The inverse matrix for the square matrix A of order n
with coefficients of some field exists, as it is known
then and only then, when its determinant is not equal to
zero. If the matrix A has a certain type (certain
structure), then an inverse matrix A-1 should not have
exactly the same structure. Therefore, it is interesting
to describe such square matrices A, which have an
inverse matrix A-1, having the same structure as the
matrix A, under certain conditions. For example, a
subdiagonal matrix with nonzero elements on the main
diagonal has an inverse matrix over a field of
characteristic zero, having also the form of subdiagonal
matrix. Similarly, an inverse matrix towards
symmetrical or skew-symmetric matrix is also
symmetric or skew-symmetric accordingly. Also, the
matrix inverse to non-degenerate (nonsingular)
circulant will be a circulant itself, and finally, the
matrix inverse to nonsingular quasdiagonal matrix D
will be quasdiagonal itself, and will have the same
partitioned structure as D. Thus, there is a problem of
determining these types of nonsingular matrices that
have an inverse matrix of the same type as a given
matrix. In line with this problem in the present study it
is determined such type of matrices for which an
inverse matrix has the same type, at that the conditions
are identified in explicit form, ensuring the
nonsingularity of the matrix. The matrices of three
orders are shown in detail. These results allow
determining the characteristics of fields over which
there are inverse matrices of the considered types

Partially ordered set is a basic concept of modern settheoretic
mathematics. The problem of linear set
ordering with given binary relations is well-known.
Every partial order over a finite set can be linearly
ordered, but not every binary relation over this set can
be linearly ordered as well. Up to now, there is no
known formula for calculating the number of partial
orders over a given finite set. It appears that there is a
formula for calculating linearly ordered binary
relations over a finite set. This article is concerned
with derivation of this formula. The fact from work of
G.N. Titov [9] that a binary relation over a finite set is
linearly ordered if and only if any diagonal block,
derived from the binary relation matrix as a result of
setting main diagonal elements to zero, contains at
least one zero row (diagonal block of matrix means
any matrix composed of elements at the crossings of
rows and columns of a given matrix with the same
numbers), plays a key role in process of corroboration.
The main conclusion of the article is a theorem that
allows to find the number of linearly ordered binary
relations over a set of n elements using the formula. A
recurrence formula for the number of linearly ordered
(irreflexive) binary relations over a finite set of n
elements, provided in the lemma, was derived as well

In this article, we have proposed a forecast of labor
resources by sectors of the labor market of the Russian
Federation until 2018 inclusive. The probabilistic
parameters of the dynamics of labor resources were
calculated using the considered model (articles [1-4]).
Further, these probabilities were used to estimate and
parameterize the trends of employed and unemployed
(with a known last place of employment) specialists in
each industry. For each sector of the Russian economy,
the type of trend that best approximates the long-term
(more than three years) dynamics of labor resources in
this sector was selected. It is shown that the forecast
error for 1 year is less than 1% using the selected trend
models. Further, identified long-term trends were used
in forecasting. Based on the results of the forecast of
the number of people employed in the sectors of the
Russian economy, the sectors were divided into two
groups. The first group contains sectors with a
significant change in the number of employed in. The
second group contains sectors, changes in employed in
which were insignificant. At the end of the article, an
example of an analysis of two sectors from the first
group is given. Causes of changes in the dynamics of
labor resources in the researched sectors of the
economy were identified. Ones of the main sectors of
the economy were considered in this example:
"Manufacturing industries" and "Financial activities,
real estate transactions, leasing and provision of
services"

In this work, a model is developed that describes the
formation of a plasmoid and streamers 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, the streamer
model is formulated in the form of a system of parabolictype
nonlinear equations. 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 similarity of ball lightning and plasmoid is
discussed. If this similarity is confirmed, then the
number of theoretical hypotheses concerning the nature
of ball lightning, currently more than 200, can be
drastically reduced to one described in this article

Statistical methods are widely used in domestic
feasibility studies. However, for most managers,
economists and engineers, they are exotic. This is
due to the fact that modern statistical methods are
not taught in the universities. We discuss the
situation, focusing on the statistical methods for
economic and feasibility studies, ie, econometrics.
In the world of science, econometrics has a rightful
place. There are scientific journals in econometrics,
Nobel Prizes in Economics are given to series of
researches in econometrics. The situation in the field
of scientific and practical work and especially the
teaching of econometrics in Russia is disadvantaged.
Often, individual particular constructions replace
econometrics in general, such as those related to
regression analysis. The article is devoted to
econometrics as an academic discipline. Our course
begins with a discussion of the structure of modern
econometrics, the connections between applied
statistics and econometric methods. We consider
sample researches (analysis of surveys results), the
elements of econometrics numbers, and methods of
testing of statistical hypothesis about homogeneity.
We have given the concepts of regression analysis,
econometric classification methods, modern
measurement theory. The important places are
occupied by the statistics of non-numerical data
(including fuzzy sets and their links with random
sets) and the statistics of interval data. The problem
of the stability of statistical procedures with respect
to the tolerances of input data and model
prerequisites is discussed. The representations of the
econometric methods of expert research and quality
control, analysis and forecasting of time series,
econometrics of forecasting and risks are given

In this work, we develop a model describing the
propagation and branching of a streamer in a conducting
medium in external electric field. 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, the streamer model is formulated in
the form of nonlinear equations of parabolic type. In the
framework of the proposed model, the problem of the
propagation of a streamer in the form of a traveling wave
is considered, which leads to the emergence of SaffmanTaylor
streamers. For streamers of this type, the
branching problem is formulated, which has a unique
solution. The dependence of the branch point on the
parameters of the problem-the speed of the streamer, the
diffusion coefficient of the electrons and the strength of
the external electric field, is found. The branching
mechanism of the streamer head by dividing it into two
parts has been well studied and several alternative
models have been formulated for its description. The
novelty of the problem in question is that the streamer
splits into two three-dimensional channels that are
symmetric with respect to the given plane. Numerical
experiments also revealed the mechanism of branching
of the streamer in the cathode region, connected with the
separation of the main channel into several lateral
branches. It is noted, that in nature both branching
mechanisms are realized, whereas in theory the
instability of the surface of the streamer head is
investigated

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