The article deals with the solution of the NavierStokes
equations describing turbulent flows over
rough surfaces. It is known, that there is a mechanism
of turbulent mixing in natural systems, leading to an
increase in the viscosity of the continuous medium. In
this regard, we suggest methods of regularization of
the Navier-Stokes equations, similar to the natural
mechanisms of mixing. It is shown, that in threedimensional
flows over a rough surface turbulent
viscosity increases proportionally to the square of the
distance from the wall. The models of the flow,
taking into account the properties of the turbulent
environment are considered. A modification of the
continuity equation taking into account the limiting
magnitude of pressure fluctuations is proposed. It is
shown, that due to the pressure pulsation, the
incompressibility condition may be violated even for
flows with low Mach numbers. Modification of the
continuity equation taking into account turbulent
fluctuations leads to a system of nonlinear equations
of parabolic type. Modification of continuity equation
in the system of Navier-Stokes by the introduction of
turbulent viscosity allows the regularization of the
Navier-Stokes equations to solve the problems with
rapidly changing dynamic parameters. The main
result of which is obtained by numerical simulation of
the modified system of equations is the stability of the
numerical algorithm at a large Reynolds number,
which can be explained, first, a system of parabolic
type, and a large quantity of turbulent viscosity. A
numerical model of flow around plates with the rapid
change in angle of attack has been verified. We have
discovered the type of instability of the turbulent
boundary layer associated with the rapid changes in
dynamic parameters. It is shown, that the fluctuations
of the boundary layer to cause generation of sound at
a frequency of 100 Hz to 1 kHz

It is known that not every finite group can be
realized over the field of rational numbers as a
Galois group of some binomial. In this connection,
a more general question arises: suppose that there
is given a finite transitive subgroup G of the
symmetric group S on n symbols; Can this group G
be realized as a Galois group of some trinomial of
degree n over the field of rational numbers? In this
paper we prove that every transitive subgroup of
the group S can be realized in the form of the
Galois group of a certain trinomial of the degree n,
for the values n = 2, 3, 4. For n = 5 , 6 we give
examples that realize concrete Galois groups. In the
case n = 7, all the transitive subgroups of the group
S are realized, except possibly one group of the
isomorphic dihedral group D. Further calculations
will be directed to the realization of specific Galois
groups for n = 8, 9 ..., however, the number of
transitive subgroups of the group S for n = 8, 9 ...
grows very fast, so the larger the value of n, the
more difficult it is to realize not just everything but
the specific subgroup of the group S in the form of
a trinomial over Q

On the basis of the objective analysis it must be
noted that in the arsenal of managers, especially
foreign ones, there is practically no fundamentally
new methods and tools of controlling. So says the
executive director of Russian Association of
Controllers prof. S. G. Falco. However, promising
mathematical and instrumental methods of
controlling actively developed in our country. It is
necessary to implement them. For example,
managers should be used techniques which
discussed in the book by Orlov AI, Lutsenko EV,
Loikaw VI "Advanced mathematical and
instrumental methods of controlling" (2015). These
methods are based on the modern development of
mathematics as a whole - on the system interval
fuzzy math (see the same named book by Orlov AI
and Lutsenko EV, 2014). Considered methods are
developed in accordance with the new paradigm of
mathematical methods of research. It includes new
paradigms of applied statistics, mathematical
statistics, mathematical methods of economics,
methods of analysis of statistical and expert data in
management and control. In the XXI century there
were more than 10 books issued, developed in
accordance with the new paradigm of mathematical
methods of research. The systems approach to
solving specific applications often requires going
beyond the economy. Very important are the
procedures for the introduction of innovative
methods and tools. In this article we consider the
above research results in their interconnection

In this work, we examine the dynamics of relativistic
particles in the ring or spiral galaxy metric in general
relativity. On the basis of the solution of Einstein's
equations we have derived metric having axial
symmetry, comprising N centers of gravity and a
logarithmic singularity. The application received metrics
to describe the motion of particles in a spiral and ring
galaxy. On the basis of Einstein's equations solutions for
vacuum we are explained rotation of matter in spiral
galaxies. An expression for gravitation potential in the
inner region of spiral galaxies in agreement with
experimental data on the rotation of the CO and
hydrogen is described. It is established that in the metric
with N centers of gravity which are distributed on the
circumference, exist as a local motion near the center of
gravity, and motion around N gravity center as well. The
transition from one mode of motion to another is
determined by the initial distance to the circle on which
the distributed centers of gravity. A system of non-linear
parabolic equations describing the evolution of the
metric in the Ricci flow proposed. The boundary
problem for the gravitational potentials in the Ricci flow
was formulated. There are applications of the theory to
describe a spiral and ring galaxy

According to measurement theory, statistical data
are measured in various scales. The most widely
used ordinal scale, scales of intervals and relations.
Statistical methods of data analysis should
correspond to the scales in which the data is
measured. The term "correspondence" is specified
with the help of the concepts of an adequate
function and an allowable scale transformation. The
main content of the article is a description of the
average values that can be used to analyze data
measured in the ordinal scale, interval and
relationship scales, and some others. The main
attention is paid to the means for Cauchy and the
means for Kolmogorov. In addition to the mean,
from this point of view, polynomials and correlation
indices are also analyzed. Detailed mathematical
proofs of characterization theorems are given for the
first time in scientific periodicals. It is shown that in
the ordinal scale there are exactly n average values,
that can be used, namely, n order statistics. The
proof is represented as a chain of 9 lemmas. In the
scale of intervals from all Kolmogorov means, only
the arithmetic mean can be used. In the scale of
relations from all the Kolmogorov means, only the
power means and the geometric mean are
permissible. The kind of adequate polynomials in
the relationship scale is indicated

In this work, we consider two types of vortex
currents-cyclones and anticyclones in the Northern
and Southern Hemispheres. Numerical modeling of
turbulent flows of these types uses the model of the
planetary boundary layer developed by the author.
The purpose of the study is to test hypotheses about
the influence of the Coriolis force on the formation of
cyclones and anticyclones in the northern and
southern latitudes. The first hypothesis on the
direction of circulation in cyclones was verified in the
case of axisymmetric radially converging and
vertically rising turbulent flows with a natural
Coriolis parameter and viscosity. From the obtained
data of numerical experiments, it follows that the
current in the northern latitudes circulates in a counter
clockwise direction, and in the south - in a clockwise
direction, in full accordance with the observational
data. Thus, we have shown that a cyclonic flow is
formed in a turbulent radially converging flow under
the influence of the Coriolis force. The second
hypothesis on the formation of anticyclones was
verified in the case of radially divergent and vertically
descending turbulent flows. Because of numerical
experiments, it was established that in this case, the
current in the northern latitudes circulates clockwise,
and in the south - in a counter clockwise direction,
which corresponds to observations for anticyclones.
To test the effect of the cyclone (anticyclone) center
velocity on circulation, a nonstationary 3D model of
turbulent flow was developed. Within the framework
of this model, flows in cyclones and anticyclones
moving at a constant speed, as well as in shear flow,
are studied. Some types of loop protuberances on the
Sun are explained by the presence of a vortex
turbulent flow starting in the bowels of the Sun and
encompassing the chromosphere

In the study we consider the problem of determining
the motion and similarity parameter to the system of
worlds in a Riemannian space 112D with a common
field of gravity. Centrally symmetric metric,
depending on the 110 angle coordinates and the radial
coordinate and time was investigated. It is assumed
that there are intelligent beings in every world, striving
for self-knowledge. By virtue of the presence of the
world hierarchy in one of them there is a system of
complete identification of each characteristic of the
individual being with macroparameters his world. If
sentient beings in all the world to create a device to
simulate their own history in the form of a network of
computers using the available material and the
physical laws of his world, and the loss of information
when displaying one world to another is 1%, then 37-
th world played only 68.9449%. For Earthlings, it was
found that the average similarity parameter of
professional group in recognition by using
astronomical parameters is 68.75%. Therefore, we can
assume that the world system, including Earth,
contains 37 "floors." Assuming that each "floor" takes
three space dimensions, and all the "floors" connected
by a single time, we find here that the number of
dimensions of space-time of the whole system is 112.
In the article the angular motion in a Riemannian space
is considered. The effect of the separate worlds on
other worlds is simulated. It has been shown that the
physical laws in all worlds represent a single
movement covering the markers in the form of the
motion of atoms and elementary particles in a
gravitational field in the 112D

The problem of establishing of the factorization of
irreducible polynomials with integer coefficients on
prime modules p has been long of interest to
mathematicians. The quadratic and cubic reciprocity
laws solve this problem for quadratic polynomials and
binomials of the form x3-a . More general reciprocity
laws solve the formulated problem for some classes of
polynomials, for example, with Abelian Galois group,
but for polynomials with non-Abelian Galois group,
the problem is far from its complete solution. Our
study shows how using the results of Voronov G.F.,
Hasse H. and Stickelberger L., one can find conditions
that must satisfy prime number p. Gauss received a
similar result for binomial x3-2. Specific examples are
given, for instance, for the polynomial x3-x - I, also
conditions arc formulated for which a quadratic field is
immersed in non-Abelian Galois extension of degree
6. Also, conditions are given under which a
Diophantine equation: Ð°12a22-4a22-4a13a3-
27a32+18a1a2a3=D has a solution for integer values
of D

The work discusses various examples of physical
systems which state is determined by the logarithmic
law - quantum and classical statistical systems and
relativistic motion in multidimensional spaces. It was
established that the Fermi-Dirac statistics and BoseEinstein-Maxwell-Boltzmann
distribution could be
described by a single equation, which follows from
Einstein's equations for systems with central
symmetry. We have built the rate of emergence of
classical and quantum systems. The interrelation
between statistical and dynamic parameters in
supergravity theory in spaces of arbitrary dimension
was established. It is shown that the description of the
motion of a large number of particles can be reduced
to the problem of motion on a hypersphere. Radial
motion in this model is reduced to the known
distributions of quantum and classical statistics. The
model of angular movement is reduced to a system of
nonlinear equations describing the interaction of a test
particle with sources logarithmic type. The HamiltonJacobi
equation was integrated under the most general
assumptions in the case of centrally-symmetric metric.
The dependence of actions on the system parameters
and metrics was found out. It is shown that in the case
of fermions the action reaches extremum in fourdimensional
space. In the case of bosons there is a
local extremum of action in spaces of any dimension

In this article we discuss a version of the metric theory
of the fundamental interactions in which it is assumed
that the physical constants due to the presence of extra
dimensions of space-time. The estimation of the
number of physical constants based on the theory of
supergravity in 112D is that the minimum number of
constants is equal to 222, and the maximum number -
1404928. At present, the number of parameters that
characterize the elementary particles, isotopes and
chemical elements is about 150920. This number is 9.3
less than the maximum possible number of parameters
that indicate still great potential of modern science.
Functions describing the area and volume of a unit
hypersphere, embedded in a Riemannian space of
arbitrary dimension, were used to find the fundamental
physical constants. A satisfactory agreement with a
relative error of 0.03% calculated and experimental
values of the fine structure constant found out. For the
ratio of the average mass of a nucleon to the electron
mass is obtained coincidence with the experimental
value with an accuracy of 0.002%. The proposed
theory of physical constants different from that Bartini
theory that established the optimal dimension of the
space is a hypersphere 5 and 7, rather than 6 as in
Bartini theory. The problems of the compactification
of extra dimensions in describing the motion in fourdimensional
space-time are discussed