Many procedures of applied mathematical statistics
are based on the solution of extreme problems. As
examples it is enough to name methods of least
squares, maximum likelihood, minimal contrast,
main components. In accordance with the new
paradigm of applied mathematical statistics, the
central part of this scientific and practical discipline
is the statistics of non-numerical data (it is also
called the statistics of objects of non-numerical
nature or non-numeric statistics) in which the
empirical and theoretical averages are determined by
solving extreme problems. As shown in this paper,
the laws of large numbers are valid, according to
which empirical averages approach the theoretical
ones with increasing sample size. Of great
importance are limit theorems describing the
asymptotic behavior of solutions of extremal
statistical problems. For example, in the method of
least squares, selective estimates of the parameters
of the dependence approach the theoretical values,
the maximum likelihood estimates tend to the
estimated parameters, etc. It is quite natural to seek
to study the asymptotic behavior of solutions of
extremal statistical problems in the general case.
The corresponding results can be used in various
special cases. This is the theoretical and practical
use of the limiting results obtained under the
weakest assumptions. The present article is devoted
to a series of limit theorems concerning the
asymptotics of solutions of extremal statistical
problems in the most general formulations. Along
with the results of probability theory, the apparatus
of general topology is used. The main differences
between the results of this article and numerous
studies on related topics are: we consider spaces of a
general nature; the behavior of solutions is studied
for extremal statistical problems of general form; it
is possible to weaken ordinary requirements of
bicompactness type by introducing conditions of the
type of asymptotic uniform divisibility

We consider numerical solutions of the Navier-Stokes
equations describing laminar and turbulent flows in
channels of various geometries and in the cavity at
large Reynolds numbers. An original numerical
algorithm for integrating a system of nonlinear partial
differential equations is developed, based on the
convergence of the sequence of solutions of the
Dirichlet problem. Based on this algorithm, a
numerical model is created for the fusion of two
laminar flows in a T-shaped channel. A new
mechanism of meandering is established, which
consists in the fact that when the two streams merge,
a jet is formed containing the zones of return flow.
Vortex motion in a rectangular cavity is studied. It is
established that the numerical solution of the problem
with discontinuous boundary conditions loses
stability at Reynolds number Re> 2340. The
trajectories of passive impurity particles in a
cylindrical cavity are investigated. An explanation of
the behavior of tea leaves in a cup of tea in the
formation of a toroidal vortex because of circular
stirring is confirmed, which is confirms the wellknown
hypothesis of Einstein. A numerical model of
flow in an open channel with a bottom incline in a
rotating system is developed. It is shown that in both
laminar and turbulent flow under certain conditions a
secondary vortex flow arises in the channel due to the
Coriolis force, which explains the well-known Baer
law and confirms the Einstein hypothesis

The soil fertility increase issues are very relevant now. Intensive development of agriculture cannot be made effectively without complex actions for farmlands protection from different types of degradations. On the one hand, it is necessary to ensure the maximum harvest of crops, and to preserve and increase the fertility of the soil and prevent negative anthropogenic impact on the environment on the other. For an extended reproduction of soil fertility, a system of measures is necessary for introduction of mineral and organic fertilizers into the soil, agrotechnical and reclamation methods, stimulation of humus formation processes, and so on. Therefore, methods are important that allow us to estimate the planned measures in advance to improve soil fertility and to eliminate environmental damage. In the article, the estimated parameters are treated by random variables. This allows us to consider the uncertainty in terms of probability distributions. It is offered a probabilistic model of the process of reducing the price of the proposed activity. Mathematical expectation, variance, distribution density of the considered random variable probabilities as the main characteristics of the object state price are calculated. The model can be used to address issues of rational use of land, scientifically based land management organization, when drafting land reclamation project

The concept of generic polynomial appeared in
Saltmanâ€™s works at the end of the last century and it is
connected with the inverse problem of Galois theory,
which is still far from its complete solution. Let G be a
finite group and K be a field, the polynomial
f(x,t1, â€¦ , tn) with coefficients from the field K is
generic for the group G, if Galois group of this
polynomial over the field K(t1, â€¦ , tn) is isomorphic G
and if for any Galois extension L/K with Galois group
isomorphic G there are such values of parameters
ti
= ai
, i = 1,2, â€¦ , n, that the field L is the splitting
field of the polynomial f(x,a1, â€¦ , an) over K. Generic
polynomials over a given field K and a given finite
group G do not always exist, and if they exist then itâ€™s
not easy to construct them. For example, for a cyclic
group of the eight order C8 there is no generic
polynomial over the field of rational numbers Q,
although there are found specific polynomials with
rational coefficients having Galois group isomorphic
C8. Therefore, this is of interest to construct generic
polynomials for the group G in cases when G is a
direct product of groups of lower orders. In this study
we show to solve this problem in case when G is a
direct product of certain cyclic groups and there is a
type of corresponding generic polynomials. Moreover,
we give constructions over the fields of characteristic 0
and over the fields of characteristic 2

A model is developed that describes the formation of the
plasma channel and the trace when moving in a
conducting medium of various objects that are sources of
plasma - ball lightning, plasmoids, charged particles, and
so on. To describe the contribution of conduction
currents, we modified the standard electrostatic equation
considering the vortex component of the electric field.
As a result of this generalization, a system of parabolictype
nonlinear equations is formulated that describes the
formation of the plasma channel and the track behind the
moving object. In this formulation, the problem of the
formation of the lightning channel in weak electric
fields, characteristic for atmospheric discharges of cloudearth,
is solved. Numerical simulation of the motion of
plasma sources in a region with a ratio of the sizes 1/100,
1/200 makes it possible to find the shape of the channel
and the total length of the track, as well as the branching
regimes. It was previously established that there are three
streamer branching mechanisms. The first mechanism is
associated with the instability of the front, which leads to
the separation of the head of the streamer into two parts.
The second mechanism is related to the instability of the
streamer in the base region, which leads to the branching
of the streamer with the formation of a large number of
lateral streamers closing the main channel of the
streamer to the cathode. The third branching mechanism,
observed in experiments, is associated with the closure
of the space charge to the anode through the streamer
system. These branching mechanisms are also revealed
when the leader is spread. Numerical experiments have
revealed a new channel branching mechanism and a
trace behind a moving plasma object, caused by the
conductivity of the medium

In some works, the feasibility of the use of fixed and
variable electromagnetic fields of different frequencies
and tension in the production technology of sunflower
oil are shown, but there is no theoretical justification.
The possibility of electromagnetic effects is associated
with the presence of polar molecules specific to organic
systems. Without prejudice to the role of polar groups
of terrestrial circuits, this work tries to address this
challenge more comprehensively. The reason for this is
the distinctive feature of the behavior of sunflower
during its flowering. This characteristic is that the
sunflower hat during the day changes its direction in
accordance with the direction of movement of the Sun
across the sky; so called "magnetism" of their
attraction. To justify this effect, we have analyzed the
essence of emitted photons, the Sun chemical
composition and structure arrangement of seeds in a
sunflower hat. Particles of light from the Sun represent
a stream of photons - a wide range of electromagnetic
waves of frequencies that exhibit and magnetic
properties. The article shows principal macro- and
micronutrients of sunflower raw materials and divides
them into groups of para- , dia- , and ferromagnetic
materials. In sunflower seeds, there are chemical
elements: diamagnetism-C, H, N, P, S, B, Cu, Zn, J;
paramagnetism-O, K, Ca, Mg, Mo, As and
ferromagnetic-iron (Fe). As there is resultant force of
the magnetic attraction between the sunflower hat and
magnetic flow of photons from the Sun, this effect
dominates the action of paramagnetics K2O ( -28.4
24.5%), CaO (7.6-17.0)%, MgO (12.3-17.9%),
magnetized in an external magnetic field in the
direction of the field. The presence of evident effect
demonstrates that it is possible to improve a number of
technological operations in the manufacture of
sunflower oil using electrical, magnetic or
electromagnetic fields

In this work, a model is developed that describes the
formation of a stepped lightning leader in a conducting
medium. To describe the contribution of the conductivity
currents, we modified the standard electrostatic equation
taking into account the vortex component of the electric
field. As a result of this generalization, a system of
parabolic-type nonlinear equations is formulated that
describes the formation of streamers and the lightning
channel. Numerical simulation of the propagation of
ionization waves in a region with a ratio of 1/100, 1/200
allows us to identify two types of stepped streamers in
the form of waves of compression and rarefaction,
respectively. It was previously established that there are
three streamer branching mechanisms. The first
mechanism is related to the instability of the front, which
leads to the separation of the head of the streamer into
two parts. The second mechanism is associated with the
instability of the streamer in the base region, which leads
to the branching of the streamer with the formation of a
large number of lateral streamers closing the main
channel of the streamer to the cathode. In numerical
experiments, the third branching mechanism observed in
experiments connected with closing the space charge to
the anode through the streamer system was observed.
These branching mechanisms are also revealed when the
leader is propagated. The obtained results, as well as the
data of numerical experiments confirm the hypothesis of
the universality of the minimal model of the streamer, as
well as its expansion in the form proposed by the author.
Known phenomena of nature associated with the
electrical discharge - streamer, plasmoid, ball lightning
and stepped leader can be described within the
framework of the minimal model

Nonparametric estimates of the probability
distribution density in spaces of arbitrary nature are
one of the main tools of non-numerical statistics.
Their particular cases are considered - kernel density
estimates in spaces of arbitrary nature, histogram
estimations and Fix-Hodges-type estimates. The
purpose of this article is the completion of a series
of papers devoted to the mathematical study of the
asymptotic properties of various types of
nonparametric estimates of the probability
distribution density in spaces of general nature.
Thus, a mathematical foundation is applied to the
application of such estimates in non-numerical
statistics. We begin by considering the mean square
error of the kernel density estimate and, in order to
maximize the order of its decrease, the choice of the
kernel function and the sequence of the blur
indicators. The basic concepts are the circular
distribution function and the circular density. The
order of convergence in the general case is the same
as in estimating the density of a numerical random
variable, but the main conditions are imposed not on
the density of a random variable, but on the circular
density. Next, we consider other types of
nonparametric density estimates - histogram
estimates and Fix-Hodges-type estimates. Then we
study nonparametric regression estimates and their
application to solve discriminant analysis problems
in a general nature space

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

In this work, a model is developed to describe the
formation of streamers, plasmoid, and ball lightning in a
conducting medium. To describe the contribution of the
conductivity currents, we modified the standard
electrostatic equation taking into account the vortex
component of the electric field. As a result of this
generalization, a system of parabolic-type nonlinear
equations is formulated that describes the formation of
streamers, plasma long-lived formations and ball
lightning. As is known, in laboratories it is possible to
create a plasmoid with a lifetime of 300-500 ms and a
diameter of 10-20 cm, which is interpreted as a ball
lightning. With high-speed photography, a complex
structure is detected, consisting of a plasmoid and
surrounding streamers. Within the framework of the
proposed model, problems are posed about the formation
of a plasmoid and the propagation of streamers in an
external electric field. In this model, the plasmoid is
considered to be a long-lived streamer. The range of
parameters in which a plasmoid of spherical shape is
formed is indicated. It is established that there are three
streamer branching mechanisms. The first mechanism is
related to the instability of the front, which leads to the
separation of the head of the streamer into two parts. The
second mechanism is associated with the instability of
the streamer in the base region, which leads to the
branching of the streamer with the formation of a large
number of lateral streamers closing the main channel of
the streamer to the cathode. In numerical experiments,
the third branching mechanism observed in experiments
connected with the branching of the plasmoid in the
cathode region with the closure of the space charge to
the anode through the streamer system was observed.
The results of modeling the evolution of globular
clusters in a scale of hundreds of milliseconds are given.
Plasma exchange recharge modes leading to the
formation of a positive or negative charge of the system
are found