This article describes a mathematical model of transport
of salt ions in a cell with a rotating disk cation exchange
membrane at transcendent current regimes, taking into
account electroconvection. Based on this model, we had
a theoretically study of the process of transfer of salt
ions and the dependence of the thickness of the
diffusion layer from the fall of potential. This article is
a continuation of [8] and [9], it conducted a numerical
analysis of boundary value problem for a system of
equations Nernst-Planck-Poisson and Navier-Stokes
equations, modeling the transport of salt ions in a
cylindrical cell with a rotating disc cation exchange
membrane based on electroconvection. It is shown there
is an electroconvection vortex in the center of the
membrane disc. The solution flows around this vortex
and forms a stagnation zone in front of it. With the
increase in the size of the fall of potential, the
electroconvective vortex decreases and at some value,
the electroconvective vortex disappears. The study was
conducted in the 1000 s when the angular velocity of 30 turns in a minute and change of the potential difference
of 0.2V to 1.4V with a step 0.1. As a result, in this
study it is shown that the thickness of the diffusion
layer is practically linearly dependent on the fall of
potential. The linear dependence of the thickness of
diffusion layer from the fall of potential, in the first
approximation, is disturbed by a slight deflection curve,
the causes of which are needed to be found by means of
extra experiments
In this work, a model is developed that describes the
formation of a stepped lightning leader 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, a system of
parabolic-type nonlinear equations is formulated that
describes the formation of streamers and the lightning
channel. Numerical simulation of the propagation of
ionization waves in a region with a ratio of 1/100, 1/200
allows us to identify two types of stepped streamers in
the form of waves of compression and rarefaction,
respectively. It was previously 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 closing the space charge to
the anode through the streamer system was observed.
These branching mechanisms are also revealed when the
leader is propagated. The obtained results, as well as the
data of numerical experiments confirm the hypothesis of
the universality of the minimal model of the streamer, as
well as its expansion in the form proposed by the author.
Known phenomena of nature associated with the
electrical discharge - streamer, plasmoid, ball lightning
and stepped leader can be described within the
framework of the minimal model
This question is about the speed of wave
propagation in electromagnetic environment.
Electromagnetic environment (field) is the space
that fills the whole Universe, occupied by the
electromagnetic particles-photons. At the heart of
the special relativity theory, the constancy of the
speed of light in vacuum is affirmed. According to
modern concepts, the speed of light in vacuum is the
maximum speed of the particle motion and
propagation of interactions. However, light is the
narrow range of electromagnetic radiation –
(4÷8)·1014 Hz, therefore experimentally measured
speed of light is referred to this frequency range.
The fact that this speed of electromagnetic waves
can theoretically be non permanent – physicists have
pondered for a long time and this question is
periodically excited in the scientific literature. The
author of this article also had an impression that the
speed of light, in which he understands distribution
speed of waves of a wide range of frequencies in the
electromagnetic environment, is not a constant. The
article attempts to prove it. Many photons of
different frequencies move simultaneously in
different directions in a photonic electromagnetic
field in environment. They are involved in the
formation of a wave of compression –
decompression in this field under the influence of
the antenna radiated photons. It is approved that the
speed of photons of different frequencies can change
within a wide range from 1,285·103
m/s (ν = 1024
Hz) to 1,285·1012 m/s (ν = 106 Hz) and, therefore,
the speed of wave propagation in the
electromagnetic environments that are filled by
photons of the same frequency or a narrow
frequency range can change widely from 8,58·102
m/s to 8,58·1011 m/s and be significantly different
from the experimentally discovered speed of light.
Interplanetary space in different parts of the
Universe can be represented by different spectra of
photons and therefore they will have different speed
of propagation of electromagnetic waves
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 the intellectual system called "Eidos", which is a software toolkit for the automated system-cognitive analysis (ASC-analysis), we use initially proposed by its developers measure of the reliability of the models, which is essentially a fuzzy multiclass generalization of the classical F-measure (it is proposed to call it the L-measure). In this article, L-measure is mathematically described and its application is demonstrated on a simple numerical example
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
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
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
The article concentrates on the matters of current interest in the sphere of product flows. The object of research is the relocation of product flows from the supply sphere, represented by supply and sales organizations or other commercial-intermediary agencies, to the sphere of business enterprise. The ultimate goal of the production and economic system modeling is the preparation for managerial decision-making. The choice of the model depends on the purposes of the modeling, management functions, automation manufacturing step, applied mathematical tools technique. The article considers the main characteristics of the flow, which while retaining their individuality at the same time depend on each other and function logically in the economic space. The advantages and disadvantages of the material inventory and flows management in micrologistic intraproductive systems are being analyzed. External and internal environment, taken as a basis for the real logistical process modeling, determine the type of the principal stock regulation system and the type of the corresponding mathematical model. Methods and models of the stock theory, the primary objective of which is to determine the most important incoming product flow parameters of the system, are still in demand and their primary goal is to adapt the manufacturing company to the consumers’ needs
In the article we build a mathematical model of elec-tro-diffusion of ions in the diffusion layer of a mem-brane system complicated by the occurrence of the previous slow homogeneous chemical reaction with the condition of electrical neutrality of the solution. We have set a two-point boundary value problem and developed a method to solve it; we have given an algorithm and a numerical method for solving it in Comsol 3.5 environment. The formula for limiting kinetic current was derived. Some of the model’s capabilities to describe the properties of the system are given
The article presents the simulation of non-linear spatial-temporal color oscillations in Yang-Mills theory in the case of SU (2) and SU (3) symmetry. We examined three systems of equations derived from the Yang-Mills theory, which describes the transition to chaotic behaviour. These transitions are caused by nonlinear vibrations of colour, depending on the model parameters - the coupling constants and the initial wave amplitude. Such transitions to chaotic behaviour by increasing the parameters are characteristic of hydrodynamic turbulence. A model of spatial-temporal oscillations of the Yang-Mills theory in the case of three and eight colors. The results of numerical simulation show that the nonlinear interaction does not lead to a spatial mixing of colors as it might be in the case of turbulent diffusion. Depending on the system parameters there is a suppression of the amplitude of the oscillations the first three of five colors or vice versa - the first three five other colors. The kinetic energy fluctuations or shared equally between the color components, or dominated by the kinetic energy of repressed groups of colors. Note that the general property of physical systems described by nonlinear equations in the Yang-Mills theory and hydrodynamics is particularly strong in the formation of quark-gluon plasma and hadrons jets, when the Yang-Mills is involved in the formation of hydrodynamic flow. Note that there is a relationship between the Einstein and Yang-Mills theory, on the one hand, Einstein's equations and hydrodynamics - on the other. All of this points to the existence in the nature of a general mechanism of formation of a special type of turbulence - geometric turbulence