The Euler function is very important in number theory
and in Mathematics, however, the range of its values in
the natural numbers has not been written off. The
greatest value of the Euler function reaches on Prime
numbers, furthermore, it is multiplicative. The value of
the Euler function is closely associated with the values
of the Moebius function and the function values of the
sum of the divisors of the given natural number. The
Euler function is linked with systems of public key
encryption. The individual values of the Euler function
behave irregularly because of the irregular distribution
of primes in the natural numbers. This tract is
illustrated in the article with charts; summatory
function for the Euler function and its average value
are more predictable. We prove the formula of
Martinga and, based on it, we study the approximation
accuracy of the average value of the Euler function
with corresponding quadratic polynomial. There is a
new feature associated with the average value of the
Euler function and calculate intervals of its values. We
also introduce the concept of density values of the
Euler function and calculate its value on the interval of
the natural numbers. It can be noted that the results of
the behavior of the Euler function are followed by the
results in the behavior of functions of sums of divisors
of natural numbers and vice versa. We have also given
the results of A.Z.Valfish and A.N.Saltykov on this
subject

There is a discussion about the question of the
mechanism of the action of the magnetic field of the
Earth and the Sun on the human body. It is noted that in
the 21st century an international conference on the
subject "Man and electromagnetic fields" is regularly
held, as well as the international congress "Weak and
superweak fields and radiation in biology and
medicine". This indicates the importance of studying
the effect of electromagnetic fields on the human body.
Participants in these conferences and congresses give a
lot of experimental data on the influence of various
factors on various biological objects. However, there is
no theoretical justification for the influence of these
fields on the human body. In this connection, in order to
solve this problem, the article analyzes the atomic
composition of the human body. It is shown that the
human body more than 60% consists of hydrogen
atoms. On the example of a hydrogen atom, the
interaction of the magnetic moment of an electron of an
atom with an external magnetic field is considered. This
leads to a precession motion of the electron's orbit.
Taking into account the fact that photons rotate around
electrons in atoms and the temperature is determined by
the bulk density of photon energy, the appearance of
precessional electron motion will lead to an increase in
the frequency of oscillation of photons and,
consequently, to an increase in their energy and body
temperature. This is confirmed by the fact that the body
temperature changes during the day, and, it is minimal
in the morning and increases by the evening. The
chemical elements of the human body, related to
different groups of magnets, are analyzed. It is noted
that the external magnetic field has the greatest
influence on the state of the human body through a
ferromagnet - iron. It is concentrated in the blood, up to
60% in hemoglobin. It is a complex iron-containing
blood protein, an integral part of erythrocyte - red blood
cells, oxygen carriers. Under conditions of an increase
in the intensity of the external magnetic field or the
immobile state of the body, the orientation of the
individual erythrocytes will increase due to their iron
atoms in the direction of the external field. This will
lead to the pooling of erythrocytes into clusters, that is,
to the formation of unique magnetic domains with a significant increase in the viscosity of the blood and a
decrease in its circulatory ability. The last is confirmed
by the fact that in people suffering from cardiovascular
diseases, heart attacks and strokes most often occur in
the early morning especially during periods of solar
magnetic storms

The article continues the cycle of their studies
associated with the formulation and development of
methods of construction of nonnegative solutions of
inverse problems for dynamic systems. In practice, we
have developed and tested mathematical models of
dynamic systems. The basis of these models was based
on the apparatus of linear algebra, mathematical
analysis, mathematical programming, differential
equations, optimization methods, optimal control
theory, probability theory, stochastic processes,
operations research, game theory, statistical analysis.
The inverse problem in various models of
mathematical Economics was considered rare. These
tasks were sufficiently well investigated in the study of
physical processes. As shown by the analysis of the
theoretical and applied studies of economic processes
they represent considerable interest for practice.
Therefore, the article considered the inverse problem
of the mathematical model, as shown already
introduced the results of other mathematical models,
are of considerable interest in applied and theoretical
research. In this article the authors formulated and
investigated the inverse problem for dynamical
systems zero-order and the model of Keynes. For their
solution, the authors propose to build a system of
algebraic equations, then, using methods of quadratic
programming, to find the best average of mean square
estimation of the model parameter, which are defined
in MS Excel

It was shown before [1,2], that variants of intensity of
γ-quantas of axion origin, induced by the variants of
the magnetic field in the the tacho wedge through the
termomagnetic Ettinshausen-Nernst effect, cause
variations of solar luminance and ultimately
characterise the changes of active and calm state of the
Sun. It is shown in the article in which way the areas
of sunspots are generated by the action of global
dynamo in the convective zone, or in other words,
which fundamental processes connect the sunspots and
solar cycles with the large-scaled magnetic field of the
Sun

Chemical processes are often connected with use or
formation of condensed dispersed phase (CDP).
Dispersed particles can change mobility of charges, as
well as other parameters of the low-temperature plasma.
The aim of this work is to study the effect of magnetic
field on the processes of dispersed particles formation in
argon-oxygen plasma containing iron and carbon atoms
at atmospheric pressure. The equilibrium composition of
iron and carbon atoms containing mixture simulated at
temperatures of 1000-5000K for optimization of the
plasma-forming gas composition. It is shown that in case
of oxygen excess, the CDP particles contain only iron
oxides. The literature data about the phase transition
processes in a low-temperature plasma, as well as the
data about the processes with participation of
ferromagnetic particles in a constant magnetic field
analyzed. The results of investigations of the dispersed
particles forming in argon-oxygen plasma of arc
discharge in the presence and in the absence of the
magnetic field are shown. The formed disperse phase
was deposited on the substrates and studied by the
electron microscopy and X-ray methods. It was found
that with the lack of oxygen the size of the iron-oxide
particles created in the arc discharge containing iron and
carbon is affected by magnetic field: in a magnetic field
of 10 mT the particles are larger than in its absence

In the present article, we investigate the metric of the
crystal space in the general theory of relativity and in the
Yang-Mills theory. It is shown that the presence of a
lattice of gravitational ether has observable macroscopic
consequences. Earlier, the influence of the gravity of the
celestial bodies of the solar system on the electrical
conductivity, inductance, the rate of radioactive decay of
atomic nuclei, on seismic activity, the magnetic field and
the motion of the pole of our planet, and on the rate of
biochemical reactions was established. In all cases, a
similar behavior of the physicochemical characteristics
of materials and processes is observed, depending on the
universal parameters characterizing the seasonal
variations of the gravitational field of the solar system.
The relationship between lattice parameters and the
properties of materials, elements, atomic nuclei, and
elementary particles is discussed. Possible metrics of the
crystal space are constructed: a metric that depends on
the Weierstrass function, derived in the Yang-Mills
theory and analogous metrics found in Einstein's theory.
Such metrics, which have a central symmetry, can be
used to justify the structure of elementary particles, the
properties of atomic nuclei, atoms and matter. Periodic
metrics are constructed that admit an electromagnetic
field, as well as metrics associated with the assumed
structure of the crystal space. These metrics are of
particular interest, since the properties of the substance
are related to the metric parameters. We proposed the
model of electron beam as a streamer of preons

Classic quantitative measure of the reliability of the models: F-measure by van Rijsbergen is based on counting the total number of correctly and incorrectly classified and not classified objects in the training sample. In multiclass classification systems, the facility can simultaneously apply to multiple classes. Accordingly, when the synthesis of the model description is used for formation of generalized images of many of the classes it belongs to. When using the model for classification, it is determined by the degree of similarity or divergence of the object with all classes, and a true-positive decision may be the membership of the object to several classes. The result of this classification may be that the object is not just rightly or wrongly relates or does not relate to different classes, both in the classical F-measure, but rightly or wrongly relates or does not relate to them in varying degrees. However, the classic F-measure does not count the fact that the object may in fact simultaneously belongs to multiple classes (multicrossover) and the fact that the classification result can be obtained with a different degree of similarity-differences of object classes (blurring). In the numerical example, the author states that with true-positive and true-negative decisions, the module similarities-differences of the object classes are much higher than for false-positive and false-negative decisions. It would therefore be rational to the extent that the reliability of the model to take into account not just the fact of true or false positive or negative decisions, but also to take into account the degree of confidence of the classifier in these decisions. In classifying big data we have revealed a large number of false-positive decisions with a low level of similarity, which, however, in total, contribute to reducing the reliability of the model. To overcome this problem, we propose a L2-measure, in which instead of the sum of levels of similarity we use the average similarity by different classifications. Thus, this work offers measures of the reliability of the models, called L1-measure and the L2 measure, mitigating and overcoming the shortcomings of the F-measures; these measures are described mathematically and their application is demonstrated on a simple numerical example. In the intellectual system called "Eidos", which is a software toolkit for the automated system-cognitive analysis (ASC-analysis), we have implemented all these measures of the reliability of the models: F, L1 and L2

In 1893, the French mathematician J. Adamar
raised the question: given a matrix of fixed order
with coefficients not exceeding modulo this value,
then what is the maximum modulo value can take
the determinant of this matrix? Adamar fully
decided this question in the case when the
coefficients of the matrix are complex numbers and
put forward the corresponding hypothesis in the
case when the matrix coefficients are real numbers
modulo equal to one. Such matrices satisfying the
Hadamard conjecture were called Hadamard
matrices, their order is four and it is unknown
whether this condition is sufficient for their
existence. The article examines a natural
generalization of the Hadamard matrices over the
field of real numbers, they are there for any order.
This paper proposes an algorithm for the
construction of generalized Hadamard matrices,
and it is illustrated by numerical examples. Also
introduces the concept of constants for the natural
numbers are computed values of this constant for
some natural numbers and shown some
applications of Hadamard constants for estimates
on the top and bottom of the module of the
determinant of this order with arbitrary real
coefficients, and these estimates are in some cases
better than the known estimates of Hadamard. The
results of the article are associated with the results
of the con on the value of determinants of matrices
with real coefficients, not exceeding modulo units

In the article we investigate the multicriteria task
arising at the organization of distributed calculations
in a corporate network. As a mathematical tool to
solve the problem we use prefractal graphs, which
naturally reflect the structure of relationships in
global and corporate networks. The corporate network
with the distributed computing system at the solution
of a particular task has to be reliable, quickly and
qualitatively to make decisions. And every computer
in the network should be a part in the solution of the
problem, since it is fixed for a certain function. The
problem is reduced to cover the prefractal graphs with
disjoint simple paths along the edges and vertices.
On the set of all admissible coverings we constructed
a vector-target function with specific criteria. All
these criteria have a specific meaningful
interpretation, allowing organizing the calculation of
maximum reliability, with minimum time information
processing and loading balancing between the
network elements. In the article we constructed
polynomial algorithms for finding optimal solutions
according to specific criteria. For the criteria which
are not optimizing the allocated coverings, estimates
of the lower and upper bounds are given. For all the
algorithms we constructed and substantiated
estimation of computational complexity, confirming
the advantage of using algorithms on prefractal
graphs to classical algorithms on graphs

The article continues the cycle of their studies
associated with the formulation and development of
methods of construction of nonnegative solutions of
inverse problems for dynamic systems. In this article
the authors formulated and investigated inverse
problems for dynamic systems: model of Samuelssonâ€“
Hicks. The technique of constructing non-negative
solutions of the studied inverse problems. This method
is based on the following scheme of the solution. First,
we have to identify the formulation of the direct
problem, then the formulation of the inverse. This
work investigates how correct the mathematical
models describing the dynamic economic system are.
Further, in the specified tabular solutions of the direct
problem, we have built a system of algebraic equations
containing the unknown estimated parameters of the
studied model. Then posed inverse problem is reduced
to solution of a problem of quadratic programming, the
solutions of which are defined in MS Excel. The
theoretical material is accompanied by the specific
example