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

Macrodeficiency of polymeric materials, including the
pervasive one, is a consequence of the development of
initial microdefects which appear in polymers both
due to external factors and during their processing.
This article solves the task of identification the
interrelation and interdependence of structural
microdefects of various polymeric materials,
consisting in the estimation of the structural
parameters under uniaxial deformation. It is
experimentally shown that the process of changing
microdefects in the material bulk begins to flow more
intensely during deformation of the material, having
anisotropic structures in the bulk. Change of the
parameters of microdefects is not observed compared
to the starting material at small quantities of the
deformation. In the area of deformation corresponding
to the transition of the material from isotropic to
anisotropic state parameters of microdefects do not
depend on the degree of deformation of the sample,
and abrupt changes of the parameters of microdefects
are observed after completing the formation of
anisotropic patterns. It is shown experimentally the
identity of the parameters of microdefects in the bulk
material (rubber-based natural caoutchouc) during
deformation with the quantities of local deformations
at the edge of the artificially created macrodefects.
Under uniaxial deformation more than 280-300% and
the growth of the quantities of local deformations
come out at the edge of the artificially created
macrodefects and normalized linear dimension k and
decrease the total number n of microdefects

In the article we consider illustrative images of rare
pumpkin cultures presented in stamps of different
countries. In this work, we analyze stamps which
represent rare pumpkin cultures. Stamps are reflection
of the economy; they portray agriculture and elements
of technology of agricultural production.
Cucurbitaceae family includes a large number of
species that are very different from each other in the
form of fruits, in purpose, in origin. The objective of
our study was to conduct a visual analysis of rare
pumpkin crops, which are used in the world; to reflect
the research progress on the introduction and the
achievement of breeding work. The article provides an
analysis of several cultures: momordika (Momordica
charantia L.), Kiva (Kiwano), horned melon, cucumber
Antilles, (Cucumis metuliferus L.); Akantositsios
Naudin (Acanthosicyos naudinianus L.); momordica
balsamina (Momordica Balsamina L.); koktsiniya
(Coccinia sessifolia L.); luffa cylindrical (Luffa
cylindrica L.); Lagenaria (Lagenaria siceraria
(Molina) Standl.); cucumber snake (Trichosanthes
anguina L.); antilles cucumber, anguria (Cucumis
anguria L.); african cucumber (Cucumis africanus
Lindl); wax gourd (Benincasa hispida (Thunb)
Cogn..); chayota or mexican cucumber (Sechium edule
Swartz). The analysis of the iconography of images of
rare pumpkin crops in stamps of different countries
allowed us to see the history of culture introduction of
wild pumpkin from the local and foreign flora

The article presents experimental data on the research
of gold nanoparticles (GNPs) with mean diameter 15
nm and 50 nm effect on seeds germination and growth
of 3-day-old etiolated coleoptiles and roots. Compared
with distilled water (control) initial colloidal solution
of 15 nm and 50 nm GNPs (57 Âµg/ml) had no
significant impact on seeds germination and growth of
coleoptiles and roots. However, in both cases a weak
tendency to stimulation of the coleoptile growth and
root growth inhibition was observed. Reduction of 15
nm GNPs concentration down to 10 Âµg/ml and 1
Âµg/ml had not effect on the growth of the seedlings,
but stimulated seed germination up to twofold.
Similar concentrations of 50 nm GNPs exerted the
stimulating effect on seed germination (twofold) and
the growth of root and coleoptiles. Seedlings grew
especially intensive in colloidal GNPs solution with
Au concentration of 10 Âµg/ml. Temperature rise of 2
degrees (from 25Â°C to 27Â°C) resulted in growth
increase of control 3-day-old seedlings and opposite
effect of 50 nm GNPs: coleoptiles and roots growth
fell behind control seedlings growth by 16-17%.
However, on the 4th day, the relative growth
slowdown of control seedlings occurred at 27Â°C and
growth stimulation effect under the influence of 50
nm GNPs appeared again. Over time, the stimulating
effect of 50 nm GNPs decreased: at the end of October it weakened, and in November â€“ beginning of
December, it was no observed for the roots, and there
was a decline in coleoptiles growth. However, in all
cases the effect of stimulation of seeds germination
persisted under the influence of 50 nm GNPs,
weakening by December. We hypothesize the
molecular mechanisms of biological action of GNPs

Investigation variability of morphological features of
Iris halophila an introduction in Central Yakutia.
Dependence of morphological variability of Iris
halophila on quantity of dropping-out rainfall in
initial habitats is revealed