It is proved that an artificial neural network with smooth activation functions and without bottlenecks is a Morse function for almost all, with respect to the Lebesgue measure, sets of weights.

A new version of accelerated gradient descent is proposed. The method does not require any a priori information on the objective function, uses a linesearch procedure for convergence acceleration in practice, converge according to well-known lower bounds for both convex and nonconvex objective functions, and has primal-dual properties. A universal version of this method is also described.

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

The Le Bars conjecture (2001) states that the binomial random graph *G*(*n*, 1/2) obeys the zero–one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge.

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.

In the paper we discuss a new bound of the total variation distance in terms of *L^*2 distance for random variables that are polynominals in log-concave random vectors.

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.

Scalar real Riccati equations with coefficients expandable in convergent power series in a neighborhood of infinity are considered. Extendable solutions to equations of this kind are studied. Methods of power geometry are used to obtain conditions for asymptotic series expansions of these solutions.

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

New results are obtained on convergence to stationary measures in nonlinear Fokker-Planck-Kolmogorov equations.

Differential properties of semigroups and estimates of distances between stationary distributions of diffusions are studied.

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.