Vol 510, No 1 (2023)

Büchi arithmetics BA_n, , are extensions of Presburger arithmetic with an unary functional symbol denoting the largest power of n that divides x. Definability of a set in BA_n 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 BA_n 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.

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.

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.

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.

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.