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
We consider an approach to the transition from
continuous to discrete scale which was defined by
means of step of quantization (i.e. interval of
grouping). Applied purpose is selecting the number
of gradations in sociological questionnaires. In
accordance with the methodology of the general
stability theory, we offer to choose a step so that the
errors, generated by the quantization, were of the
same order as the errors inherent in the answers of
respondents. At a finite length of interval of the
measured value change of the scale this step of
quantization uniquely determines the number of
gradations. It turns out that for many issues gated it
is enough to point 3 - 6 answers gradations (hints).
On the basis of the probabilistic model we have
proved three theorems of quantization. They are
allowed to develop recommendations on the choice
of the number of gradations in sociological
questionnaires. The idea of "quantization" has
applications not only in sociology. We have noted,
that it can be used not only to select the number of
gradations. So, there are two very interesting
applications of the idea of "quantization" in
inventory management theory - in the two-level
model and in the classical Wilson model taking into
account deviations from it (shows that
"quantization" can use as a way to improve
stability). For the two-level inventory management
model we proved three theorems. We have
abandoned the assumption of Poisson demand,
which is rarely carried out in practice, and we give
generally fairly simple formulas for finding the
optimal values of the control parameters,
simultaneously correcting the mistakes of
predecessors. Once again we see the interpenetration
of statistical methods that have arisen to analyze
data from a variety of subject areas, in this case,
from sociology and logistics. We have another proof
that the statistical methods - single scientificpractical
area that is inappropriate to share by areas
of applications
The article begins with the letter of the chief
engineer of chemical plant near Moscow. He
requests to analyze of data by means of modern
statistical methods and give an opinion on the
presence (or absence) of the relationship between
the two methods of determining the viscosity of the
mastic. For each of the batches of mastic It was
presented two numbers - the viscosity measurement
results of the two methods. These numbers form two
paired samples. We want to install, give whether
two specific methods similar results. The true values
of viscosity in different batches are not equal. Their
difference is not allows us to combine the results of
the first measurement method in first sample, the
results of the second method - in the second sample,
as we can do in the case of testing the homogeneity
of two independent samples. For solutions to this
problem we discuss four statistical criterions, based
on a study of the differences between corresponding
values in two paired samples. We test the hypothesis
of equality 0 of median of these differences (sign
test) and of equality 0 of the mathematical
expectation of these differences. Hypothesis of
testing of equality of the distribution functions of
two paired samples is reduced to the hypothesis of
symmetry of the distribution function of these
differences with respect to 0. In the alternative of the
shift is proposed to use the Wilcoxon signed rank
criterion. In the total alternative is proposed to use
criterion of the omega-square type which is
developed by the author of this article
Classic quantitative measure of the reliability of the models: F-measure by van Rijsbergen is based on counting the total number of correctly and incorrectly classified and not classified objects in the training sample. In multiclass classification systems, the facility can simultaneously apply to multiple classes. Accordingly, when the synthesis of the model description is used for formation of generalized images of many of the classes it belongs to. When using the model for classification, it is determined by the degree of similarity or divergence of the object with all classes, and a true-positive decision may be the membership of the object to several classes. The result of this classification may be that the object is not just rightly or wrongly relates or does not relate to different classes, both in the classical F-measure, but rightly or wrongly relates or does not relate to them in varying degrees. However, the classic F-measure does not count the fact that the object may in fact simultaneously belongs to multiple classes (multicrossover) and the fact that the classification result can be obtained with a different degree of similarity-differences of object classes (blurring). In the numerical example, the author states that with true-positive and true-negative decisions, the module similarities-differences of the object classes are much higher than for false-positive and false-negative decisions. It would therefore be rational to the extent that the reliability of the model to take into account not just the fact of true or false positive or negative decisions, but also to take into account the degree of confidence of the classifier in these decisions. In classifying big data we have revealed a large number of false-positive decisions with a low level of similarity, which, however, in total, contribute to reducing the reliability of the model. To overcome this problem, we propose a L2-measure, in which instead of the sum of levels of similarity we use the average similarity by different classifications. Thus, this work offers measures of the reliability of the models, called L1-measure and the L2 measure, mitigating and overcoming the shortcomings of the F-measures; these measures are described mathematically and their application is demonstrated on a simple numerical example. In the intellectual system called "Eidos", which is a software toolkit for the automated system-cognitive analysis (ASC-analysis), we have implemented all these measures of the reliability of the models: F, L1 and L2
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
Particle dynamics in metrics with logarithmic potential
The work considers the problem of modeling the
motion of particles in a unified field theory to 6D, in
theory, supergravity in the 112D and metric galaxies.
We have investigated a centrally symmetric metric in
the 112-dimensional Riemannian space, which
depends on the radial coordinate, time, and 110 angles.
We present a system of equations describing the
angular movement on a hypersphere of any dimension
N. It is shown that the motion on the hypersphere
depends on the 2 (N-1) of singular points. We have
installed general nature of relativistic motion on a
hypersphere when it is displayed on the plane and in
three-dimensional space. It is shown that the motion
determined by the reflection from the singular points
that of motion on the plane in some cases leads to
thickening of the trajectories in the neighborhood of
sides of the rectangle. The 6D investigated 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 centers of origin. It is found that in this
system there is a motion with condensation paths
around the sides of the rectangle, due to scattering of
test particles gravity sources. We set the general nature
of angular motion on a hypersphere and radial
movements in 6D in the metric of a logarithmic
potential. It is proved that similar solutions with
logarithmic potential exist in galaxies metric in the
metric of Einstein's theory of gravity. The article also
describes the connection of the solutions to the
nonlinear electrodynamics, and with a theory of quark
interactions and Yang-Mills theory
In this article, we investigate the problem of creation of
matter in the collision of particles, presented by
singularities of the gravitational field. A system of nonlinear
parabolic equations describing the evolution of the
axially symmetric metrics in the Ricci flow derived. A
model describing the creation of matter in the collision
and merger of the particles in the Ricci flow proposed. It
is shown that the theory that describes the Ricci flow in
the collision of black holes is consistent with EinsteinInfeld
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 which contains two
singularities simulating particles of finite mass. It is
shown that the static metric with two singularities
corresponding to in Newton's theory of gravity two
particles moving around the center of mass in circular
orbits in a non-inertial frame of reference, rotating with a
period of two-body system rotation. We have
numerically investigated the change of the metric in the
collision of particles with subsequent expansion. In
numerical experiments, we have determined that the
collision of the particles in the Ricci flow leads to the
formation of two types of matter with positive and
negative energy density, respectively. When moving
singularities towards each other in the area between the
particles the matter is formed with negative energy
density, and in the region behind the particles - with
positive density. In the recession of the singularities, the
matter with positive energy density is formed in the area
between the particles. The question of the nature of
baryonic matter in the expanding universe is discussed
In this article, we investigate the restricted problem of
many bodies with a logarithmic potential in the general
theory of relativity. We consider the metric having
axial symmetry and containing a logarithmic
singularity. In numerical calculations, we studied the
properties of the gravitational potential in the problem
of establishing a static condition in which multiple
singularities retain the initial position on the axis of the
system. This is achieved due to relativistic effects,
which have no analogues in Newton's theory of
gravitation. The motion of relativistic particles in a
logarithmic potential sources distributed on the surface
of a torus simulated. It is shown that the trajectory of
the particles in these systems form a torus covered with
needles. It was found, that the Ricci flow in the general
theory of relativity could be born three kinds of matter -
positive and negative energy density, as well as the
color of matter, the gravitational potential of which is
complex. It has been shown that this type of material is
associated with the manifestation of the quantummechanical
properties, which is consistent with the
hypothesis of the origin of Schrodinger quantum
mechanics. It is assumed that the most likely candidate
for the role of the color of matter is the system of
quarks as to describe the dynamics of quarks using the
logarithmic potential, and the quarks themselves are not
observed in the free state
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
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