We obtain a macroscopic criterion of fragile fracture (limit curve) when creating an isolated defect in the form of “narrow” undercut, when conformal mapping of the exterior of a unit circle on the plane with de-effect in the form of a recess defined by cut fiber-foam series. It is shown that in this case, the limit curve has the form identical to the case when the defect is set to "narrow" ellipse. The same crack oriented along either the compressive stress or tensile perpendicular stress. From here, we can suggest that the shape and geometric properties of a sufficiently "narrow" defect do not affect the values of the critical loads required to start its distribution
The article presents a project of the Yang-Mills
amplifier. Amplifier model is a multilayer spherical
shell with increasing density towards the center. In
the center of the amplifier is the core of high-density
material. It is shown that in such a system, the
amplitude of the Yang-Mills waves rises from the
periphery to the center of several orders of
magnitude. The role of the Yang-Mills field in the
processes occurring in the nuclei of galaxies, stars
and planets is discussed. The data modeling to
strengthen the Yang-Mills field in the bowels of the
planet, with an atomic explosion, and in some special
devices such as the voltaic pile. To describe the
mechanism of amplification chromodynamics field
used as accurate results in Yang-Mills theory and
numerical models developed based on an average and
the exact equations as well. Among the exact
solutions of the special role played by the centralsymmetric
metric describing the contribution of the
Yang-Mills field in the speed of recession of galaxies.
Among the approximate numerical models can be
noted the eight-scalar model we have developed for
the simulation of non-linear color oscillations and
chaos in the Yang-Mills theory. Earlier models were
investigated spatio-temporal oscillations of the YangMills
theory in the case of three and eight colors. The
results of numerical simulation show that the
nonlinear interaction does not lead to a spatial mixing
of colors as it might be in the case of turbulent
diffusion. Depending on the system parameters there
is a suppression of the amplitude of the oscillations
the first three by five colors or vice versa. The kinetic
energy fluctuations or shared equally between the
color components, or dominated by the kinetic energy
of repressed groups of colors. In the present study, we
found that amplification chromodynamic field leads
to a sharp increase in the amplitude of the suppressed
color, which can lead to an increase in entropy,
excitation of nuclear reactions and decays particles
The article presents a model for choosing a variety of
alternative solutions, in which we have a subset of
turns or more alternative options, based on the use of
the Bayesian approach, based on the formulated
concept of security functions as a priori estimate of the
effects of the decision. This reduces the projected
parameters and, therefore, increases the values of
security. Thus, the considered indicators of data
protection reflect the essence of Bayesian approach to
decision making and management of GIS, so it allows
to generate optimal decision rules
The article deals with mathematical models of
management decision-making to select the option to
protect the AU, based on sufficient statistical
information about attacks on the AU. The amount of a
priori uncertainty about the choice of protection option
in GIS was described with Boltzmann's entropy.
Introduction of the value within Shannon’s definition
of mutual information is called the context random
variables, it allows removing the uncertainty regarding
the actions of the enemy, and it enables decisionmakers
to choose protection options. The model of
decision for choosing the type of protection of the AIS
presented in the article is based on sufficient statistical
information about the attacks to the system
components. In the ideal case, for decision-making,
we use large sample statistical data that provides high
accuracy control system for protection of information.
Based on the available amount of information
available to the IPA, against the acts of SIN, it is
possible to choose a decision on the choices you make
The article is devoted to the nonparametric point and
interval estimation of the characteristics of the
probabilistic distribution (the expectation, median,
variance, standard deviation, variation coefficient) of
the sample results. Sample values are regarded as the
implementation of independent and identically
distributed random variables with an arbitrary
distribution function having the desired number of
moments. Nonparametric analysis procedures are
compared with the parametric procedures, based on
the assumption that the sample values have a normal
distribution. Point estimators are constructed in the
obvious way - using sample analogs of the
theoretical characteristics. Interval estimators are
based on asymptotic normality of sample moments
and functions from them. Nonparametric asymptotic
confidence intervals are obtained through the use of
special output technology of the asymptotic relations
of Applied Statistics. In the first step this technology
uses the multidimensional central limit theorem,
applied to the sums of vectors whose coordinates are
the degrees of initial random variables. The second
step is the conversion limit multivariate normal
vector to obtain the interest of researcher vector. At
the same considerations we have used linearization
and discarded infinitesimal quantities. The third step
- a rigorous justification of the results on the
asymptotic standard for mathematical and statistical
reasoning level. It is usually necessary to use the
necessary and sufficient conditions for the
inheritance of convergence. This article contains 10
numerical examples. Initial data - information about
an operating time of 50 cutting tools to the limit
state. Using the methods developed on the
assumption of normal distribution, it can lead to
noticeably distorted conclusions in a situation where
the normality hypothesis failed. Practical
recommendations are: for the analysis of real data we
should use nonparametric confidence limits
In various applications it is necessary to analyze
some expert orderings, ie clustered rankings of
examination objects. These areas include technical
studies, ecology, management, economics, sociology,
forecasting, etc. The objects may make samples of
the products, technologies, mathematical models,
projects, job applicants and others. We obtain
clustered rankings which can be both with the help
of experts and objective way, for example, by
comparing the mathematical models with
experimental data using a particular quality criterion.
The method described in this article was developed
in connection with the problems of chemical safety
and environmental security of the biosphere. We
propose a new method for constructing a clustered
ranking which can be average (in the sense,
discussed in this work) for all clustered rankings
under our consideration. Then the contradictions
between the individual initial rankings are contained
within clusters average (coordinated) ranking. As a
result, ordered clusters reflects the general opinion
of the experts, more precisely, the total that is
contained simultaneously in all the original
rankings. Newly built clustered ranking is often
called the matching (coordinated) ranking with
respect to the original clustered rankings. The
clusters are enclosed objects about which some of
the initial rankings are contradictory. For these
objects is necessary to conduct the new studies.
These studies can be formal mathematics
(calculation of the Kemeny median, orderings by
means of the averages and medians of ranks, etc.) or
these studies require involvement of new information
from the relevant application area, it may be
necessary conduct additional scientific research. In
this article we introduce the necessary concepts and
we formulate the new algorithm of construct the
coordinated ranking for some cluster rankings in
general terms, and its properties are discussed
The article discusses the use of automatic systemic-cognitive analysis (ASC-analysis), its mathematical model is a system of information theory and software tools – an intellectual system called "Eidos" for the solution of some problems of ampelography: 1) digitization of scanned images of the leaves and creation of their mathematical models; 2) the formation of mathematical models of specific leaves using the spreading of information theory; 3) the formation of models of generalized images of leaves of various sorts; 4) comparing an image of a specific leaf with a generalized image of the leaf of different varieties and finding a quantitative degree of similarity and differences between them, i.e. the identification of the varieties on the leaf; 5) quantification of the similarities and differences of the varieties, i.e. cluster-constructive analysis of generalized images of the leaves of different varieties. We propose a new approach to digitizing images of leaves, based on using the polar coordinate system, the center of gravity of the image and its external contour. Before scanning images we may use transformation to standardize the position of the still images, their sizes and rotation angle. Therefore, the results of digitization and ASC-analysis of the images might be invariant (independent) relatively to their position, size and rotation. The specific shape of the contour of the leaf is regarded as noise information on the variety, including information about the true shape of the leaf of the class (clean signal) and noise, which distort this true form, originating in a random environment. Software tools of ASC-analysis – intellectual "Eidos" system ensures noise reduction and the selection of the signal about the true shape of the leaf of each variety on the basis of a number of noisy concrete examples of the leaves of this variety. This creates a one way form of a leaf of each class, free from their concrete implementations, i.e., the "Eidos" of these images (in the sense of Plato) is a prototype or archetype (in the Jungian sense) of the images
The article presents a project of the capacitor in the
Yang-Mills theory. Model capacitor represents the
equipotential surfaces separated by a space. To
describe the mechanism of condensation
chromodynamics field used numerical models
developed based on an average of the Yang-Mills
theory. In the present study, we used eight-scalar
component model that in the linear case is divided
into two groups containing three or five fields
respectively. In contrast to classical electrodynamics,
a static model of the Yang-Mills is not divided into
independent equations because of the nonlinearity of
the model itself. However, in the case of a linear
theory separation is possible. It is shown that in this
particular case, the Yang-Mills theory is reduced to
Poisson theory, which describes the electrostatic and
magnetostatic phenomena. In the present work it is
shown that in a certain region of the parameters of the
capacitor of the Yang-Mills theory on the functional
properties of the charge accumulation and retention of
the field is similar to the capacitor of the electrostatic
field or a magnet in magnetostatics. This means that
in nature there are two types of charges, which are
sources of macroscopic Yang-Mills field, which are
similar to the properties of electric and magnetic
charges in the Poisson theory. It is shown that in
Yang-Mills only one type of charge may be
associated with the distribution density of the
substance, while another type of charge depends on
the charge distribution of the first type. This allows us
to provide an explanation for the lack of symmetry
between electric and magnetic charges
The article presents the theorem of Chebyshev on the
distribution of primes, considering functions that
approximated prime numbers. We have also
considered a new function, which is quite good for
approximation of prime numbers. A review of the
known results on distribution of prime numbers is
given as well
In this article, we present the fundamental theorem of
arithmetic and its role. We consider various rings for
its performance