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Volume 525, Nº 1 (2025)
MATHEMATICS
MATHEMATICAL AND NEURAL NETWORK MODELS OF CONTROL AND OPTIMIZATION OF ECONOMIC DEVELOPMENT
Resumo
The article presents a concept for developing artificial intelligence (AI) based on a hybrid model that uses, on the one hand, verified mathematical models to calculate and forecast potential trend trajectories of long-term socio-economic development, and on the other hand, short- and medium-term models of crisis phenomena for training a neural network with its subsequent use to determine the real economic situation and develop an appropriate optimal policy for managing current socio-economic development. It is proposed to build AI on the basis of the Kolmogorov–Arnold neural network, which is increasingly used to create AI designed to solve physical and engineering problems, including those related to the management of various dynamic processes. The advantage of the hybrid model proposed for the first time is its complete transparency in relation to identifying cause-and-effect relationships between the main factors and output in the process of socio-economic development.
Doklady Mathematics. 2025;525(1):5-11
5-11
NON-ESSENTIAL EXPANSIONS OF WEAKLY O-MINIMAL THEORIES OF FINITE CONVEXITY RANK
Resumo
We study constant expansions of weakly o-minimal theories of finite convexity rank having less than 2ω countable models. We prove that any non-essential expansion (expansion by finitely many new constants) of a weakly o-minimal Ehrenfeucht theory of finite convexity rank preserves Ehrenfeuchtness. We also establish that the countable spectrum of such an expanded theory is not decreased.
Doklady Mathematics. 2025;525(1):12-23
12-23
ON CONDITIONS FOR DOBRUSHIN’S CENTRAL LIMIT THEOREM FOR NON-HOMOGENEOUS MARKOV CHAINS
Resumo
A new sufficient condition is proposed in the problem of Central Limit Theorem in the array scheme for non-homogeneous Markov Chains, with a possibility that the minimum of Markov-Dobrushin’s ergodic coefficient may be closer to zero than in Dobrushin’s main condition, or even equal to zero.
Doklady Mathematics. 2025;525(1):24-30
24-30
MODAL LOGICS WITH THE COMPLEMENT MODALITY
Resumo
Modal logic with two operators, one for the accessibility relation on a Kripke model and the other for its complement, was studied first by Humberstone in 1983 [5], who axiomatized it using an infinite number of axioms. In this paper we suggest a finite axiomatization of this logic, and also axiomatize the corresponding logics of some natural classes of Kripke frames.
Doklady Mathematics. 2025;525(1):31-39
31-39
WEAK SOLVABILITY OF THE INITIAL BOUNDARY VALUE PROBLEM FOR THE INHOMOGENEOUS INCOMPRESSIBLE VOIGT MODEL WITH FULL TIME DERIVATIVE
Resumo
The paper is devoted to proving the existence of a weak solution to the initial-boundary value problem for the non-homogeneous incompressible Voigt fluid motion model with full derivative in the rheological relation. For the proof, a problem approximating the original one is considered, and its solvability is proved using the Leray-Schauder theorem. After that, passing to the limit in the approximation problem as the approximation parameter tends to zero, it is shown that, up to a subsequence, the solutions of the approximation problem weakly converge to a weak solution of the original problem.
Doklady Mathematics. 2025;525(1):40-46
40-46
SINGLE POINT PENALIZATION FOR SYMMETRIC LEVY PROCESSES
Resumo
We consider a one-dimensional symmetric Levy process ξ(t), t ≥ 0, that has local time, which we denote by L(t, x), and construct the operator A + μ δ(x − a), μ > 0, where A is the generator of ξ(t), and δ(x − a) is the Dirac delta function at a ∈ ℝ. We show that the constructed operator is the generator of (Ut)t ≥ 0 – C0-semigroup on L2(ℝ), which is given by (Ut f)(x) = E f (x − ξ(t)) eμ L(t,x−a), f ∈ L2(ℝ) ∩ Cb(ℝ), and prove the Feynman–Kac formula for the delta function-type potentials. Furthermore, we construct a family of penalized distributions {QT,xμ}T ≥ 0 of form QT,xμ = eμ L(T,x−a) / Eeμ L(T,x−a) PT,x, where PT,x is the measure of the process ξ(t), t ≤ T. We show that this family weakly converges to a Feller process as T → ∞, study the Feynman–Kac semigroup generated by this Feller process and prove a limit theorem for the distribution of ξ(T) under QT,x.
Doklady Mathematics. 2025;525(1):47-51
47-51
52-56
PSEUDOANALYTIC SOLUTIONS OF SINGULARLY PERTURBED EQUATIONS IN BANACH ALGEBRAS
Resumo
The small parameter method proposed by A. Poincare allows constructing solutions to singularly perturbed problems in the form of series by degrees of a small parameter, which converge, as a rule, asymptotically. At the same time, the decomposition theorems he proved, in the regular case guarantee the existence of solutions analytically dependent on the parameter. The regularization method of S. A. Lomov reduces a singularly perturbed problem to a regularly perturbed one and makes it possible to build solutions in the form of series converging in the usual sense. This paper studied the Burgers type differential equation given in Banach algebra.
Doklady Mathematics. 2025;525(1):57-61
57-61
REGULARIZED HYDRODYNAMIC EQUATIONS IN A PROBLEM OF TURBULENT FLOW MODELING IN A PIPE
Resumo
The problem of the development of hydrodynamic instability in a pipe at moderate Reynolds numbers for the flow of a viscous incompressible fluid is considered. Numerical simulation is performed on the basis of regularized or quasi-hydrodynamic equations. The simulation was carried out by the finite volume method implemented by the authors within the framework of an open source software package using parallel technologies. Within the framework of this approach, it is shown in a direct numerical experiment that random perturbations of the inlet velocity in a pipe attenuate for subcritical Reynolds numbers and lead to the formation of a turbulent regime for supercritical values. This result shows the prospects of using regularized equations of hydrodynamics as a new alternative model in calculations of laminar-turbulent transition in incompressible liquid and gas in pipeline systems, including estimates of the pipe resistance coefficient.
Doklady Mathematics. 2025;525(1):62-70
62-70
ABOUT ONE KINETIC MODEL FOR DESCRIBING TURBULENT FLOWS TAKING INTO ACCOUNT HEAT TRANSFER
Resumo
A closed system of equations is obtained to describe spatially two-dimensional flows taking into account turbulent heat transfer. Additional equations for pulsation moments and the analysis of the coefficient of turbulent thermal conductivity are derived based on the kinetic equation previously used to derive a quasi-gas dynamic system of equations. The results of simulation the problem of the turbulent flow of a hot plane jet are presented.
Doklady Mathematics. 2025;525(1):71-77
71-77
ON ASYMPTOTIC PROPERTIES OF DISTANCE CORRELATION WITH CENSORED RESPONSE
Resumo
We consider asymptotic properties of the empirical correlation coefficient with survival response, based on the famous distance correlation and initially proposed in [1]. We show that this empirical coefficient is consistent and asymptotically normal for corresponding correlation measure and, if the survival time and uncensored feature are independent, establish its convergence to a Gaussian chaos.
Doklady Mathematics. 2025;525(1):78-82
78-82
83-90
91-97
98-101
ARROW’S SINGLE-PEAKED DOMAINS
Resumo
The paper studies structured preferences domains that avoid configuration with three different third elements in three elements restrictions (Arrow’s single-peaked domains). The number of Arrow’s single-peaked domains and the number of non-isomorphic classes of Arrow’s single-peaked domains are found. We present a forbidden submatrix characterization for matrix representation of Arrow’s single-peaked preferences.
Doklady Mathematics. 2025;525(1):102-108
102-108
TO THE THEORY OF POLYNOMIAL MODELS OF DISCRETE DYNAMIC SYSTEMS
Resumo
The paper develops the framework of Kronecker products over the field of real numbers and the field of predicates. Procedures and logical structures of algorithms transforming Kronecker vectors into vectors in extended spaces are constructed. The application of the developed framework for mathematical modeling of polynomial discrete dynamic systems is considered and a method for their transformation to a quasilinear form is developed. The latter is used to generate data ensembles with given numerical characteristics and randomized forecasting. Discrete polynomial systems with feedback are considered. Their solution in the form of a multidimensional power series is constructed and its convergence is studied.
Doklady Mathematics. 2025;525(1):109-129
109-129
ON ESTIMATES ON THE EXPONENT IN THE CONSTRUCTION OF FINITELY PRESENTED INFINITE SEMIGROUP
Resumo
The work is devoted to improving the exponent in the construction of a finitely defined infinite nil semigroup. The construction addresses the Shevrin–Saper question, posed in particular in the Sverdlov Notebook (3.81b) [3]. Additionally, the construction is one of the first examples of Burnside-type constructions in the finitely defined case. The development is part of a series of works [4–8]. The initial version of the construction was carried out for the exponent 9, i.e., an infinite finitely defined nil semigroup with the identity x9 = 0 was constructed.
Doklady Mathematics. 2025;525(1):130-134
130-134
135–143
ПРОЦЕССЫ УПРАВЛЕНИЯ
OBSERVATION OF THE FLOW OF OBJECTS ON A GEODETIC ARC IN ℝ3 IN CONDITIONS OF COUNTERACTION
Resumo
This paper proposes methods for an observer f to track a flow of objects ti moving at constant velocity along a smooth geodesic trajectory T, composed of a segment and an arc Λ. The observer detects pairs of objects at the moments they pass through predetermined fixed control points evenly distributed along T. An unfriendly object, after being detected, may direct a dangerous mini-object toward the observer. The work presents algorithmic approaches for tracking objects moving along straight-line segments and smooth arcs.
Doklady Mathematics. 2025;525(1):144-148
144-148