Doklady Mathematics最新文献

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

This paper is devoted to the localisation of random 3-CNF formulas that are polynomially solvable by the resolution algorithm. It is shown that random formulas with the number of clauses proportional to the square of the number of variables, are polynomially solvable with probability close to unity when the proportionality coefficient exceeds the found threshold.

Criteria for a Banach limit to belong to the discrete or continuous part of the set of Banach limits are presented. The diameters and radii of these parts are found.

We propose ways of acting an observer f when tracking an object (t) moving in ({{mathbb{R}}^{3}}) along the shortest trajectory (mathcal{T}) bypassing a collection ({ {{G}_{i}}} ) of convex sets. The object has high-speed miniobjects threatening the observer. The tracking methods depend on the geometric properties of ({{G}_{i}}) and (mathcal{T}). The observer’s task is to track the motion of the object over as long a segment of (mathcal{T}) as possible.

The tasks of analyzing and visualizing the dynamics of viscous incompressible flows of complex geometry based on traditional grid and projection methods are associated with significant requirements for computer performance necessary to achieve the set goals. To reduce the computational load in solving this class of problems, it is possible to apply algorithms for constructing artificial neural networks (ANNs) using exact solutions of the Navier–Stokes equations on a given set of spatial regions as training sets. An ANN is implemented to construct flows in regions that are complexes made up of training sets of standard axisymmetric domains (cylinders, balls, etc.). To reduce the amount of calculations in the case of 3D problems, invariant flow manifolds of lower dimensions are used. This makes it possible to identify the structure of solutions in detail. It is established that typical invariant regions of such flows are figures of rotation, in particular, ones homeomorphic to the torus, which form the structure of a topological bundle, for example, in a ball, cylinder, and general complexes composed of such figures. The structures of flows obtained by approximation based on the simplest 3D unsteady vortex flows are investigated. Classes of exact solutions of the incompressible Navier–Stokes system in bounded regions of ({{mathbb{R}}_{3}}) are distinguished based on the superposition of the above-mentioned topological bundles. Comparative numerical experiments suggest that the application of the proposed class of ANNs can significantly speed up the computations, which allows the use of low-performance computers.

The logistic equation with delay and diffusion and with nonclassical boundary conditions is studied. The stability of a nontrivial equilibrium state is investigated, and the resulting bifurcations are studied numerically.

To satisfy the conditions of Jacobi’s theorem on the last multiplier, the existence of an invariant measure and a sufficient number of independent first integrals are needed. In this case, the system can be locally integrated by quadratures. There are examples of systems for which the existence of partial first integrals is sufficient for the possibility of integration by quadratures. Moreover, integration by quadratures occurs at the level of partial first integrals. In this paper, Jacobi’s theorem on the last multiplier is extended to the general situation when the first integrals include partial ones.

We define a divisible completion of the solvable Baumslag-Solitar group (BS(1,n)) and prove that under certain restrictions on n the elementary theory of this completion is decidable.