Volume 519, Nº 1 (2024)
MATHEMATICS
ON THE STABILITY OF STRICT EQUILIBRIUM OF PLASMA IN TWO-DIMENSIONAL MATHEMATICAL MODELS OF MAGNETIC TRAPS
Resumo
The article presents an analysis of instabilities known from previous works in a two-dimensional mathematical model of plasma configuration equilibrium using the example of a toroidal magnetic trap “Galatea–Belt” straightened into a cylinder and possessing plane symmetry. It is established that the previously observed large values of the two-dimensional velocity of disturbances in the plane of the cylinder cross-section arise on the periphery of the configuration near its conventional boundary, do not grow with time, and are due to arbitrarily small values of density, which is not determined by the idealized model of strict equilibrium. By varying the density, it is possible to influence stability. Three-dimensional (corrugated along the axis of the cylinder) disturbances grow with time in accordance with the traditional Lyapunov instability. Calculations allow us to determine the dependence of its quantitative characteristics on the problem parameters.
Doklady Mathematics. 2024;519(1):3-6
3-6
STABILIZED SCHEME FOR CALCULATING RADIATION TRANSFER IN THE 𝑃1 APPROXIMATION
Resumo
We consider interpolation-characteristic scheme approximating approximation to the radiative transfer equations corresponding to the 𝑃1 model. The explicit finite-difference scheme is corrected by special term adjusting the rate of radiation energy transfer. Such correction can reduce the influence of non-physical effects when calculating radiative heat transfer in a medium with non-uniform opacity.
Doklady Mathematics. 2024;519(1):7-13
7-13
INFINITE ALGEBRAIC INDEPENDENCE OF POLYADIC SERIES WITH PERIODIC COEFFICIENTS
Resumo
Consider sequences of integers 𝑎𝑛(𝑘,𝑗), 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 such that 𝑎𝑛(𝑘,𝑗) = 𝑎𝑛(𝑘+,𝑇𝑗)𝑗 , 𝑗 = 1, … , 𝑚, 𝑘 = 1, … , 𝑇𝑗, 𝑛 = 0, 1, …, and consider the series 𝐹𝑗,𝑘(𝑧) = ∞ ∑ 𝑛=0 𝑎(𝑘,𝑗)𝑛 𝑛!𝑧𝑛, 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚. The conditions are established under which the set of series 𝐹𝑗,𝑘(𝑧), 𝑘 = 2, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 and the Euler series Φ(𝑧) = ∞ ∑ 𝑛=0 𝑛!𝑧𝑛 are algebraically independent over ℂ(𝑧) and for any algebraic integer γ ≠ 0, their values at the point γ are infinitely algebraically independent.
Doklady Mathematics. 2024;519(1):14-17
14-17
COMPARISON OF THE COSTS FOR GENERATING THE TOLLMIEN-SCHLICHTING WAVES AND OPTIMAL DISTURBANCES USING OPTIMAL BLOWING–SUCTION
Resumo
The problem of generating the Tollmien-Schlichting waves (leading eigenmodes) and optimal disturbances with a given accuracy using optimal blowing–suction is considered with the example of Poiseuille flow in a duct of square cross-section and streamwise-harmonic blowing-suction through the duct walls. The problem is reduced to solving optimal control problems for the linearized governing equations of viscous incompressible media. It is shown for the first time that generating the optimal disturbances using blowing–suction is much more expensive than generating the leading modes.
Doklady Mathematics. 2024;519(1):18-21
18-21
AN EXACT QUADRATIC ALGORITHM FOR THE SHORTEST TREE TRANSFORMATION
Resumo
The article proposes a new exact quadratic algorithm in complexity that solves the problem of the shortest transformation (“alignment”) of one loaded tree into another, taking into account arbitrary prices of operations on trees. Three operations are considered: adding vertex deletions to an edge or root of a tree and shifting a subtree with deletions.
Doklady Mathematics. 2024;519(1):22-27
22-27
THE REALITY OF SPECTRAL SHIFT FUNCTIONS FOR CONTRACTIONS AND DISSIPATIVE OPERATORS
Resumo
Recently the authors of this note solved a famous problem that remained open during many years and proved that for arbitrary contractions on Hilbert space with trace class difference there exists an integrable spectral shift function, for which an analogue of the Lifshits—Krein trace formula holds. Similar results were also obtained for pairs of dissipative operators. It turns out that in contrast with the case of self-adjoint or unitary operators, it can happen that there is no real-valued integrable spectral shift function. In this note we state results that give sufficient conditions for the existence of a real-valued integrable spectral shift function for pairs of contractions and pairs of dissipative operators.
Doklady Mathematics. 2024;519(1):28-32
28-32
STUDY OF THE BIAS OF N-PARTICLE ESTIMATES OF THE MONTE CARLO METHOD IN PROBLEMS WITH PARTICLE INTERACTION
Resumo
The paper gives a theoretical and numerical justification of the bias with the 𝑂(1/𝑁) order for the 𝑁-particle statistical estimates of the functionals of the solution of nonlinear kinetic equations for the model with interaction of particle trajectories. An estimate of the coefficient in the corresponding bias formula is obtained.
Doklady Mathematics. 2024;519(1):33-38
33-38
CONSTRUCTION OF SMOOTH “SOURCE-SINK” ARCS IN THE SPACE OF DIFFEOMORPHISMS OF A TWO-DIMENSIONAL SPHERE
Resumo
It is well known that the mapping class group of the two-dimensional sphere 𝕊2 is isomorphic to the group ℤ2 = {−1, +1}. At the same time, the class +1 (−1) contains all orientation-preserving (orientationreversing) diffeomorphisms and any two diffeomorphisms of the same class are diffeotopic, that is, they are connected by a smooth arc of diffeomorphisms. On the other hand, each class of maps contains structurally stable diffeomorphisms. It is obvious that in the general case, the arc connecting two diffeotopic structurally stable diffeomorphisms undergoes bifurcations that destroy structural stability. In this direction, it is particular interesting in the question of the existence of a connecting them stable arc ”— an arc pointwise conjugate to arcs in some of its neighborhood. In general, diffeotopic structurally stable diffeomorphisms of the 2-sphere are not connected by a stable arc. In this paper, the simplest structurally stable diffeomorphisms (“source-sink” diffeomorphisms) of the 2-sphere are considered. The non-wandering set of such diffeomorphisms consists of two hyperbolic points: the source and the sink. In this paper, the existence of an arc connecting two such orientation-preserving (orientation-reversing) diffeomorphisms and consisting entirely of “source-sink” diffeomorphisms is constructively proved.
Doklady Mathematics. 2024;519(1):39-45
39-45
ON NUMERICAL BEAMFORMING FOR ACOUSTIC SOURCE IDENTIFICATION BASING ON SUPERCOMPUTER SIMULATION DATA
Resumo
The paper is devoted to the method of numerical beamforming for processing spatio-temporal data obtained from supercomputer simulation of aeroacoustics problems, in order to localize a distributed acoustic source formed by interaction of turbulent flow and an aircraft or its elements in flight mode, and to determine its amplitude-frequency characteristics. Mathematically, the proposed method is based on solving the inverse problem of restoring the right-hand side in the Helmholtz equation for sources of monopole and dipole types. Compared to an analogue intended for the analysis of experimental measurements, the new method has significant advantages and allows generalization to the case of correlated sources. In the paper, the capabilities of the method are demonstrated by solving the problem of identifying an acoustic source that is generated by an upswept aircraft wing with deployed high-lift devices in landing mode.
Doklady Mathematics. 2024;519(1):46-52
46-52
ANALYTIC METHOD FOR SOLVING ONE CLASS OF NONLINEAR EQUATIONS
Resumo
The paper proposes an analytical approximate method for calculating multidimensional integrals of analytic integrand functions, using the approximation of the latter by a power series. This approach lets transforming the original system of nonlinear equations with integral components into a system of equations with a polynomial left-hand side. To solve equations of this class we developed an analytical method. A sequential recurrent procedure has been developed for the analytical solution of this class of nonlinear equations.
Doklady Mathematics. 2024;519(1):53-56
53-56
MODAL LOGICS OF ALMOST SURE VALIDITIES AND ZERO-ONE LAWS IN HORN CLASSES
Resumo
In this paper we develop a method to study Horn classes of Kripke frames from a probabilistic perspective. We consider the uniform distribution on the set of all 𝑛-point Kripke frames. A formula is almost surely valid in a Horn class F if the probability that it is valid in the F-closure of a random 𝑛-point frame tends to 1 as 𝑛 → ∞. Such formulas constitute a normal modal logic. We show that for pseudotransitive and pseudoeuclidean closures this logic equals S5, and the zero-one law holds.
Doklady Mathematics. 2024;519(1):57-64
57-64
CONCERNING ONE SUPPLEMENT TO UNIFICATION METHOD OF N.N. KRASOVSKII IN DIFFERENTIAL GAMES THEORY
Resumo
The paper deals with the game problem of approach for a conflict-controlled system in a finite-dimensional Euclidean space at a fixed moment of time. The problem of approximate calculation of the solvability sets is studied for the considered approach game. An approach is proposed for approximate calculation of solvability sets on the basis of a unification model, which supplements the unification method of N.N. Krasovskii in the theory of differential games.
Doklady Mathematics. 2024;519(1):65-71
65-71
COMPUTER SCIENCE
INFOBUSINESS AS A NEW DIGITAL PHENOMENON IN THE SOCIO-ECONOMIC SPHERE OF RUSSIA: POSSIBILITIES OF MODELING AND REGULATION
Resumo
Digital technologies are known to have high potential for economic growth, to promote social and political communications. They are platforms for the implementation of new business ideas. New digital phenomenon ”— the information business ”— is investigated in the article. How does information business affect the economy and society? This question determines the main topic for discussion. The following methods were used in the article: system analysis, content analysis, sociological survey, comparative analysis, cluster analysis, fuzzy logic methods, mathematical modeling. The purpose of the article is to determine the relationship and spread of new digital technologies and phenomena in the socio-economic and political spheres of Russia. There are such results as recommendations on the transformation of socio-economic relations, modeling and state regulation of processes in the information environment in the article. At the same time, mathematical methods for studying the tasks are proposed.
Doklady Mathematics. 2024;519(1):72-84
72-84