Doklady Mathematics最新文献

In many applications, multidimensional integrals over the unit hypercube arise, which are calculated using Monte Carlo methods. The convergence of the best of them turns out to be quite slow. In this paper, fundamentally new cubature formulas with superpower convergence based on improved Korobov grids and a special variable substitution are proposed. A posteriori error estimates are constructed, which are nearly indistinguishable from the actual accuracy. Examples of calculations illustrating the advantages of the proposed methods are given.

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum with the identity operator multiplied by a positive square-integrable function is a monotone operator. The multivalued mapping is Lipschitz continuous with respect to the phase variable. We prove the existence of a solution and the density in the corresponding topology of the solution set of the initial inclusion in the solution set of the inclusion with a convexified multivalued mapping. For these purposes, new distances between maximal monotone operators are introduced.

Based on the geometric characteristics of a tetrahedron, quantitative estimates of its degeneracy are proposed and their relationship with the condition number of local bases generated by the edges emerging from a single vertex is established. The concept of the tetrahedron degeneracy index is introduced in several versions, and their practical equivalence to each other is established. To assess the quality of a particular tetrahedral partition, we propose calculating the empirical distribution function of the degeneracy index on its tetrahedral elements. An irregular model triangulation (tetrahedralization or tetrahedral partition) of three-dimensional space is proposed, depending on a control parameter that determines the quality of its elements. The coordinates of the tetrahedra vertices of the model triangulation tetrahedra are the sums of the corresponding coordinates of the nodes of some given regular mesh and random increments to them. For various values of the control parameter, the empirical distribution function of the tetrahedron degeneracy index is calculated, which is considered as a quantitative characteristic of the quality of tetrahedra in the triangulation of a three-dimensional region.

A stability estimate for the solution of a source problem for the stationary radiative transfer equation is given. It is supposed that the source has an isotropic distribution. Earlier, stability estimates for this problem were found in a partial case of the emission tomography problem with a vanishing scattering operator and for the complete transfer equation under additional difficult-to-check conditions imposed on the absorption coefficient and the scattering kernel. In this work, we suggest a new fairly simple approach for obtaining a stability estimate for the problem under the consideration. The transfer equation is considered in a circle of the two-dimension space. In the forward problem, it is assumed that incoming radiation is absent. In the inverse problem of recovering the unknown source, data on solutions of the forward problem related to outgoing radiation are given on a portion of the boundary. The obtained result can be used to estimate the total density of distributed radiation sources.

In this paper, we construct a strongly continuous operator group generated by a one-dimensional Dirac operator acting in the space (mathbb{H} = {{left( {{{L}_{2}}[0,pi ]} right)}^{2}}). The potential is assumed to be summable. It is proved that this group is well-defined in the space (mathbb{H}) and in the Sobolev spaces (mathbb{H}_{U}^{theta }), (theta > 0), with a fractional index of smoothness θ and boundary conditions U. Similar results are proved in the spaces ({{left( {{{L}_{mu }}[0,pi ]} right)}^{2}}), (mu in (1,infty )). In addition, we obtain estimates for the growth of the group as (t to infty ).

A distributed renewable resource of any nature is considered on a two-dimensional sphere. Its dynamics is described by a model of the Kolmogorov–Petrovsky–Piskunov–Fisher type, and the exploitation of this resource is carried out by constant or periodic impulse harvesting. It is shown that, after choosing an admissible exploitation strategy, the dynamics of the resource tend to limiting dynamics corresponding to this strategy and there is an admissible harvesting strategy that maximizes the time-averaged harvesting of the resource.

An infinite horizon optimal control problem with general endpoint constraints is reduced to a family of standard problems on finite time intervals containing the conditional cost of the phase vector as a terminal term. A new version of the Pontryagin maximum principle containing an explicit characterization of the adjoint variable is obtained for the problem with a general asymptotic endpoint constraint. In the case of the problem with a free final state, this approach leads to a normal form version of the maximum principle formulated completely in the terms of the conditional cost function.

A new correlative-grid approximation of a homogeneous random field is introduced for an effective numerical-analytical investigation of the superexponential growth of the mean particle flow with multiplication in a random medium. The complexity of particle trajectory realization is independent of the correlation scale. Test computations for a critical ball with isotropic scattering showed high accuracy of the corresponding mean flow estimates. For the correlative-grid approximation of a random density field, the possibility of Gaussian asymptotics of the mean particle multiplication rate as the correlation scale decreases is justified.

We consider Bernstein inequality for the Riesz derivative of order (0 < alpha < 1) of entire function of exponential type in the uniform norm on the real line. The interpolation formula is obtained for this operator; this formula has non-equidistant nodes. By means of this formula, the exact Bernstein inequality is found for all (0 < alpha < 1), namely, the extremal entire function and the sharp constant are written out.

The paper presents the results of mathematical modeling of plasma transport in a spiral magnetic field using new experimental data obtained at the SMOLA trap created at the Budker Institute of Nuclear Physics of the Siberian Branch of the Russian Academy of Sciences. Plasma confinement in the trap is carried out by transmitting a pulse from a magnetic field with helical symmetry to a rotating plasma. A new mathematical model is based on a stationary plasma transport equation in an axially symmetric formulation. The distribution of the plasma concentration obtained by numerical simulation confirmed the confinement effect obtained in the experiment. The dependences of the integral characteristics of the plasma on the depth of corrugation of the magnetic field, diffusion, and plasma potential are obtained. The mathematical model is intended to predict plasma confinement parameters in designing traps with a spiral magnetic field.