Vol 24, No 2 (2024)

In this article, we have defined new weighted integral operators. We formulated a lemma in which we obtained a generalized identity through these integral operators. Using this identity, we obtain some new generalized Simpson's type inequalities for $(h,m)$-convex functions. These results we obtained using the convexity property, the classical Hölder inequality, and its other form, the power mean inequality. The generality of our results lies in two fundamental points: on the one hand, the integral operator used and, on the other, the notion of convexity. The first, because the ''weight''  allows us to encompass many known integral operators (including the classic Riemann and Riemann – Liouville), and the second, because, under an adequate selection of the parameters, our notion of convexity contains several known notions of convexity. This allows us to show that many of the results reported in the literature are particular cases of ours.

In this work, we consider the operations over Abelian integers of rank $n$. By definition, such numbers are elements of the complex field and have the form of polynomials with integer coefficients from the $n$th degree primitive root of 1. In contrast, the degrees of such polynomials are not greater than Euler's totient function $\varphi(n)$. We provide an example to show that there are infinitely many Abelian integers inside any zero-centered circle on the complex plane. In this work, for considered operations we give in particular the algorithm of calculation of the inverse for the Abelian integer of rank $n$. It allows us to analyze not only the rings of such numbers but also the fields of Abelian integers. Natural arithmetics for such algebraic structures leads us to study  the polynomials with integer Abelian coefficients. Thus, in the presented work we also investigate the problem of finding roots of such polynomials. As a result, we provide an algorithm that finds the integer Abelian roots of the polynomials over the ring of Abelian integers. This algorithm is based on the proposed statement that all roots of the polynomial are bounded by some domain. The computer calculations confirm the statistical truth of the statement.

The present work is devoted to the construction of asymptotically optimized equations of the hyperbolic boundary layer in thin shells of revolution in the vicinity of the dilation wave front at shock edge loading of the tangential type. These equations are derived by asymptotically integrating  of the exact three-dimensional theory elasticity equations in the special coordinate system. This system defines the boundary layer region. The wave front has a complicated form, dependent on the shell curvature and therefore its asymptotical model is constructed. This geometrical model of the front defines it via the turned normals to the middle surface. Also, these turned normals define the geometry of the hyperbolic boundary layer applicability region. Constructed asymptotically optimised equations are formulated for the asymptotically main components of the stress-strain state: the longitudinal displacement and the normal stresses. The governing equation for the longitudinal displacement is the hyperbolic equation of the second order with the variable coefficients. The asymptotically main part of this equation is defined as the hyperbolic boundary layer in plates.

One of the current and widely used non-destructive testing methods for monitoring and determining the elastic properties of materials is nanoindentation. In this case, to interpret the test results, a non-trivial task arises of constructing an adequate mathematical model of the indentation process. As a rule, in many cases, analytical formulas are used that are obtained from an elastic linear formulation of problems about the introduction of a non-deformable stamp into a homogeneous elastic half-space. Currently, the numerical formulation of the problem makes it possible to obtain and use a numerical solution obtained taking into account the complete plastic nonlinear behavior of the material. In this work, a study of contact problems on the introduction of a spherical and conical indenter into an elastoplastic homogeneous half-space was carried out. To verify the numerical solution, the problem of introducing a spherical and conical indenter into an elastic homogeneous half-space was also solved and compared with known analytical solutions. Issues of convergence and tuning of numerical methods, the influence of plasticity and the applicability of analytical solutions are explored. Problems are solved numerically using the finite element method in the Ansys Mechanical software package.

The subject of mathematical study and modelling in this paper is an inbound call center that receives calls initiated by customers. A closed exponential queueing network with customer retrials and impatient customers is used as a stochastic model of call processing. A brief review of published results on the application of queueing models in the mathematical modeling of customer service processes in call centers is discussed. The network model is described. The possible customer states, customer routing, parameters, and customer service features are given. The allocation of customers by network nodes at a fixed time fully describes the situation in the call center at that time. The state of the network model under study is represented by a continuous-time Markov chain on finite state space. The model is studied in the asymptotic case under the critical assumption of a large number of customers in the queueing network. The mathematical approach used makes it possible to use the passage to the limit from a Markov chain to a continuous-state Markov process. It is proved that the probability density function of the model state process satisfies the Fokker – Planck – Kolmogorov equation. Using the drift coefficients of the Fokker – Planck – Kolmogorov equation, a system of ordinary differential equations for calculating the expected number of customers in each network node over time can be written. The solution of this system allows for predicting the dynamics of the expected number of customers at the model nodes and regulating the parameters of the call center operation. The asymptotic technique used is applicable both in transient and steady states. The areas of implementation of research results are the design of call centers and the analysis of their workload.