A Generalization of the Blackwell–Ryll-Nardzewski Measurable Selection Theorem

One of the main forms of the measurable selection theorem is connected with the existence of the graph of a measurable mapping in a given measurable set 𝑆 in the product of two measurable spaces 𝑋 and 𝑌 . Such a graph enables one to pick a point in the section 𝑆𝑥 for each 𝑥 in 𝑋 such that the obtained mapping will be measurable. The indicated selection is called a measurable selection of the multi-valued mapping associating to the point 𝑥 the section 𝑆𝑥 , which is a set in 𝑌 . The classical theorem of Blackwell and Ryll-Nardzewski states that a Borel set 𝑆 in the product of two complete separable metric spaces contains the graph of a Borel mapping (hence admits a Borel selection) provided that there is a transition probability on this product with positive measures for all sections of 𝑆 . The main result of this paper gives a generalization to the case where only one of the two spaces is complete separable and the other one is a general measurable space whose points parameterize a family of Borel probability measures on the first space such that the sections of the given set 𝑆 in the product have positive measures.

INEQUALITIES REFINING RELATIONSHIPS BETWEEN MIXED MODULES OF SMOOTHNESS IN METRICS OF $L_p$ AND $L_{\infty}$

The problem of estimating the moduli of smoothness of functions from Lq in terms of their moduli of smoothness from Lp is well known. The first stage in the estimation of moduli of smoothness was the study of the properties of functions from the Lipschitz classes and obtaining the corresponding embeddings in the works of Titchmarsh, Hardy, Littlewood, Nikol’skii. The classical Hardy-Littlewood embedding for Lipschitz spaces can be obtained as a consequence of the Ulyanov’s inequality for the moduli of continuity of a function of one variable. In the works of Ulyanov, the modulus of smoothness of natural order was considered. The introduction of fractional moduli of smoothness made it possible in the works of Potapov, Simonov, Tikhonov to strengthen the Ulyanov’s inequality. Later, the same authors were able to generalize Ulyanov’s inequality to functions of two variables, obtaining estimates for mixed moduli of smoothness. The sharpness of these inequalities was proved in the case when 1 < 𝑝 < 𝑞 < ∞ or 1 = 𝑝 < 𝑞 = ∞. In this article, we study mixed moduli of smoothness of fractional orders of a function of two variables. Inequalities are obtained that refine the previously known estimates of the Ulyanov type inequalities between mixed moduli of smoothness in the metrics Lp and Lq for values 1 < 𝑝 < 𝑞 = ∞. The accuracy of the obtained estimates is investigated. The relationship between these and previously known estimates has been studied.

The Informative Power of all Possible Linear Functionals and the Mean-Square Error in the Diccretization of Solutions of the Diriclet Problem for the Laplace Equation in the Circle

The Dirichlet problem for the Laplace equation in the case of a circle belongs to the classical ones and in various aspects has been the subject of study in various fields of mathematics. Among them are such topics as - "Boundary properties of analytic functions", in the study of which powerful methods of function theories were created and honed, - The Banach problem on the existence of a basis for a class of functions consisting of continuous in a closed circle and analytic in, - Numerical methods, since this problem as a mathematical model describes many real processes. In this article, we consider the discretization problem of solutions of the Dirichlet problem for the Laplace equation in a circle from finite numerical information obtained from the boundary function as a result of applying all possible linear functionals. The optimal order of discretization error is found and the corresponding optimal operator of discretization is constructed. The problem of constructing probabilistic measures on functional classes is also considered. Probabilistic measures on the Korobov 𝐸𝑟 (0, 2𝜋) and Nikolsky 𝐻𝑟 2 (0, 2𝜋) classes are introduced. Two-sided estimates of the mean-square error of discretization the solution of the problem by operator (𝑇𝑁 𝑓) (𝛼, 𝜃) are established.

Stabilization of combustion of kerosene-air mixture by hydrogen injection

On aircrafts at high flight speeds, the ramjet engine (ramjet) is of greatest interest at present, the efficiency of which can be increased by using the thermochemical fuel conversion (TFC) technology. Some aspects of the operation of such an engine can be simulated numerically. In this work, a theoretical and experimental study of the processes in the combustion chamber is carried out, and a method is proposed for stabilizing the combustion of a keroseneair mixture by injecting molecular hydrogen into a direct-flow combustion chamber. An experimental setup created at TsAGI was used, on which the influence of the hydrogen temperature, the location and size of the mass supply area on the combustion process was studied. For a theoretical study, the corresponding numerical technology was applied based on the Navier-Stokes equations using the Spal-rt-Allmares turbulence model, implemented in a complex of computer programs developed at the Research Institute of Mechanics of Moscow State University. Numerical modeling solved two problems. The first concerned the conditions for the ignition of molecular hydrogen supplied to the flow-type igniter, and the second - the conditions for the stabilization of the combustion of a kerosene-air mixture by a hydrogen flame. Based on the results of calculations, it was found that the ignition process is facilitated by an increase in the temperature of hydrogen and the power of its injection. The position and size of the hydrogen source is less influential. In the course of a comprehensive study by the method of a full-scale and computational experiment, the characteristic features of the flow structure in the channel, including the formation of a detonation wave, were revealed, and the possibility of controlling combustion by hydrogen injection was shown.

About the solvability of the problem of nonlinear optimization of oscillatory processes with the appearance of special controls

The article deals with the cases when special controls appear in nonlinear optimization of oscillatory processes, when the function of external influence depends non-linearly on the control parameter. Quality-managed process is to minimize the integral of the functional, that is, in the final moment of time square deviation of a controlled process from a given desired state was minimal. The study was conducted using a generalized solution of the boundary value problem, which more or less adequately describes the actual process. According to the well-known method of optimal control theory, the increment of the functional is calculated and the Pontryagin function is written out, which is studied for the maximum in the range of acceptable values of the control function. The optimal control conditions are written out in the form of equality and differential inequality, which must be fulfilled simultaneously. The case when the maximum principle degenerates is studied. It is established that the desired control satisfies an infinite-dimensional system of linear integral Fredholm equations of the first kind, the solvability of which is studied by operator methods. It is proved that the operator equation has infinitely many solutions and an algorithm for their construction is developed. Then the functional is minimized on the set of solutions of the operator equation. This problem has at least one solution that is the desired special optimal control.

Calculations for one illustration of the Pythagorean Theorem

On the sides of a right-angled isosceles triangle, squares external to this triangle are built. The task is to construct a straight line on a plane that divides a figure consisting of these three squares into two equal-sized (that is, equal area) figures. Some parameters determined by this line are calculated, in particular, the areas of parts of the original squares in half-planes. The corresponding results are illustrated by various drawings, tables and graphs.