# Computer Sciences

## The Geometric Form of Automaton Mappings, Recurrent and Z-recurrent Deﬁnition of Sequences

For automaton mappings we present a method to construct geometric images, a method for complexity estimate by geometric forms, a method of Z-recurrent deﬁnition of sequences. A method for complexity estimate for ﬁnite sequences by recurrent and Z-recurrent numerical indicators is proposed. Numerical indicators of recurrent and Z-recurrent deﬁnitions of sequences are systematized into the spectrum of recurrent deﬁnitions with 5 levels of numerical indicators.

## The Sperner Property for Polygonal Graphs Considered as Partially Ordered Sets

A ﬁnite poset is said to have the Sperner property if at least one of its maximum antichains is formed from elements of the same height. A polygonal graph is a directed acyclic graph derived from a circuit by some orientation of its edges. The reachability relation of a polygonal graph is a partial order. A criterion is presented for posets associated with polygonal graphs to have the Sperner property.

## On Applications of Wavelets in Digital Signal Processing

Discrete Wavelet transform associated with the Walsh functions was deﬁned by Lang in 1998. The article describes an application of Lang’s transform and some its modiﬁcations in analysis of ﬁnancial time series and for the compression of fractal data. It is shown that for the processing of certain signals the studied discrete wavelet transform has advantages over the discrete transforms Haar, Daubechies and the method of zone coding.

## Analyticity Conditions of Characteristic and Disturbing Quasipolynomials of Hybrid Dynamical Systems

Hybrid dynamical systems (HDS) are connected by means of the boundary conditions and the constraint’s conditions systems of ordinary differential equations and partial differential equations with the corresponding initial conditions. Check the stability of HDS can be performed on the basis of the "fast"algorithm for the application which requires analytic characteristic and disturbing quasipolynomials of HDS in the right half-plane and near the imaginary axis.

## An Approach to Fuzzy Modeling of Digital Devices

In the article the problem of fuzzy binary logic modeling for digital devices (DD) is investigated. In contrast to the similar classic problem of logical simulation, it is assumed that inputs signals of DD are fuzzy signals. In the real of DD for each input (0 or 1) there is a certain voltage range. If an input signal is out of the range, the correct signal identification is not guaranteed. The fuzziness of input signals means that there observed values can be either within of the defined range, or out of it.

## Quantum Computers and Quantum Algorithms. Part 2. Quantum Algorithms

The paper discusses principles of construction for quantum algorithms and their main features. Distinction of quantum parallelism from classical methods of high-performance computing is shown. Quantum algorithms design strategy is presented based on quantum circuits. Methods of programming for implementation of quantum algorithms using high-level languages are proposed. An approach to implement unitary transformations based on the oracle method is described.

## Algebraic Properties of Abstract Neural Network

The modern level of neuroinformatics allows to use artificial neural networks for the solution of various applied problems. However many neural network methods put into practice have no strict formal mathematical substantiation, being heuristic algorithms. It imposes certain restrictions on development of neural network methods of the solution of problems. At the same time there is a wide class of mathematical models which are well studied within such disciplines as theory of abstract algebras, graph theory, automata theory.

## Quantum Computers and Quantum Algorithms. Part 1. Quantum Computers

The paper presents the principles of operation of quantum computers. Competitive advantages of quantum computing are shown and some variants of a construction of an ideal quantum computer proposed. We analyze also the computational process in a quantum computer from the point of view of the complexity of algorithms. Implementation of nodes of a quantum computer is exemplified based on quantum communication schemes. The operation of Bloch sphere and visualization of the state of the qubit are described. Major obstacles to the creation of quantum computers are considered.

## The Development of Software Components for Streaming Audio Content Filtering Through the Use of Hidden Markov Models

The results of the development of efficient algorithms for streaming voice recognition using stochastic models based on the use of hidden Markov models are shown in this work. The article provides basic theoretical information for the hidden Markov model of the discrete system and the necessary parameters to define it are distinguished. Also there are three main tasks considered that need to be solved for the successful application of hidden Markov models in speech recognition systems.

## T-irreducible Extensions for Starlike Trees

We deal with a sort of optimal extensions of graphs, so called T-irreducible extensions. T-irreducible extension of a graph G is an extension of G obtained by removing a maximal set of edges from the trivial extension of G. A difficult starlike tree is a starlike tree that has at least one difficult node. T-irreducible extensions for nondifficult starlike trees were constructed by M. B. Abrosimov, T-irreducible extensions for palms (one of subclasses of starlike trees) were constructed by S. G. Kurnosova.