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Partial boolean functions taking their values 0, 1 and -- with a probability equal to 1/3 are considered. The lower and upper bounds on the length of minimum DNF representation of such functions are obtained in the paper.

The paper considers analysis methods of texts in natural language. It proposes 1) the model cause-and-effect relation in Russian text; 2) application of derived relations as expert statements in the algorythm of extremal situations forecasting.

We know a lot of cases, when topological entropy of continious map determine by structure of periodic orbits. The classical result in this field is the Sharkovskiy ordering. Sharkovskiy $\left [ 6 \right ]$ enter mapping, that turn manifold of natural numbers in to ordered set as follows:
$$3 \prec 5 \prec 7 \prec 9 \prec 11 \cdots \prec 2 \cdot 3 \prec 2 \cdot 5 \prec \cdots \prec 2^{2} \cdot 3 \prec 2^{2}\cdot 5 \prec \cdots \prec 2^{3} \prec 2^{2} \prec 2 \prec 1$$

A three-stage scheme for synthesis of semimetric adjustment algorithms in data-mining problems is proposed. Within its framework algorithms designed to modify the distance between a single pair of objects at the given value are studied. These ajustment algorithms guarantee preservation of the simetric properties. The proposed adjustment procedures constructed within the three-stage scheme framework are completed within the first two stages, and in the specials case - within the first stage.

In work the spatial problem of a comuter tomography with use of tomograms which lay on system of mutually perpendicular planes with use blending approximations is solved. The offered method allows to smooth experimental data (tomograms) and has rather high accuracy. In article computing experiment for restoration of internal structure of a brain of the person with the help of operators blending approximations by Bernstein's polynoms is shown.

The materials represented in this article carry, mainly, surveying character. The aim of the paper is complete enough presentation of possibilities of mathematical apparatus of algorithmic complexity and probability for application in machine learning. Nevertheless, some new results are presented: theorems about exact compressors and decompressors, approach to determination of the moment of stopping of learning procedure on the basis of complexity analogue of the Bayes rule et al.

In this paper we propose a new approach to constructing the Delaunay Triangulationfor the case of multidimensional spaces $(d > 2)$. Analysing the modern state, it is possible to draw a conclusion, that the ideas for the existing effective algorithms developed for the case of $d ≤ 2$ are not simple to generalize on a multidimensional case, without the loss of efficiency. We offer for the solving this problem an effective algorithm that satisfies all the given requirements.

Canonical m-handle decomposition of three-dimensional handlebody was constructed for m-functions. It is obtained a criterion on homotopy equivalence of m-functions without internal critical points on handlebody in terms of the generators of the fundamental group of the surface.

The branch of analysis, connected with the study of functions of an infinite-dimensional argument, has been recently intensively developing. This is due both to internal causes and to applications to theoretical and mathematical physics. It is well known which role in classical mathematical physics is played by distributions — linear continuous functionals on smooth basic functions. Therefore, it is natural to develop such a theory in infinite-dimensional spaces.

Mathematical model of consolidation of soil has been improved taking into account their salinity and chemical erosion. Numerical solution of the corresponding three-dimensional boundary value problem has been found by the radial basis functions method.

The system under consideration consists of two unreliable lines. The first line is more effective then the second one. Both lines have Poisson failure flow. If the lines are in good working order, failure rate of the first line is $\alpha _{1}$, failure rate of the second line is $\beta _{1}$. If one of the lines is down, failure rate of the other equals $\alpha_{2}$ or $\beta_{2}$ espectively. The system is serviced by a single repairman.

Kolmogorov’s definition of algorithmic complexity of constructive objects [7] is applicable not only to character strings, but also to the functions, function collections, and many other objects. If we define Kolmogorov complexity of a family of recursive functions, it will be its entropy characteristics. Introduced by different ways measures of a complexity of families of functions are of great theoretical and practical interest.

This article discusses the pencil of ordinary differential operators generated on $[0,1]$ by a linear differential expression of $n$-th order with constant coefficients, polynomially depending on spectral parameter $\lambda$ , and the two-point boundary conditions of a general form with coefficients which are polynomials of the spectral parameter $\lambda$.