# Computer Sciences

## On stability theory of autonomous angular stabilization system for combined dynamical systems

Studied the effect on the stability of the longitudinal acceleration discretely-continuum model of single-channel angular stabilization system with of delayed argument. Methods of construction asymptotic stability areas and analysis of impulse transition functions are developed. The critical values of the longitudinal acceleration are defined.

## Characterization of graphs with a small number of additional arcs in a minimal 1-vertex extension

A graph G∗ is a k-vertex extension of a graph G if every graph obtained from G∗ by removing any k vertices contains G. k-vertex extension of a graph G with n+k vertices is called minimal if among all k-vertex extensions of G withn+k vertices it has the minimal possible number of arcs. We study directed graphs, whose minimal vertex 1-extensions have a specific number of additional arcs. A solution is given when the number of additional arcs equals one or two.

## Analysis of closed unreliable queueing networks with batch movements of customers

Closed unreliable queueing network with batch movements is considered. The main result of the paper is the steady state distribution for given type queueing networks.

## On upper bound of vertex distinguishing word length on vertex labeled graph

The problem of vertex distinguishing on vertex labeled graphs is considered. Two vertices are called distinguishable if associated languages over the alphabet of labels are different. A linear upper bound of vertex distinguishing word length equal to half the number of vertices is obtained.

## T-irreducible extension for union of paths and cycles

A graphH with nodes is an extension of a graph G with nnodes if each maximal subgraph of H contains G. Trivial extension of a graph G is the connection of graph G and the singleton graph (i.e. we add one node to the graph G and this node join with each node of G). T-irreducible extension of graph G is an extension of the graph G which is obtained by removing maximal set of edges from the trivial extension of G. One of T-irreducible extensions is constructed for an arbitrary union of cycles and paths.

## Ordered automata and tolerant images of FDA

Finite deterministic automaton (FDA) with partially ordered (an ordered automaton) sets of states, input and output symbols is described in the article. The mapping of FDA on an ordered automaton, which is named "p-morphism" is defined. It is shown that so called tolerant images, which are constructed with the help of compatible tolerances on the set of states of FDA, are particular case of ordered automata, which are connected with the original automaton by a p-morphism.

## Using parallel computing technologies for modeling of metallic photonic crystals

This article presents opportunities of using parallel computing technologies Message Passing Interface and Open Computing Language for modeling of metallic photonic crystals with the method of Green's functions and integral equations. The efficiency of these technologies is analized and the results are presented.

## Numerical Modelling and the Analysis of Impact of Distortions on OFDM/QAM-signal

In this work mathematical models of communication channels with various interferences, their influence on constellation diagrams’

points in systems with OFDM/QAM signals are considered, recommendations about channel monitoring are made.

## Minimal Edge Extensions of Palm Trees

Minimal edge extension of graphs can be regarded as a model of optimal edge fault tolerant implementation of a system. The problem

of finding the minimal edge extensions of an arbitrary graph is NP-complete, that’s why it is of interest to find classes of graphs for

which it is possible to build a minimal edge extension analytically. This paper is about of the one-edge extensions of a graphs from

a special class named palm trees. In this paper presents a kind of one-edge extension for some palm trees and the proof that it is

minimal.

## On the Error of Approximation by Means of Scenario Trees with Depth 1

Let¤n denote the set of scenario trees with depth 1 and n scenarios. LetX = (0 · x1 < . . . < xn · 1) and let¤n(X) denote

the set of all scenario trees of depth 1 with the scenarios X = (0 · x1 < . . . < xn · 1). Let G be a probability distribution

defined on [0, 1] and H be a subset of measurable functions defined on [0, 1]. Let dH,X(G) = inf ˜G∈¤n(X) dH(G, ˜ G) and

dH(G) = inf ˜G∈¤n

dH(G, ˜ G), where dH(G, ˜ G) := suph∈H

¯¯¯

R h dG − R h d˜G

¯¯¯

. The main goal of the paper is to estimate