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卷 516, 编号 1 (2024)

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MATHEMATICS

On reconstruction of Kolmogorov operators with discontinuous coefficients

Bogachev V., Shaposhnikov S.

摘要

We obtain broad sufficient conditions for reconstructing the coefficients of a Kolmogorov operator by means of a solution to the Cauchy problem for the corresponding Fokker–Planck–Kolmogorov equation. 

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):5-8
pages 5-8 views

Exact estimates of functions in Sobolev spaces with uniform norm

Kazimirov D., Sheypak I.

摘要

For functions from the Sobolev space W°n[0;1] and an arbitrary point a(0;1), the best estimates are obtained in the inequality f(a)An,0,(a)fnL[041]. The connection of these estimates with the best approximations of splines of a special kind by polynomials in L10;1 and with the Peano kernel is established. Exact constants of embedding the space W°n[0;1] in L0;1 are found.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):9-14
pages 9-14 views

On undecidability of subset theories of some unars

Karlov B.

摘要

This paper is dedicated to studying of the algorithmic properties of unars with an injective function. We prove that the theory of every such unar admits quantifier elimination if the language is extended by a countable amount of predicate symbols. Necessary and sufficient conditions are established for the quantifier elimination to be effective, and a criterion of decidability of theories of such unars is formulated. Using this criterion we build a unar such that its theory is decidable, but the theory of the unar of its subsets is undecidable.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):15-20
pages 15-20 views

Induced forests and trees in Erdös–Rényi random graph

Akhmejanova M., Kozhevnikov V.

摘要

We prove concentration in the interval of size o 1/p for the size of the maximum induced forest (of bounded and unbounded degree) in Gn, p forCε / n < p < 1 ε for arbitrary fixed ε > 0. We also show 2-point concentration of the size of the maximum induced forest (and tree) of bounded degree in the binomial random graph Gn, p for p = const

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):21-25
pages 21-25 views

Generalized solution of a mixed problem for a wave equation with a non-smooth right-hand side

Lomov I.

摘要

Under minimal conditions on the right side of the wave equation, a generalized solution of the mixed problem is constructed. The solution is presented as a series from the Fourier method, its sum is found. The form of a generalized solution of a mixed problem for an inhomogeneous telegraphic equation is given.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):26-30
pages 26-30 views

On kernels of invariant Schrödinger operators with point interactions. Grinevich–Novikov problem

Malamud M., Marchenko V.

摘要

According to Berezin and Faddeev, a Schrödinger operator with point interactions +j=1mαjδ(xxj),X={xj}1m3,{αj}1m, is any self-adjoint extension of the restriction x of the Laplace operator  to the subset {fH2(R3):f(xj)=0,1jm} of the Sobolev space H2(3). The present paper studies the extensions (realizations) invariant under the symmetry group of the vertex set X={xj}1m of a regular m-gon. Such realizations HB are parametrized by special circulant matrices Bm×m. We describe all such realizations with non-trivial kernels. А Grinevich–Novikov conjecture on simplicity of a zero eigenvalue of the realization HB with a scalar matrix B=αI and an even  is proved. It is shown that for an odd m non-trivial kernels of all the realizations  with scalar  are two-dimensional. Besides, for arbitrary realizations BαI the estimate dimkerBm1 is proved, and all the invariant realizations of the maximal dimension dimkerB=m1 are described. One of them is the Krein realization, which is the minimal positive extension of the operator x.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):31-37
pages 31-37 views

A joint logic of problems and propositions

Melikhov S.

摘要

In a 1985 commentary to his collected works, Kolmogorov informed the reader that his 1932 paper On the interpretation of intuitionistic logic “was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types – propositions and problems.” We construct such a formal system as well as its predicate version, QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The axioms of QHC are obtained as a result of a simultaneous formalization of two well-known alternative explanations of intiuitionistic logic: 1) Kolmogorov's problem interpretation (with familiar refinements by Heyting and Kreisel) and 2) the proof interpretation by Orlov and Heyting, as clarified and extended by Gödel.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):38-50
pages 38-50 views

Description of turbulent flows using a kinetic model

Chetverushkin B., Lutsky А., Shilnikov Е.

摘要

In this paper, a closed system of equations for describing turbulent flows is obtained. Additional equations for cross pulsation moments are derived on the basis of a balanced kinetic equation, with the help of which a quasi gas-dynamic system of equations was previously obtained. The results of the calculation of the spatially two-dimensional problem of the mixing layer of two streams are presented.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):51-58
pages 51-58 views

Continued fractions in hyperelliptic fields with an arbitrarily large period length

Platonov V., Fedorov G.

摘要

The article proves the following statement: in any hyperelliptic field L defined over the field of algebraic numbers K which having non-trivial units of the ring of integer elements of the field L, there is an element for which the period length of the continued fraction is greater any pre-given number.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):59-64
pages 59-64 views

Invariants of seventh-order homogeneous dynamical systems with dissipation

Shamolin M.

摘要

New cases of integrable dynamical systems of the seventh order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a three-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):65-74
pages 65-74 views

On an extremal problem for compactly supported positive definite functions

Manov A.

摘要

This article considers an extremal problem for positive definite functions on n with a fixed support and a fixed value at the origin (the class

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):75-78
pages 75-78 views

On an invariant of pure braids

Manturov V., Nikonov I.

摘要

Using the recoupling theory, we define a representation of the pure braid group and show that it is not trivial.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):79-82
pages 79-82 views

Maximum induced trees in sparse random graphs

Buitrago Oropeza J.

摘要

We prove that a.a.s. for any ε > 0 and ne23e2+εp=o(1) the maximum size of an induced subtree of the binomial random graph Gn,p is concentrated in 2 consecutive points.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):83-86
pages 83-86 views

On the Boyarsky–Meyers estimate for the gradient of the solution to the Dirichlet problem for the second order elliptic equation with drift. The case the critical Sobolev exponen

Alkhutov Y., Chechkina A.

摘要

The increased integrability of the gradient of the solution to the Increased integrability of the gradient o the solution to the homogeneous Dirichlet problem for the Poisson equation with lower terms in a bounded Lipschitz domain is established. A proof of the unique solvability of this problem is also given.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):87-92
pages 87-92 views

Inversion problem for Radon transforms defined on pseudoconvex sets

Anikonov D., Konovalova D.

摘要

This paper is devoted to some questions of inversion for the classical and generalized integral Radon transform. The main question is to determine information about the integrand functions if the values of some integrals are known. A feature of the work of the authors of this message is an analysis of the case when the function is integrated according to hyperplanes in finite-dimensional Euclidean space, and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. At the same time, the number of independent variables describing known integrals are smaller than those of the unknown integrand. We consider discontinuous integrands defined specifically introduced pseudo-convex sets. A Stefan-type problem is posed about finding surfaces discontinuities of the integrand function. The work provides formulas based on the application special integro-differential operators to known data and allowing you to solve the assigned tasks.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):93-97
pages 93-97 views

Multi-vortices and lower bounds for the attractor dimension of 2d Navier-Stokes equations

Kostianko A., Ilyin A., Stone D., Zelik S.

摘要

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations, which does not use Kolmogorov flows, is presented. Using this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):98-102
pages 98-102 views

COMPUTER SCIENCE

On ml methods for network powered by computing infrastructure

Smeliansky R., Stepanov E.

摘要

The paper considers the application of machine learning methods for optimal resource management for Network Powered by Computing (NPC) – a new generation computing infrastructure. The relation between the proposed computing infrastructure and the GRID concept is considered. It is shown how machine learning methods applied to computing infrastructure management make it possible to solve the problems of computing infrastructure management that did not allow the GRID concept to be fully implemented. As an example, the application of multi-agent optimization methods with reinforcement learning for network resources management is considered. It is shown that the application of multi-agent machine learning methods makes it possible to increase the speed of distribution of transport flows and ensure optimal NPC network channel load according to the criterion of uniform load distribution, and that such management of network resources is more effective than a centralized approach.

Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ. 2024;516(1):103-112
pages 103-112 views