Abstract:
We introduce certain modified Meyer-K"onig and Zeller operators $ M_{n;r} $ in the space of $r $-th times differentiable functions $ f $ and we study strong differences $ H_{n;r}^q(f) $ for them. This note is motivated by results on strong approximation connected with Fourier series ([7]).

Abstract:
We study some approximation properties of the Picard and the Gauss-Weierstrass singular integrals of functions of two variables belonging to some exponential weighted spaces. In Sec. 3.1 and 3.2 we give two theorems on the degree of approximation and two theorems of the Voronovskaya type. In Sec. 3.3 the Bernstein inequalities for these singular integrals are proved. Some similar results for the Picard and the Gauss-Weierstrass singular integral of functions of one variable were given in [1] - [3].

Abstract:
We give the Voronovskaya theorem for some operators of the Szasz-Mirakjan type defined in the space of functions continuous on [0,+infinity) and having the polynomial grouth at infinity. Some approximation properties of these operators are given in [2], [4].

Abstract:
We introduce certain modified Szasz - Mirakyan operators in exponential weighted spaces of functions of one variable. We give theorems on the degree of approximation and the Voronovskaya type theorem.

Abstract:
We consider Szasz-Mirakyan operators in polynomial and exponential weighted spaces of functions of two variables. We give Voronowskaya type theorem and theorem on convergence of certain sequences.

Abstract:
The set S_{F}(x_{0};T) of states y reachable from a given state x_{0} at time T under a set-valued dynamic x’(t)∈F(x (t)) and under constraints x(t)∈K where K is a closed set, is also the capture-viability kernel of x_{0} at T in reverse time of the target {x_{0}} while remaining in K. In dimension up to three, Saint-Pierre’s viability algorithm is well-adapted; for higher dimensions, Bonneuil’s viability algorithm is better suited. It is used on a large-dimensional example.

The introduction of an exponential or power law gradient
in the interstellar medium (ISM) allows to produce an asymmetric
evolution of the supernova remnant (SNR) when the framework of the thin layer
approximation is adopted. Unfortunately both the exponential and power law
gradients for the ISM do not have a well defined physical meaning. The physics
conversely is well represented by an isothermal self-gravitating disk of
particles whose velocity is everywhere Maxwellian. We derived a law of motion
in the framework of the thin layer approximation with a control parameter of
the swept mass. The photon’s losses, which are often neglected in the thin layer
approximation, are modeled trough velocity dependence. The developed framework
is applied to SNR 1987A and the three observed rings are simulated.

Abstract:
A first new luminosity function of galaxies can be built starting from a left truncated beta probability density function, which is characterized by four parameters. In the astrophysical conversion, the number of parameters increases by one, due to the addition of the overall density of galaxies. A second new galaxy luminosity function is built starting from a left truncated beta probability for the mass of galaxies once a simple nonlinear relationship between mass and luminosity is assumed; in this case the number of parameters is six because the overall density of galaxies and a parameter that regulates mass and luminosity are added. The two new galaxy luminosity functions with finite boundaries were tested on the Sloan Digital Sky Survey (SDSS) in five different bands; the results produce a “better fit” than the Schechter luminosity function in two of the five bands considered. A modified Schechter luminosity function with four parameters has been also analyzed.

Abstract:
We argued that the standard field scalar potential couldn’t be widely used for getting the adequate galaxies’ curve lines and determining the profiles of dark matter their halo. For discovering the global properties of scalar fields that can describe the observable characteristics of dark matter on the cosmological space and time scales, we propose the simplest form of central symmetric potential celestial-mechanical type, i.e. U(φ) = –μ/φ. It was shown that this potential allows get rather satisfactorily dark matter profiles and rotational curves lines for dwarf galaxies. The good agreement with some previous results, based on the N-body simulation method, was pointed out. A new possibility of dwarf galaxies’ masses estimation was given, also.