A linear continuous nonzero operator $G {:}\allowbreak\ X \to Y$ is a Daugavet center if every rank-$1$ operator $T {:}\allowbreak\ X \to Y$ fulfills $\|G+T\|=\|G\|+\|T\|$. We introduce the notions of a $G-$strong Daugavet operator and a $G-$narrow operator which are the generalizations of the concepts of strong Daugavet and narrow operators for Daugavet centers. We also consider examples of $G-$narrow operators.

In given work meromorphic functions $f(z)$ in $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$ are concerned. For these functions the family $\{f(\lambda z)\}_{\lambda\in\mathbb{C}^*}$ is normal. Such functions (with additional restriction, namely, presence of a pole or removable singularity in zero) were studied by A.Ostrovsky. He received its representation in terms of zeros and poles. Later А.Eremenko assumed that Ostrovsky's result is true in the general case. In this work we give the detailed proof of this result.

In this paper we describe a set of all spectral functions of a vector polynomial sequence. The necessary and sufficient conditions for a given matrix-valued function to be a spectral function are established. The necessary and sufficient conditions for a given matrix-valued function to be the correlation function of a vector polynomial sequence are obtained. A criteria in terms of the correlation function for an arbitrary set of elements of a Hilbert space to be a vector polynomial sequence is obtained, as well.

Approximate solutions of the kinetic Boltzmann equation for hard spheres are built which describe the transitional regime between two accelerating and packing flows. Sufficient conditions for the minimization of the discrepancy between the sides of this equation in some special metric are obtained.

The electrostatic problem for a spherical segment which is shielded by sectional spheres is solved by a method of regularization. The main part of matrix operator of the problem is inverted using the solution of Abel integral equations. The Fredholm system of the second kind of algebraic equations has been obtained. Some generalizations and numerical results of the problem are considered.

Based on the method of harmonic analysis, developed by A.Nagel and W.Rudin for the theory of unitarily invariant function spaces, the expansion of plurisubharmonic functions on $\mathbb{C}^n\ (n\geq2)$ into series of homogeneous holomorphic and anti-holomorphic polynomials. Proposed approach naturally combines well known methods of harmonic analysis, developed by L.A. Rubel and B.A. Taylor for functions of one complex variable and by A.Nagel, W.~udin, W.Stoll, R.Kujala and P.Noverraz for functions of several complex variables.

Approximation of the infinitely differentiable functions by the partial sums of the based on the atomic function $up(x)$ generalized Taylor series is considered in the paper. The rate of approximation for some classes of the infinitely differentiable functions is estimated.

We extend some recent (2009, 2010) results devoted to the Lyapunov stabilization for the switched systems in the strict-feedback form. More specifically, we prove that a multi-input and multi-output triangular switched system with an unknown switching signal, with right-invertible input-output links, and with dynamics, which is affine in external disturbances is globally uniformly input-to-state stabilizable with respect to the disturbances.

Formulations and solutions of the one-dimensional nonlinear and two-dimensional linearized problems describing the pulse wave propagation in arteries are presented. For several models of arterial beds that are presented by the systems of tubes with different topology, the comparative study of the one-dimensional and two-dimensional models is carried out. The modeling of pathologies, related to the stenosis and microcirculatory problems is conducted. It is shown that the wave-intensity analysis method allows determination of localization of the pathology.