s$, is carried out using analytical integration over the variable $\rho =\sqrt{r^{2} -d^{2} } $, where $r$ and $d$ are the distances from the observation point to the integration point and to the boundary of the domain, respectively. To do this, the integrand is represented as the sum of two products, each of which consists of two factors, namely: a function smooth in а near-boundary domain containing the Jacobian of the transition from the integration variable $s$ to the variable $\rho $, and a weight function containing a singularity at $r=0$ and uniformly absolutely integrable in the near-boundary region. Smooth functions are approximated with the help of the piecewise quadratic interpolation over the variable $\rho $, and then analytical integration becomes possible. Analytical integration over $\rho $ is carried out on a section of the boundary fixed in width, containing the projection of the observation point, and on the rest of the boundary, the integrals over $s$ are calculated using the Gauss formulas. Integration over the parameter of $C_{0} $-semigroup formed by time shift operators is also  carried out analytically. To do this, the $C_{0} $-semigroup is approximated using the piecewise quadratic interpolation over its parameter. It is proved that the proposed approximations have stable cubic convergence in the Banach space of continuous functions with the uniform norm, and such convergence is uniform in the closed near-boundary region. The results of computational experiments on finding of  the normal derivative of solutions of the second initial-boundary problem of heat conduction in a unit circle with a zero initial condition are presented, confirming the uniform cubic convergence of the proposed approximations of  the normal derivative of the simple layer heat potential.

The paper is devoted to the study of one class of infinite systems of nonlinear two-dimensional equations with convex and monotone nonlinearity. The studied  class of nonlinear systems of algebraic equations has both theoretical and practical significance, in particular, in the study of discrete analogs of problems in dynamic theory of $p$-adic open-closed strings, in the kinetic theory of gases, in mathematical biology in the study of space-time distribution of epidemics. Existence and uniqueness theorems for a positive solution in a certain class of non-negative and bounded matrices are proved. Some qualitative properties of the solution are revealed. The obtained results supplement and generalize some of the previously obtained ones. Illustrative examples of the corresponding matrices and nonlinearities (including those of an applied nature) that satisfy all the conditions of the formulated theorems are given.

Only finite groups are considered. $\frak F$-projectors and $\frak F$-covering subgroups, where $\frak F$ is a certain class of groups, were introduced into consideration by W.~Gaschutz as a natural generalization of Sylow and Hall subgroups in finite groups. Developing Gaschutz's idea, V. A. Vedernikov and M. M. Sorokina defined $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, where $\omega$ is a non-empty set of primes, and established their main characteristics. The purpose of this work is to study the properties of $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, establishing their relation with other subgroups in groups. The following tasks are solved: for a non-empty $\omega$-primitively closed homomorph $\frak F$ and a given set $\pi$ of primes, the conditions under which an $\frak F^{\omega}$-projector of a group coincides with its $\pi$-Hall subgroup are established; for a given formation $\frak F$, a relation between $\frak F^{\omega}$-covering subgroups of a group $G=A\rtimes B$ and $\frak F^{\omega}$-covering subgroups of the group $B$ is obtained. In the paper classical methods of the theory of finite groups, as well as methods of the theory of classes of groups are used.

The kinematics and dynamics of P. L. Chebyshev's “paradoxical” mechanism are considered. The point of interest in the dynamics of the “paradoxical” mechanism is connected with the fact that its configuration space contains six singular points. These points are successively passed through a full turn of the handle. Holonomic constraints which are imposed on the system become linearly dependent at singular points. Thus it is impossible to apply the standard methods of derivation of the motion equations at singular points. The properties of the “paradoxical” mechanism are based on the properties of the lambda mechanism. P.  L. Chebyshev designed many mechanisms for particular types of motion by using the construction of the lambda mechanism. For example, it is possible to obtain an anti-rotation mechanism, a mechanism with two swings per revolution of the handle, or a mechanism with a stop of the driven link with certain parameters of the lambda mechanism. The trajectory of the vertex of the lambda mechanism in the “paradoxical” mechanism is located between two circles and touches each circle at three points. Therefore singular points arise in the configuration space. It is shown in the article that the configuration space consists of two “curves” that intersect at a nonzero angle in the neighbourhood of a singular point. In order to get a numerical and analytical model of the “paradoxical” mechanism, the main formulas from the works of P. L. Chebyshev are given. “Paradoxical” mechanism is represented as a combination of a lambda-mechanism and a singular pendulum, which motions are limited by two holonomic constraints. The equations of motion are written out and the reaction forces are found. It is shown that with a small increase in the length of the double pendulum rod the configuration space splits into two non-intersecting curves. The smaller the link perturbation becomes, the larger the Lagrange multipliers around singular configurations become.

A mechanical system consisting of a pipeline connected at one end to the combustion chamber of an aircraft engine, and with a sensor designed to measure the pressure in the combustion chamber at the other end is investigated in the work. The sensitive element of the sensor, which transmits information about pressure, is an elastic plate. A mathematical model of a pressure measurement system, taking into account the transfer of heat flow through a pipeline with a working medium (gas or liquid) from the engine to the elastic element, is proposed. To describe the vibrations of the sensitive element of the sensor, a linear model of a solid deformable body is considered, taking into account the temperature distribution over the thickness of the elastic element. Using the small parameter method, a coupled system of asymptotic partial differential equations was obtained that describes the joint dynamics of the gas-liquid medium in the pipeline and the elastic sensitive element of the pressure sensor. The cases of hinged and rigid fastening of the ends of the sensing element were studied. Based on the Bubnov – Galerkin method, the problem is reduced to the study of a coupled system of ordinary differential equations. Using the computer algebra system Mathematica 12.0, numerical experiments were carried out for specific parameters of the mechanical system.

Asymptotic integration methods have been used to model the propagation of a shear wave beam along a nonlinear-elastic cylindrical shell of the Sanders – Koiter model. The shell is assumed to be made of a material characterized by a cubic dependence between stress and strain intensities, and the dimensionless parameters of thinness and physical nonlinearity are considered to have the same order of smallness. The multiscale expansion method is used, which makes it possible to determine the wave propagation speed from the equations of the linear approximation, and in the first essentially nonlinear approximation, to obtain a nonlinear quasi-hyperbolic equation for the main term of the expansion of the shear displacement component. The derived equation is a cubically nonlinear modification of the Lin – Reisner – Tsien equation modeling unsteady near-sonic gas flow and can be transformed into the modified Khokhlov – Zabolotskaya equation used to describe narrow beams in acoustics. The solution of the derived equation is found in the form of a single harmonic with slowly changing complex amplitude, since in deformable media with cubic nonlinearity the effect of self-induced wave essentially prevails over the effect of generation of higher harmonics. As a result, a perturbed nonlinear Schrödinger equation of defocusing type is obtained for the complex amplitude, for which there is no possibility of modulation instability development. In terms of the elliptic Jacobi function, an exact physically consistent solution, periodic along the dimensionless circumferential coordinate, is constructed.

The paper presents the engineering estimate of the dispersion level of the values of such fused quartz glass physical and mechanical properties as density and elasticity modulus. The value scatter of the considered parameters over material volume leads to  negative effect appearance —  working natural frequency splitting of gyroscope resonator. Evaluation is carried out on the basis of the consideration of the fused quartz glass micro-structure and variability of its parameters. As a result, resonator frequency splitting values from calculated scattering levels of the structural material properties are presented.

The car insurance fraud level assessment is an urgent and complex task, which is largely due to the activities of fraudulent groups. For the confident management of insurance companies in the anti-fraud strategy, a tool to assess the current state of the claim’s portfolio is needed. Modern machine learning methods make it possible to carry out such an assessment using data on policyholders and insurance cases. When applying these approaches, a number of problems arise that do not allow achieving the required quality of fraud detection. These include class imbalance and the so-called concept drift, which arises as a result of changes in the scenarios of fraudsters’ schemes and the subjectivity of the expert assessment of a specific insurance case. This study proposes an approach to improve model metrics for detecting fraud in a claims portfolio. A numerical experiment conducted on two open data sets demonstrated a significant improvement in the detection rate of insurance fraud compared to classical modeling. Specifically, there was an increase in the completeness of fraud detection by 49 and 19 percentage points for the two datasets, respectively.