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
In this study, we investigate the problem of the emission
of gravitational waves produced in collisions of particles
submitted to the singularities of the gravitational field. A
system of non-linear parabolic equations describing the
evolution of the axially symmetric metrics in the Ricci
flow derived. A model describing the emission of
gravitational waves in the collision and merger of the
particles in the Ricci flow proposed. It is shown that the
theory of the Ricci flow describes the problem of black
holes merge, consistent with Einstein-Infeld theory,
which describes the dynamics of the material particles
provided by the singularities of the gravitational field. As
an example, we consider the metric having axial
symmetry and comprising two singularities simulating
particles of finite mass. We have numerically
investigated the change of the metric in the collision and
merger of the particles. The initial and boundary
conditions using the exact solution of the static problem,
so the collision persist particularly metrics caused by the
presence of particles. In numerical experiments
determined that the collision of the particles in the Ricci
flow leads to the formation of gravitational waves,
similar in structure to the waves, registered in the LIGO
experiment. Consequently, we can assume that the
observed gravity waves caused mainly by transients
associated with the change in the metric of a system. The
influence of the parameters of the problem - the speed
and mass of the particles, on the amplitude and intensity
of the emission of gravitational waves was numerically
simulated. We have found chaotic behavior of
gravitational potentials at the merger of the singularities
in the Ricci flow
As we know, currently, around the north pole of Saturn there is a large-scale hexagonal flow, with characteristic scales of length and speed - 120 m / s and 14,500 km respectively. This trend observed for more than 35 years, is the subject of many experimental and theoretical studies. In this study, we propose a model and discuss the numerical solutions of the equations describing turbulent flow in the planetary boundary layer around the north pole of Saturn. It has been shown that a small violation of the axial symmetry in geostrophic shear leads to the development of hexagonal patterns in a turbulent boundary layer. In addition, under the influence of Coriolis forces and turbulent eddy viscosity gradient in a turbulent boundary layer formed jet pressed to the bottom edge of the layer. These results are used to simulate the observed hexagonal flow around the north pole of Saturn. It is assumed that the small amplitude geostrophic flow is described by a sum of zero and the sixth current harmonic functions, which leads to the excitation current at the upper boundary of the planetary boundary layer. It is found that such excitation enhanced in the boundary layer and reaches a maximum in the jet pressed to the bottom border. This jet, circulating on the hexagon coincides with the region of origin of the cloud cover, which is registered in the experiments. This excitation mechanism hexagonal flow around the north pole of Saturn is confirmed by numerical calculations of three-dimensional non-stationary planetary boundary layer
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
One of the "points of growth" of applied statistics is
methods of reducing the dimension of statistical
data. They are increasingly used in the analysis of
data in specific applied research, such as sociology.
We investigate the most promising methods to
reduce the dimensionality. The principal
components are one of the most commonly used
methods to reduce the dimensionality. For visual
analysis of data are often used the projections of
original vectors on the plane of the first two
principal components. Usually the data structure is
clearly visible, highlighted compact clusters of
objects and separately allocated vectors. The
principal components are one method of factor
analysis. The new idea of factor analysis in
comparison with the method of principal
components is that, based on loads, the factors
breaks up into groups. In one group of factors, new
factor is combined with a similar impact on the
elements of the new basis. Then each group is
recommended to leave one representative.
Sometimes, instead of the choice of representative
by calculation, a new factor that is central to the
group in question. Reduced dimension occurs during
the transition to the system factors, which are
representatives of groups. Other factors are
discarded. On the use of distance (proximity
measures, indicators of differences) between
features and extensive class are based methods of
multidimensional scaling. The basic idea of this
class of methods is to present each object as point of
the geometric space (usually of dimension 1, 2, or 3)
whose coordinates are the values of the hidden
(latent) factors which combine to adequately
describe the object. As an example of the
application of probabilistic and statistical modeling
and the results of statistics of non-numeric data, we
justify the consistency of estimators of the dimension of the data in multidimensional scaling,
which are proposed previously by Kruskal from
heuristic considerations. We have considered a
number of consistent estimations of dimension of
models (in regression analysis and in theory of
classification). We also give some information about
the algorithms for reduce the dimensionality in the
automated system-cognitive analysis
In the article, we describe and illustrate a method of
mathematical modeling in relation to process of decision-making
in the conditions of risk and uncertainty
on the example of building of agricultural object
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 Samuelsson–
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
We have considered the formation of the Russian
scientific school in the field of econometrics,
obtained its obtained scientific results, the
possibilities of their use in solving problems of the
economy, the organization of production and
controlling of industrial companies and
organizations, as well as in teaching. As
econometrics we consider a scientific and an
academic discipline devoted to the development and
application of statistical methods to study economic
phenomena and processes, in short, statistical
methods in economics. Therefore, we can say that a
lot of domestic books and articles, in particular, the
works by the author of this publication from the
beginning of the 70s, are the parts of econometrics.
However, in this article we consider only the works,
in the titles of which we can see the word of
"econometrics". In our country the term
"econometrics" has become popular since the mid
90s. However, many publications and training
courses are still developed in the western outdated
paradigm. They do not conform to the new paradigm
of mathematical methods of economics, the new
paradigm of applied statistics and mathematical
statistics, mathematical methods of research. Russian
science school in the field of econometrics operates
within the scientific school in the field of probability
theory and mathematical statistics based by A.N.
Kolmogorov. Russian science school is developed in
accordance with the new paradigm of mathematical
methods. It is necessary to examine the main results
of Russian scientific schools in the field of
econometrics. We present the information on the
institutional design of national scientific schools in
econometrics, in particular, on the activities of the
Institute of High Technologies statistics and
econometrics
In practice, there were developed and tested some
mathematical models of balance relationships (balance
model), economic growth, expanding economy, labour
market, theories of consumption, production, competitive
equilibrium models of the economy in conditions of
imperfect competition and others. The basis of these
models were based on 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 quite rare. These tasks were sufficiently
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 considered in the study
inverse problems of the mathematical model, as it is
shown by the already introduced results of other
mathematical models, are of considerable interest in
applied and theoretical research. In this article, the
authors have formulated and investigated an inverse
problem for a model of economic growth. For its
solution the authors propose to build a system of
algebraic equations, using a reproduction model of
national income; then, using methods of quadratic
programming, to find the best average quadratic
estimates of the model parameter
The article is dedicated to a numerical investigation of
a plane problem of the oscillation amplitude of a
buried source, depending on the frequency and motion
speed in various isotropic media. Three types of the
medium are considered: a two-layer package with a
rigidly fixed base, a two-layer package with a
mechanically free base, a half-space. The source, in the
form of a stress jump simulating a rigid inclusion of
small dimensions, moves in the interface plane at a
constant speed. Homogeneous boundary value
problems are considered in a moving coordinate
system associated with a source. The solution method
is based on the usage of integral Fourier transforms,
the method of direct contour integration and
algorithms for constructing symbols of Green's
matrices. The method of direct contour integration
significantly simplifies calculations in comparison
with the traditional approaches to the calculation of
Fourier integrals. We have presented calculations of
nine amplitude-frequency and amplitude-velocity
characteristics for different combinations of medium
and source types, that give an exhaustive qualitative
and quantitative description of the solutions for
boundary value problems in a wide range of velocities
and frequencies. Comparative analysis of calculations
showed a primary influence of the type of an elastic
medium on the investigated characteristics, as well as
the large influence of the source type. Which, in turn,
revealed some substantial connections between the
boundary value problems with a moving source and
the corresponding problems with a stationary source