Vol 25, No 4 (2023)

In this paper an approach to estimation of the Lebesgue constant for the Lagrange interpolation process with nodes in the zeros of Chebyshev polynomials of the first kind is done. Two-sided estimation of this constant is carried out by using the logarithmic derivative of the Euler gamma function and of the Riemann zeta function. The choice of interpolation nodes is due to the fact that with a fixed number of Chebyshev nodes, the Lebesgue constant tends to its minimum value, thus reducing the error of algebraic interpolation and providing less sensitivity to rounding errors. The expressions for the upper and the lower bounds of this constant are represented as finite sums of an asymptotic alternating series. Based on the expressions obtained, these boundaries are calculated depending on the number of nodes of the interpolation process. The error of each of the boundaries’ value is estimated based on the first discarded term in the corresponding asymptotic series. The results of the calculations are presented in tables showing deviations of the Lebesgue constant from its lower and upper estimated bounds. Dependence of the values’ errors on the number of Chebyshev nodes is depicted in these tables as well. It is numerically shown that with an increase in the number of these nodes, the estimation boundaries rapidly get close to each other. The presented results can be used in the theory of interpolation to estimate the norm of the operator matching a function to its interpolation polynomial and to estimate a deviation of the constructed perturbed polynomial from the unperturbed one.

One of the constructions for obtaining flows on a manifold is building a superstructure over a cascade. In this case, the flow is non-singular, that is, it has no fixed points. C. Smale showed that superstructures over conjugate diffeomorphisms are topologically equivalent. The converse statement is not generally true, but under certain assumptions the conjugacy of diffeomorphisms is tantamount to equivalence of superstructures. Thus, J. Ikegami showed that the criterion works in the case when a diffeomorphism is given on a manifold whose fundamental group does not admit an epimorphism into the group Z. He also constructed examples of non-conjugate diffeomorphisms of a circle whose superstructures are equivalent. In the work of I. V. Golikova and O. V. Pochinka superstructures over diffeomorphisms of circles are examined. It is also proven in this paper that the complete invariant of the equivalence of superstructures over orientation-preserving diffeomorphisms is the equality of periods for periodic points generating their diffeomorphisms. For the other side, it is known from the result of A.G. Mayer that the coincidence of rotation numbers is also necessary for conjugacy of orientation-preserving diffeomorphisms. At the same time, superstructures over orientation-changing diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of circles are topologically conjugate. Work of S. Kh. Zinina and P. I. Pochinka proved that superstructures over orientation-changing Cartesian products of diffeomorphisms of circles are equivalent if and only if the corresponding diffeomorphisms of tori are topologically conjugate. In this paper a classification result is obtained for superstructures over Cartesian products of orientation-preserving diffeomorphisms of circles.

The stability problem of a scalar functional differential equation is a classical one. It has been most fully studied for linear equations. Modern research on modeling biological, infectious and other processes leads to the need to determine the qualitative properties of the solutions for more general equations. In this paper we study the stability and the global limit behavior of solutions to a nonlinear one-dimensional (scalar) equation with variable delay with unbounded and bounded right-hand sides. In particular, our research is reduced to a problem on the stability of a non-stationary solution of a nonlinear scalar Lotka-Volterra-type equation, on the stabilization and control of a non-stationary process described by such an equation. The problem posed is considered depending on the delay behavior: is it a bounded differentiable function or a continuous and bounded one. The study is based on the application of the Lyapunov-Krasovsky functionals method as well as the corresponding theorems on the stability of non-autonomous functional differential equations of retarded type with finite delay. Sufficient conditions are derived for uniform asymptotic stability of the zero solution, including global stability, for every continuous initial function. Using the theorem proven by one of the co-authors on the limiting behavior of solutions to a non-autonomous functional differential equation based on the Lyapunov functional with a semidefinite derivative, the properties of the solutions’ attraction to the set of equilibrium states of the equation under study are obtained. In addition, illustrative examples are provided.

This paper considers such nonholonomic mechanical systems as Chaplygin skate, inhomogeneous Chaplygin sleigh and Chaplygin sphere moving in the gravity field along an oscillating plane with a slope varying with the periodic law. By explicit integration of the equations of motion, analytical expressions for the velocities and trajectories of the contact point for Chaplygin skate and Chaplygin sleigh are obtained. Numerical parameters of the periodic law for the inclination angle change are found, such that the velocity of Chaplygin skate will be unbounded, that is, an acceleration will take place. In the case of inhomogeneous Chaplygin sleigh, on the contrary, numerical parameters of the periodic law of the inclination angle change are found, for which the sleigh velocity is bounded and there is no drift of the sleigh. For similar numerical parameters and initial conditions, when the sleigh moves along a horizontal or inclined plane with the constant slope, the velocity and trajectory of the contact point are unbounded, that is, there is a drift of the sleigh. A similar problem is solved for the Chaplygin sphere; its trajectories are constructed on the basis of numerical integration. The results are illustrated graphically. The control of the slope of the plane, depending on the angular momentum of the sphere, is proposed for discussion. Regardless of the initial conditions, such control can almost always prevent the drift of the sphere in one of the directions.

The use of fuel cell power plants is a promising area in the generation of electricity. The path to their widespread use is hindered by their high cost and the availability of the fuel used. To solve this problem, effective energy conversion systems operating on diesel fuel are being developed. The main goal is to create a device (a fuel processor), which would convert diesel fuel into a hydrogen-containing gas. The device consists of several components: a nozzle for injecting liquid fuel having the form of drops into superheated steam, a mixing and vaporization zone for diesel fuel, an air supply area, and a reaction zone including a catalyst. The selection of temperature for the vaporization process should be made in such a way that, on the one hand, liquid droplets do not come into contact with the catalyst surface, and, on the other hand, gas-phase reactions are not initiated in the mixing zone. Developing such a device requires not only conducting laboratory experiments and studying the process catalyst, but also optimizing the basic physical parameters of the device. These parameters are its linear dimensions, operational temperature, reactant flow rates, and many others. Carrying out such a study is impossible without using methods of mathematical modeling. This significantly reduces the time and cost of work. This paper presents a digital model of an air-hydrocarbon mixture generator in an axisymmetric formulation. The dynamics of subsonic multiphase flow of water vapor carrying drops of liquid diesel fuel, the process of diesel fuel evaporation and mixing with water vapor and air are studied. The mathematical model is implemented in the ANSYS Fluent package (academic license of SSCC SBRAS). A series of calculations for various mixture feed temperatures are performed to optimize the main parameters. For the established optimal temperature, modeling of the mixture mixing process with air is carried out.