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卷 510, 编号 1 (2023)
МАТЕМАТИКА
ON INTERPRETATIONS OF PRESBURGER ARITHMETIC IN BÜCHI ARITHMETICS
摘要
Büchi arithmetics BAn, \(n \geqslant 2\), are extensions of Presburger arithmetic with an unary functional symbol \({{V}_{n}}(x)\) denoting the largest power of n that divides x. Definability of a set in BAn is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We consider the interpretations of Presburger Arithmetic in the standard model of BAn and show that each such interpretation has an internal model isomorphic to the standard one. This answers a question by A. Visser on the interpretations of certain weak arithmetical theories in themselves.
3-7
ELEMENTARY INVARIANTS FOR QUANTIFIED PROBABILITY LOGIC
摘要
Let QPL be the two-sorted probabilistic language proposed in [8], which expands the well-known ‘polynomial’ language described in [3, Section 6] by adding quantifiers over events. We show that all atomless spaces have the same QPL-theory, and this theory is decidable. Also we introduce the notion of elementary invariant for QPL and use it for obtaining exact complexity upper bounds for some interesting probabilistic theories.
8-12
NONSTATIONARY VENTTSEL PROBLEM WITH VMOx LEADING COEFFICIENTS
摘要
We obtain some new results on strong solvability in the Sobolev spaces of the linear Venttsel initial-boundary value problems to parabolic equations with discontinuous leading coefficients.
13-17
AN ANALOGUE OF MAHLER’S TRANSFERENCE THEOREM FOR MULTIPLICATIVE DIOPHANTINE APPROXIMATION
摘要
Khintchine’s and Dyson’s transference theorems can be very easily deduced from Mahler’s transference theorem. In the multiplicative setting an obstacle appears, which does not allow deducing the multiplicative transference theorem immediately from Mahler’s theorem. Some extra considerations are required, for instance, induction by the dimension. In this paper we propose an analogue of Mahler’s theorem which implies the multiplicative transference theorem immediately.
18-22
REFINED MATHEMATICAL MODEL OF ECONOMIC DYNAMICS UNDER HIGH INFLATION AND UNSTABLE DEVELOPMENT
摘要
Previously, the authors proposed two mathematical models that describe the dynamics of economic development rates and forecasting inflation rates. The events that took place after February 2022 introduced an element of significant turbulence into the processes of macroeconomic dynamics in Russia, which required some correction of the previous models. This is reflected in this article. The processes simulated on the basis of new models show that in order to restore economic growth and reduce inflation in the medium term, a significant reduction in the interest rate and a steady increase in the money supply are required. It is shown that in the current era of geopolitical changes and the collapse of many liberal market dogmas, the role of the state and planning in managing economic processes is increasing.
23-28
TRANSCENDENCE OF p-ADIC VALUES OF GENERALIZED HYPERGEOMETRIC SERIES WITH TRANSCENDENTAL POLYADIC PARAMETERS
摘要
It is established that if \({{\alpha }_{1}}, \ldots ,{{\alpha }_{m}}\) are polyadic Liouville numbers, and the number \(\xi \) is a positive integer or Ξ is a polyadic Liouville number and if \({{\Psi }_{0}}\left( z \right) = \sum\limits_{n = 0}^\infty {{{{\left( {{{\alpha }_{1}}} \right)}}_{n}} \ldots {{{\left( {{{\alpha }_{m}}} \right)}}_{n}}{{z}^{n}}} ,\) \({{\Psi }_{1}}(z)\, = \,\sum\limits_{n = 0}^\infty {{{{\left( {{{\alpha }_{1}} + 1} \right)}}_{n}} \ldots {{{\left( {{{\alpha }_{m}} + 1} \right)}}_{n}}{{z}^{n}}} \), then there are infinitely many primes p such that the at least one of the p-adic integers \({{\Psi }_{0}}\left( \xi \right),\) \({{\Psi }_{1}}\left( \xi \right)\) (respectively, \({{\Psi }_{0}}\left. {\left( {\text{\Xi }} \right)} \right),\) \({{\Psi }_{1}}\left( {\text{\Xi }} \right)\) is transcendental.
29-32
THE PROBLEM OF THE FLOW OF ONE TYPE OF NON-NEWTONIAN FLUID THROUGH THE BOUNDARY OF A MULTI-CONNECTED DOMAIN
摘要
In this paper, the existence of a weak solution of the initial boundary value problem for the equations of motion of a viscoelastic non-newtonian fluid in a multi-connected domain with memory along the trajectories of a non-smooth velocity field and an inhomogeneous boundary condition. The study assumes the approximation of the original problem by Galerkin-type approximations followed by a passage to the limit based on a priori estimates. The theory of regular Lagrangian flows is used to study the behavior of trajectories of a non-smooth velocity field.
33-38
ON ONE CONDITION FOR THE DISCRETENESS OF THE SPECTRUM AND THE COMPACTNESS OF THE RESOLVENT OF A NONSECTORIAL STURM–LIOUVILLE OPERATOR ON THE SEMIAXIS
摘要
The spectral properties of the Sturm–Liouville operator on the semi-axis with the complex-valued potential with the range exceeding the half-plane, has been little studied. The operator in this case can be non-sectorial, the numerical range can coincide with the entire complex plane. In this situation we propose the conditions ensuring the discreteness of the spectrum and the compactness of the resolvent.
39-42
ON THE ACCURACY OF DISCONTINUOUS GALERKIN METHOD CALCULATING GAS-DYNAMIC SHOCK WAVES
摘要
The results of a numerical calculation of gas-dynamic shock waves that arise when solving the Cauchy problem with smooth periodic initial data are presented using three variants of the DG (Discontinuous Galerkin) method, in which the solution is sought in the form of a piecewise linear discontinuous function. It is shown that the methods DG1A1 and DG1A2, for which the Cockburn limiter with parameters A1 = 1 and A2 = 2 are used for monotonization, have approximately the same accuracy in the influence areas of shocks (arising as a result of gradient catastrophes within the computational domain), while the nonmonotonic DG1 method, in which this limiter is not used, has a significantly higher accuracy in these areas, despite noticeable non-physical oscillations on shocks. With this in mind, the combined scheme obtained by the joint application of the DG1 and DG1A1 methods monotonously localizes the shocks and maintains increased accuracy in the areas of their influence.
43-51
ON ONE APPROACH TO THE ASSESSMENT OF A TRIANGULAR ELEMENT DEGENERATION IN A TRIANGULATION
摘要
A quantitative estimate of a triangular element quality is proposed - the triangle degeneration index. To apply this estimate, the simplest model triangulation is constructed, in which the coordinates of the nodes are formed as the sum of the corresponding coordinates of the nodes of some given regular grid and random increments to them. For different values of the parameters, the empirical distribution function of the triangle degeneration index is calculated, which is considered as a quantitative characteristic of the quality of triangular elements in the constructed triangulation.
52-56