# Mathematics

## On a Refinement of the Asymptotic Formula for the Lebesgue Constants

For the Lebesque constant of the classical Lagrange polynomial defined in the even number of nodes of interpolation, strict two-sided estimation is received. On this basis, an undefined value O(1) is refined in the well-known asymptotic equality for the Lebesque constant. Two actual problems in the interpolation theory associated with the optimal choice of O(1) are solved.

## An Estimate from Above of the Number of Invariant Straight Lines of n-th Degree Polynomial Vector Field

It is shown that the n-th degree polynomial vector field in the plane has at most 2n + 1 (2n + 2) invariant straight lines when n is even (odd) and n ≥ 3 if it has a singular point for which n + 1 invariant straight lines and n parallel invariant straight lines with a certain angular coefficient are incident.

## On Some Integral Properties of Modified Bessel Functions

New integral equaties for modified Bessel functions of an arbitrary complex order are presented. The properties of Lebedev – Skalskaya integral transforms are investigated.

## Асимптотическое отношение гармонических мер сторон разреза

The article is devoted to the geometry of solutions to the chordal Löwner equation which is based on the comparison of singular solutions and harmonic measures for the sides of a slit in the upper half-plane generated by a driving term. An asymptotic ratio for harmonic measures of slit sides is found for a slit which is tangential to a straight line under a given angle, and for a slit with high order tangency to a circular arc tangential to the real axis.

## Isoperimetry Coefficient for Simplex in the Problem of Approximation of Derivatives

We introduce the isoperimetry coefficient σ(G) = |∂G|n/(n−1)/|G| of region G ⊂ Rn. In terms of this the error δ_{Δ}(f) estimates for the gradient of the piecewise linear interpolation of functions of class C^{1}(G), C^{2}(G), C^{1,α}(G), 0 < α < 1, are obtained. The problem of obtaining such estimates is nontrivial, especially in the multidimensional case. Here it should be noted that in the two-dimensional case, for functions of class C^{2}(G), the convergence of the derivatives is provided by the classical Delaunay condition.

## Approximation of Control for Singularly Perturbed System with Delay with Geometric Constraints

The control problem for the singularly perturbed system with delay with indeterminate initial conditions and geometric constraints on the control resources according to the minimax criterion is considered. A limiting problem is formulated for which a specially selected quality functional is chosen. We propose the procedure for initial approximation construction of a control response in the control minimax problem.

## Almost Contact Metric Structures Defined by a Symplectic Structure Over a Distribution

The distribution D of an almost contact metric structure (ϕ, ξ, η, g) is an odd analogue of the tangent bundle. In the paper an intrinsic symplectic structure naturally associated with the initial almost contact metric structure is constructed. The interior connection defines the parallel transport of admissible vectors (i.e. vectors belonging to the distribution D) along admissible curves. Each corresponding extended connection is a connection in the vector bundle (D, π,X) defined by the interior connection and by an endomorphism N : D → D.

## Some Liouville-type Theorems for the Stationary Ginsburg – Landau Equation on Quasi-model Riemannian Manifolds

In this paper we find the conditions for validity of Liouville-type theorems for bounded solutions of the stationary Ginsburg – Landau equation and quasilinear elliptic inequality −Δu > uq, q > 1, on quasi-model Riemannian manifolds.