Doklady Mathematics最新文献

Consider sequences of integers (a_{n}^{{(k,j)}}), (k = 1, ldots ,{{T}_{j}}), (j = 1, ldots ,m), such that (a_{n}^{{(k,j)}} = a_{{n + {{T}_{j}}}}^{{(k,j)}}), (j = 1, ldots ,m), (k = 1, ldots ,{{T}_{j}}), (n = 0,1, ldots ), and consider the series ({{F}_{{j,k}}}(z) = sumnolimits_{n = 0}^infty a_{n}^{{(k,j)}}n!{{z}^{n}}), (k = 1, ldots ,{{T}_{j}}), (j = 1, ldots ,m). The conditions are established under which the set of series ({{F}_{{j,k}}}(z)), (k = 2, ldots ,{{T}_{j}}), (j = 1, ldots ,m) and the Euler series (Phi (z) = sumnolimits_{n = 0}^infty n!{{z}^{n}}) are algebraically independent over (mathbb{C}(z)) and, for any algebraic integer (gamma ne 0), their values at the point (gamma ) are infinitely algebraically independent.

In recent joint papers the authors of this note solved a famous problem remained open for many years and proved that for arbitrary contractions with trace class difference there exists an integrable spectral shift function, for which an analogue of the Lifshits–Krein trace formula holds. Similar results were also obtained for pairs of dissipative operators. Note that in contrast with the case of self-adjoint and unitary operators it may happen that there is no real-valued integrable spectral shift function. In this note we announce results that give sufficient conditions for the existence of an integrable real-valued spectral shift function in the case of pairs of contractions. We also consider the case of pairs of dissipative operators.

We consider interpolation-characteristic schemes approximating the radiative transfer equation corresponding to the ({{P}_{1}}) model. The model equations are modified by adjusting the rate of radiation energy transfer. This correction can reduce the influence of nonphysical effects in calculating radiative heat transfer in a medium with nonuniform opacity.

An analytical approximate method for calculating multidimensional integrals of analytic functions is proposed, in which the integrand is approximated by a power series. This approach transforms the original system of nonlinear equations with integral components into a system of equations with a polynomial left-hand side. To solve equations of this class, an analytical method based on abstract power series is developed. A recurrent procedure is developed for the analytical solution of this class of nonlinear equations.

The article proposes a new exact algorithm of quadratic complexity that solves the problem of the shortest transformation (“alignment”) of one weighted tree into another, taking into account arbitrary costs of operations on trees. Three operations are considered: adding vertex deletions to an edge or root of a tree and shifting a subtree with deletions.

The paper deals with the game problem of approach for a conflict-controlled system in a finite-dimensional Euclidean space at a fixed moment of time. The approximate calculation of the solvability sets for the considered approach game is studied. A method is proposed for approximate calculation of solvability sets on the basis of a unification model, which supplements Krasovskii’s unification method in the theory of differential games.

It is well known that the mapping class group of the two-dimensional sphere ({{mathbb{S}}^{2}}) is isomorphic to the group ({{mathbb{Z}}_{2}} = { - 1, + 1} ). At the same time, the class +1(–1) contains all orientation-preserving (orientation-reversing) diffeomorphisms and any two diffeomorphisms of the same class are diffeotopic, that is, they are connected by a smooth arc of diffeomorphisms. On the other hand, each class of maps contains structurally stable diffeomorphisms. It is obvious that in the general case, the arc connecting two diffeotopic structurally stable diffeomorphisms undergoes bifurcations that destroy structural stability. In this direction, it is particular interesting in the question of the existence of a connecting them stable arc – an arc pointwise conjugate to arcs in some of its neighborhood. In general, diffeotopic structurally stable diffeomorphisms of the 2-sphere are not connected by a stable arc. In this paper, the simplest structurally stable diffeomorphisms (source–sink diffeomorphisms) of the 2-sphere are considered. The non-wandering set of such diffeomorphisms consists of two hyperbolic points: the source and the sink. In this paper, the existence of an arc connecting two such orientation-preserving (orientation-reversing) diffeomorphisms and consisting entirely of source-sink diffeomorphisms is constructively proved.

An addition to the article “A new spectral measure of complexity and its capabilities for detecting signals in noise” is presented.

We perform a comparative analysis of the accuracy of second-order TVD (Total Variation Diminishing), third-order RBM (Rusanov–Burstein–Mirin), and fifth-order in space and third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) difference schemes for solving a special Cauchy problem for shallow water equations with discontinuous initial data. The exact solution of this problem contains a centered rarefaction wave, but does not contain a shock wave. It is shown that in the centered rarefaction wave and its influence area, the solutions of these three schemes converge with different orders to different invariants of the exact solution. This leads to a decrease in the accuracy of these schemes when they used to calculate the vector of base variables of the considered Cauchy problem. The P-form of the first differential approximation of the difference schemes is used for theoretical justification of these numerical results.

An upper estimate for the maximum width of a forbidden foothold zone that a multi-legged walking robot can overcome in static stability mode is presented. By using mathematical models of six- and four-legged robots, it is shown that the estimate cannot be improved. For this purpose, foot placement sequences are formed for which the estimate is attained. The dependence of the maximum width of the zone on the body length is found for the six-legged robot model.