The article discusses various examples of dynamical
systems in which the motion is determined by the
logarithmic law - quark systems, hydrodynamic
systems, galaxies. Set the general nature of angular
motion on a hypersphere in a space of arbitrary
dimension and radial movement 6D in the metric of a
logarithmic potential. We investigate the 6D metric
describing the case of motion with two centers of
symmetry. It is shown that in such a metric exists a
class of exact solutions, logarithmically dependent on
the gravity center coordinates. It was established that
in spiral galaxies the orbital motion is due to the
logarithmic potential, which is the exact solution of the
field equations of Einstein's theory of gravity. The
most well-known and widespread in nature case is
turbulent flow over a smooth or rough surface, in
which the mean velocity depends logarithmically on
the distance from the wall. We derivate the logarithmic
velocity profile in turbulent flow from the NavierStokes
equations. An analogy of the logarithmic
velocity profile and the logarithmic law in the case of
erosion of materials under impacts been proposed. In
electrodynamics, Ampere's law, which describes the
interaction of current-carrying conductors, is a
consequence of the logarithmic dependence of the
vector potential of the distance from the conductor
axis. There is, however, an alternative derivation of
Ampere law of the Riemann hypothesis about the
currents due to the motion of charges
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 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 the article we present a spatial structure of largescale
transport systems. The model of a transport
network can be presented in the form of a graph, with
a set of the nodes corresponding to elements of a
network and a set of edges – to sections of roads the
connecting these nodes. As the model of a card of
roads, it is offered to use prefractal graphs which
naturally reflect structure of communications when
reviewing a transport network in different scales (the
states, regions, areas). Prefractal graphs allow
describing structural dynamics of the studied system
in the discrete time. One of the most widespread
scenarios of structural dynamics is the growth of
structure. The statement of tasks of the organization
of transport routes contains requirements criteria to
finding of optimal solutions. Often these requirements
and criteria are contradicting each other. It leads to
appearance of a multicriteria problem definition.
The multicriteria problem definition on a class of
prefractal graphs is considered. The optimum
algorithm of separation of the greatest maximum
paths by the given criterion is constructed and
estimates by remaining criteria are given. In operation
computing complexity of the constructed algorithm of
separation of the greatest maximum paths on a
prefractal graph is calculated and advantage of
operation of algorithm on last before algorithm of
separation of the greatest maximum paths on normal
graphs is justified. The constructed algorithm on
prefractal graphs has polynomial complexity
Applied Statistics - the science of how to analyze
the statistical data. As an independent scientificpractical
area it develops very quickly. It includes
numerous widely and deeply developed scientific
directions. Those who use the applied statistics and
other statistical methods, usually focused on specific
areas of study, ie, are not specialists in applied
statistics. Therefore, it is useful to make a critical
analysis of the current state of applied statistics and
discuss trends in the development of statistical
methods. Most of the practical importance of
applied statistics justifies the usefulness of the work
on the development of its methodology, in which the
field of scientific and applied activities would be
considered as a whole. We have given some brief
information about the history of applied statistics.
Based on Scientometrics of Applied Statistics we
state that each expert has only a small part of
accumulated knowledge in this area. We discuss five
topical areas in which modern applied statistics
develops, ie five "points of growth": nonparametric,
robustness, bootstrap, statistics of interval data, and
statistics of non-numerical data. We discuss some
details of the basic ideas of a non-numerical
statistics. In the last more than 60 years in Russia,
there has been a huge gap between official statistics
and the scientific community of experts on statistical
methods
Fuzzy sets are the special form of objects of nonnumeric
nature. Therefore, in the processing of the
sample, the elements of which are fuzzy sets, a
variety of methods for the analysis of statistical data
of any nature can be used - the calculation of the
average, non-parametric density estimators,
construction of diagnostic rules, etc. We have told
about the development of our work on the theory of
fuzziness (1975 - 2015). In the first of our work on
fuzzy sets (1975), the theory of random sets is
regarded as a generalization of the theory of fuzzy
sets. In non-fiction series "Mathematics.
Cybernetics" (publishing house "Knowledge") in
1980 the first book by a Soviet author fuzzy sets is
published - our brochure "Optimization problems
and fuzzy variables". This book is essentially a
"squeeze" our research of 70-ies, ie, the research on
the theory of stability and in particular on the
statistics of objects of non-numeric nature, with a
bias in the methodology. The book includes the
main results of the fuzzy theory and its note to the
random set theory, as well as new results (first
publication!) of statistics of fuzzy sets. On the basis
of further experience, you can expect that the theory
of fuzzy sets will be more actively applied in
organizational and economic modeling of industry
management processes. We discuss the concept of
the average value of a fuzzy set. We have
considered a number of statements of problems of
testing statistical hypotheses on fuzzy sets. We have
also proposed and justified some algorithms for
restore relationships between fuzzy variables; we
have given the representation of various variants of
fuzzy cluster analysis of data and variables and
described some methods of collection and
description of fuzzy data
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
This work presents a new approach to the countries’
credit rating definition, based on the advanced mathematical
models, such as neural network model, multiple
regression, cluster analysis and discriminant analysis.
A range of the analyses such as discriminant, cluster,
multiple regression models and a neural network
were performed on the following economic figures:
GDP per capita, GDP value, annual growth rate of
GDP, FDI - foreign investment, rate of unemployment,
consumer price inflation index, the size of government
debt in percentage of GDP. The results, obtained for
each model were combined in the countries’ credit
rating estimation system called "7M"
The federal program on essential drugs provision
(EDP) is one of the most significant and socially
important state projects; it is directed to the reduction
of morbidity and mortality together with the
improvement of life quality of the society and its
social climate. In accordance with the federal law “On
social state assistance” from 17.07.1999 №178- FL,
the essence of the program is that medical recipes are
dispensed for preferential medicines to be received by
federal program participants. The medical-economic
control (MEC) of the drugs designation and provision
of federal benefit recipients is performed basing on the
automated registries examination of released drugs.
The number of passed and failed examination recipes
is determined according to the registers processing
results. A certain percentage of the accepted for
payment prescriptions is a subject for MEC. For the
purpose of the recipes selection for testing, the paper
proposes the mathematical models of criteria
application and MEC-planning. The game model of
organization and MEC performance in health care
organizations is build basing on the theory of games.
The considered play model suggests that the health
services quality examination need to be adjusted and
some strategies are to be improved. The solution on
the planning of checked recipes number allows to
perform the inspection of all the health care
organizations, involved in EDP program
The article deals with the numerical solution of the
Navier-Stokes equations describing turbulent flow in
a rectangle cavity or in a cuboid with one open face at
high Reynolds numbers. 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. We proposed the models based on the
properties of the turbulent environment. For this we
modified the continuity equation taking into account
the pressure fluctuations. It is shown that the
incompressibility condition is can be violated due to
pressure fluctuation even for flows with low Mach
numbers. Modification of continuity equation by the
introduction of turbulent viscosity allows the
regularization of the Navier-Stokes equations to solve
the problems with rapidly changing dynamic
parameters. It was shown that the modification of the
continuity equation taking into account turbulent
fluctuations leads to a system of nonlinear equations
of parabolic type. A numerical model of turbulent
flow in the cavity with the rapid change in the
parameters of the main flow developed. Discovered
type of instability of the turbulent flow associated
with the rapid changes in the main flow velocity. In
numerical simulations found that due to the
acceleration of the main flow there is the unsteady
vortex flow in the cavity, which is characterized by
the integral of energy not vanishing with time,
vibrations that have a certain period, depending on
the turbulent viscosity