Doklady Mathematics最新文献

Classical inversion formulas for the integral Radon transform assume that the integrand is smooth. However, this restriction does not fully correspond to application of results in probing theory, which is the main area of application of the Radon transform. It would be more natural to assume that jump discontinuities are admissible for integrands. The paper presents several inversion formulas proved by the authors for piecewise continuous functions. The formulas are compared, and preliminary recommendations on their use for numerical algorithms are given.

The problem of interpolation in the mean of a function by an integro quadratic spline given integrally averaged function values is considered. It is shown that an integro quadratic spline can be defined via a cubic interpolation spline. Since cubic interpolation splines have been studied quite well, well-known error bounds for them and some of their properties can be transferred to integro quadratic splines. Points of superconvergence of integro splines are found, i.e., points at which a spline or its derivatives provide a higher order of approximation.

An alternative definition of Weyl fractional derivatives is given, and their effect on functions from Gevrey classes is studied. For a partial differential equation with Weyl derivatives, conditions for the solvability of the Cauchy problem in Gevrey classes are found.

An improvement of Riordan’s result on the threshold probability of the occurrence of a spanning subgraph in a random graph is obtained for some classes of subgraphs. In particular, this result implies an improved bound for the maximum power of a Hamiltonian cycle in a random graph. Moreover, a sharp asymptotic threshold probability for a random graph to contain spanning subgraphs from a wide class of (k)-degenerate graphs is found.

Homogeneous Markov chains with continuous time are considered. A new approach is proposed that makes it possible to obtain accurate perturbation bounds for such chains with respect to perturbations of infinitesimal characteristics. It is shown how the results can be applied to stationary queuing systems of several classes and to some nonstationary systems.

This work examines basic categorial grammars and categorial grammars with the unique type assignment condition. For the first formalism, it is proven that determining for an arbitrary context-free language (L) whether it is generated by some grammar from this class is algorithmically undecidable. It is also proven that, for any two grammars of this class, the problem of determining the emptiness of the intersection of the languages generated by these grammars is algorithmically undecidable. For the second formalism, it is proven that, for any two categorial grammars with unique type assignment, the problem of determining language inclusion is algorithmically undecidable.

Consider the set (mathcal{E}(G,k)) of all sizes (numbers of edges) of induced subgraphs of size k in a given graph (G) on (n) vertices. For the binomial random graph (G = G(n,p)), we prove that, for each (alpha > 0) and (varepsilon ) small enough, the set (mathcal{E}(G,k)) with high probability contains a long interval for all k such that ({{(ln n)}^{{1 + alpha }}} < k < varepsilon n). We also find the asymptotic length of this interval.

Differential equations describing the behavior of continuous media with creep involve integral type operators, in accordance with Volterra’s linear theory, which is applicable to a wide range of materials with amorphous and heterogeneous structures. In these equations, the kernel of the integral operator is represented as a sum of exponentials or as a weakly singular kernel (Rabotnov function). Obtaining an analytical solution for the equations in question is problematic in some cases, so it is necessary to develop a numerical method and algorithm, taking into account the memory of the considered medium. In this paper, the equations are solved using the grid-characteristic method and dimensional splitting (for multidimensional problems). The approximation and stability of the proposed method are numerically investigated.

The three-dimensional exterior Neumann problem for the Helmholtz equation is considered. Using the potential method, we reduce it to a weakly singular boundary Fredholm integral equation of the second kind, which is solved numerically. To improve the accuracy of the numerical solution algorithm and to reduce its computational complexity, we average the kernel of the integral operator and localize its singular part during discretization by applying simple analytical expressions. Examples of using this approach in the numerical solution of the original problem are given.

In this paper, we propose a modification of the discontinuous Galerkin method using time-dependent basis functions. The use of such basis functions allows one to naturally and stably calculate strong discontinuities.