In the article we investigate the multicriteria task
arising at the organization of distributed calculations
in a corporate network. As a mathematical tool to
solve the problem we use prefractal graphs, which
naturally reflect the structure of relationships in
global and corporate networks. The corporate network
with the distributed computing system at the solution
of a particular task has to be reliable, quickly and
qualitatively to make decisions. And every computer
in the network should be a part in the solution of the
problem, since it is fixed for a certain function. The
problem is reduced to cover the prefractal graphs with
disjoint simple paths along the edges and vertices.
On the set of all admissible coverings we constructed
a vector-target function with specific criteria. All
these criteria have a specific meaningful
interpretation, allowing organizing the calculation of
maximum reliability, with minimum time information
processing and loading balancing between the
network elements. In the article we constructed
polynomial algorithms for finding optimal solutions
according to specific criteria. For the criteria which
are not optimizing the allocated coverings, estimates
of the lower and upper bounds are given. For all the
algorithms we constructed and substantiated
estimation of computational complexity, confirming
the advantage of using algorithms on prefractal
graphs to classical algorithms on graphs
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
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
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 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 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
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
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
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