Doklady Mathematics最新文献

We present a new result on the generalisation of A.G. Postnikov’s formula to the case of powers of 2. This, together with the original work of A.G. Postnikov and some structural theorems on reduced residue systems modulo prime-power ideals, is used to obtain estimates for certain character sums in algebraic number fields.

Two problems of sub-Lorentzian geometry on the Martinet distribution are studied. For the first one, the reachable set has a nontrivial intersection with the Martinet plane, while a trivial intersection occurs for the second problem. Reachable sets, optimal trajectories, and sub-Lorentzian distances and spheres are described.

For arbitrary continuous functions on the interval [0, 1], we obtain an interpolation formula based on known values of these functions on some uniform grid. No additional assumptions about the functions are required. The construction of such a formula is connected with the properties of local Bernstein polynomials and the Riemann zeta function. Numerical results for the interpolation of functions of the Riemann, Weierstrass, Besicovitch, and Takagi types are presented.

Two methods for quantitatively assessing the chirality of a set are considered. As a measure of the noncoincidence between two sets, one method uses the area of the symmetric difference between them, and the other, the Hausdorff distance between them. It is shown that these methods, generally speaking, do not provide a correct quantitative estimate for a fairly wide class of sets, such as bounded Borel sets. Using examples of flat triangles and convex quadrangles, we consider the problem of dividing geometric objects into right- and left-handed ones. For triangles, level lines of two versions of the chirality measure are calculated on the plane of angular parameters. For a spatial helix, the values of two versions of the chirality index are found by calculating the mixed product of vectors and the Hausdorff distance between two sets, respectively.

Increased integrability of the gradient of the solution to the homogeneous Dirichlet problem for the Poisson equation with lower terms in a bounded Lipschitz domain is established. The unique solvability of this problem is also proved.

Some questions concerning the inversion of the classical and generalized integral Radon transforms are discussed. The main issue is to determine information about the integrand if the values of some integrals are known. A feature of this work is that a function is integrated over hyperplanes in a finite-dimensional Euclidean space and the integrands depend not only on the variables of integration, but also on some of the variables characterizing the hyperplanes. The independent variables describing the known integrals are fewer than those in the unknown integrand. We consider discontinuous integrands defined on specifically introduced pseudoconvex sets. A Stefan-type problem of finding discontinuity surfaces of the integrand is posed. Formulas for solving the problem under study are derived by applying special integro-differential operators to known data.

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations is presented, which does not use Kolmogorov flows. By applying this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions.

A closed system of equations for describing turbulent flows is obtained. Additional equations for the cross pulsation moments (rho overline {Delta {{u}_{i}}Delta {{u}_{k}}} ) are derived using a balanced kinetic equation, which was previously used to obtain a quasi-gasdynamic system of equations. Numerical results for the problem of a two-dimensional mixing layer between two flows are presented.