# approximation

## On a Total Resource Amounts at the System with Parallel Service and MMPP Arrivals

In this paper, we consider a resource system with an unlimited resources and servers number, with parallel customers servicing, arriving at the system according to the MMPP. Using a combination of multidimensional dynamic screening methods and asymptotic analysis, it is proved that the joint asymptotic probability distribution of the total resource amounts converges to a bi-dimensional Gaussian distribution under conditions of increasing intensity of MMPP. The parameters of the asymptotic probability distribution are found.

## Simultaneous Approximation of Polynomial Functions and Its Derivatives by Feedforward Artificial Neural Networks with One Hidden Layer N. S.

In this paper we propose the algorithm for finding weights of feedforward artificial neural networks with one hidden layer to approximate polynomial functions and its derivatives with a given error. We use the rational sigmoidal function as a transfer function.

## Calculation of Meteoroids Masses by Approximating the Trajectories

At processing meteor observations the outdated and insufficiently reliable methods are commonly used. In particular, the finding outward-atmospheric meteoroid masses comes from the luminosity without providing any estimates of accuracy for calculations. However, in recent years a variety of new dynamics methods has been developed that quite good describe a motion of meteoroids in atmosphere, as well as changing their parameters.

## On an Approach to Approximate Solving of the Problem for the Best Approximation for Compact Body by a Ball of Fixed Radius

In this paper, we consider the problem of the best approximation of a compact body by a fixed radius ball with respect to an arbitrary norm in the Hausdorff metric. This problem is reduced to a linear programming problem in the case, when compact body and ball of the norm are polytops.

## On the Choice of Approximation for Number of Elements in the Sequence of Values for Simple Polynomial of Argument pq with Constraints on p and q

In this paper the theorem is obtained on one choice of the approximation for the element number in finite sequence of special kind.

## On a Limit Value of a Remainder of the Lagrange Constant Corresponding to the Lagrange Trigonometrical Polynomial

The behavior of Lebesgue constant of a trigonometrical Lagrange polynomial interpolating the periodic function in an odd number of clusters is studied. The limit value of the remainder in the known asymptotic formula for this constant is found. A special representation of a remainder allowed us to establish its strict decreasing. On this basis, for a Lebesgue constant, a non-improvable uniform bilateral logarithmic function estimate is received.

## On a Approximate Solution of the Problem of Aspherical Convex Compact Set

We examine a finite-dimensional problem of minimizing the ratio radius of the ball given a compact convex set (in an arbitrary norm) to the radius of the inscribed sphere through the choice of a common center of these balls. The article offers an approach to building the numerical method of its solution. At each step of the iterative process it is required to solve the problem of convex programming, target function of which is the difference between the radius of a circumscribed sphere, and scalable, with some positive factor, the radius of the inscribed sphere.

## Polynomials, Orthogonal on Non-Uniform Grids

Asymptotic properties of polynomials pˆn(t), orthogonal with weight ∆tj on any finite set of N points from segment [−1, 1] are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as n tends to infinity together with N is closely related to asymptotic behaviour of the Lasiandra polynomials. Furthermore are investigated the approximating properties of the sums by Fourier on these polynomials..