A new characterization of Nikolskii–Besov classes via integration by parts is obtained

A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for an nth-order FEM based on the fast discrete Fourier transform. The amount of arithmetic operations is logarithmically optimal in the theory and is rather attractive in practice. The algorithm admits numerous further applications (including the multidimensional case).

The Dotsenko-Fateev integral, an analytic function of one complex variable arising in conformal field theory, is generalized in a natural way to an analytic function of two complex variables. A system of partial differential equations and a Pfaffian system of Fuchsian type are derived for this generalized Dotsenko- Fateev integral. The Fuchsian system permits to obtain local expansions of solutions in the neighborhoods of singularities of the system.

The present article offers a simple mathematical model for forecast calculations of the synergy effect generated by the NBIC-convergence and the evaluation of its impact on the economic growth in the first half of the 21st century. We demonstrate that the NBIC-technologies (due to a powerful synergy effect) will lead to a significant acceleration of technological growth rates that will exceed the growth rates achieved at the upswing phase of the 5th Kondratieff wave (1982–2006) on the basis of the microelectronic technologies. Using the example of the USA economy, it is shown that in this country the annual technological growth rates will increase up to 3.3% (as compared with 2.3% per year in the 1980s), whereas the annual economic growth rates in the 2030s will be as high as 3.6%. Thus, we will see the replacement of the deceleration trend that was observed within the world economy in the recent forty years with an accelerating trend.

A Riccati equation with coefficients expandable into convergent power series in a neighborhood of infinity is considered. Continuable solutions to equations of this type are studied. Conditions for the expansion of these solutions into convergent series in a neighborhood of infinity are obtained by methods of power geometry.

The Riccati equation is considered. Both continuable and noncontinuable solutions of this equation are studied. Asymptotic representations of its solutions are obtained by power geometry methods

The asymptotic properties of nonoscillating solutions of Emden–Fowler-type equations of arbitrary order are considered. The paper contains the results of the study of the power and power-logarithmic asymptotics of solutions and conditions of existence of these solutions. To analyze the asymptotic behavior of solutions of the equations, methods of power geometry are used.

The set of all formulas whose n-provability in a given arithmetical theory S is provable in another arithmetical theory T is a recursively enumerable extension of S. We prove that such extensions can be naturally axiomatized in terms of transfinite progressions of iterated local reflection schemata over S. Specifically, the set of all provably 1-provable sentences in Peano arithmetic PA can be axiomatized by an epsilon(0)-times iterated local reflection schema over PA. The resulting characterizations provide additional information on the proof-theoretic strength of these theories and on the complexity of their axiomatization.

Axiom A diffeomorphisms of closed 2-manifold of genus p⩾2 whose nonwandering set contains a perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to a pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of this paper is as follows. Two diffeomorphisms from the given class are topologically conjugate on perfect spaciously situated attractors if and only if the corresponding homotopic pseudo-Anosov homeomorphisms are topologically conjugate by means of a homeomorphism that maps a certain subset of one pseudo-Anosov homeomorphism onto a subset of the other.

The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established.

A sample X_1,...,X_n consisting of шndependent identically distributed vectors in Rp with zero mean and a covariance matrix \Sigma is considered. The recovery of spectral projectors of high-dimensional covariance matrices from a sample of observations is a key problem in statistics arising in numerous applications. In their 2015 work, V.Koltchinskii and K.Lounici obtained non-asymptotic bounds for the Frobenius norm \|\hat P_r − P_r \|_2^2 of the distance between sample and true projectors and studied asymptotic behavior for large samples. More specifically, asymptotic confidence sets for the true projector \P_r were constructed assuming that the moment characteristics of the observations are known. This paper describes a bootstrap procedure for constructing confidence sets for the spectral projector \P_r of the covariance matrix \Sigmna from given data. This approach does not use the asymptotical distribution of \|\hat P_r − P_r \|_2^2 and does not require the computation of its moment characteristics. The performance of the bootstrap approximation procedure is analyzed.

We describe a general method for calculating cohomology groups of finite-dimensional spaces of nonsingular functions and calculate the real cohomology groups of the space of nonsingular curves of degree 5 in C P^5

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design of branch-and-bound algorithms solving such problems. We show in this paper that for the weighted independent set problem and a bipartite graph with n vertices and m edges an optimal solution is computed in O(nm) time and all tolerances with respect to it are computed in O(n^2) time. We design an algorithm that for an interval graph with n vertices and $m$ edges simultaneously computes an optimal weighted independent set and all corresponding tolerances in O(n+m) time.

The notion of a tolerance of an element of a combinatorial optimization problem is often used for stability analysis of an optimal solution and it is a base for design branch-and-bound algorithms solving such problems. In this paper we show that for the weighted independent set problem on trees with *n* vertices all upper and lower tolerances related to this solution can be computed with *O*(*n*) time.

Entropy balance in the one-dimensional hyperbolic quasi-gasdynamic (HQGD) system of equations is analyzed. In regular flow regimes, it is shown that the behavior of entropy in the HQGD system is determined by terms involving the natural viscosity and thermal conductivity coefficients. The total entropy production differs from the Navier–Stokes equations for viscous compressible heat-conducting gases by the second order terms with respect to a relaxation parameter. Additionally, a similar analysis of energy balance is performed for the simpler case of the barotropic HQGD system, which is of interest for some applications.

New estimates of distances between transition probabilities of diffusions are obtained

A novel approach to the fair division problem is proposed, which is based on the concept of a priori estimates and ideas of dynamical systems theory. For several problems on the division of a resource with discrete components, this approach leads to explicit constructive solutions in cases for which even the existence of solutions has not been previously known.

New results on integrability and continuity of solutions to Fokker-Planck-Kolmogorov equations are obtained

New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained.