In 2011 – 2015, the scientific community was
represented by a new paradigm of mathematical
methods of research in the field of organizational
and economic modeling, econometrics and statistics.
There was a talk about a new paradigm of applied
statistics, mathematical statistics, mathematical
methods of economics, the analysis of statistical and
expert data in problems of economics and
management. We consider it necessary to develop
organizational and economic support for solving
specific application area, such as the space industry,
start with a new paradigm of mathematical methods.
The same requirements apply to the teaching of the
respective disciplines. In the development of
curricula and working programs, we must be based
on a new paradigm of mathematical methods of
research. In this study, we present the basic
information about a new paradigm of mathematical
methods of research. We start with a brief
formulation of a new paradigm. The presentation in
this article focuses primarily on the scientific field
of "Mathematical and instrumental methods of
economy", including organizational and economic
and economic-mathematical modeling, econometrics
and statistics, and decision theory, systems analysis,
cybernetics, operations research. We discuss the
basic concepts. We talk about the development of a
new paradigm. We carry out a detailed comparison
of the old and the new paradigms of mathematical
methods of research. We give information about the
educational literature, prepared in accordance with
the new paradigm of mathematical methods of
researches
Some estimators of the probability density function
in spaces of arbitrary nature are used for various
tasks in statistics of non-numerical data. Systematic
exposition of the theory of such estimators has been
started in our articles [3, 4]. This article is a direct
continuation of these works [3, 4]. We will regularly
use references to conditions and theorems of the
articles [3, 4], in which introduced several types of
nonparametric estimators of the probability density.
We have studied linear estimators. In this article, we
consider particular cases - kernel density estimates in
discrete spaces. When estimating the density of the
one-dimensional random variable, kernel estimators
become the Parzen-Rosenblatt estimators. Under
different conditions, we prove the consistency and
asymptotic normality of kernel density estimators.
We have introduced the concept of "preferred rate
differences" and are studied nuclear density
estimators based on it. We have introduced and
studied natural affinity measures which are used in
the analysis of the asymptotic behavior of kernel
density estimators. Kernel density estimates are
considered for sequences of spaces with measures.
We give the conditions under which the difference
between the densities of probability distributions and
of the mathematical expectations of their nuclear
estimates uniformly tends to 0. Is established the
uniform convergence of the variances. We find the
conditions on the kernel functions, in which take
place these theorems about uniform convergence. As
examples, there are considered the spaces of fuzzy
subsets of finite sets and the spaces of all subsets of
finite sets. We give the condition to support the use
of kernel density estimation in finite spaces. We
discuss the counterexample of space of rankings in
which the application of kernel density estimators
can not be correct
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, the restricted problem of three and more
bodies in the Ricci flow in the general theory of
relativity considered. A system of non-linear parabolic
equations describing the evolution of the axially
symmetric metrics in the Ricci flow proposed. A model
describing the motion of particles in the Ricci flow
derived. It is shown that the theory describing the Ricci
flow in the many-body problem is consistent with the
Einstein-Infeld theory, which describes the dynamics of
the material particles provided by the singularities of the
gravitational field. As an example, consider the metric
having axial symmetry and contains two singularities
simulating particles of finite mass. It is shown that the
static metric with two singularities corresponds to
Newton's theory of the two centers of gravity, moving
around the center of mass in circular orbits in a noninertial
frame of reference, rotating with a period of
bodies. We consider the statement of the problem of
many bodies distributed at the initial time on the axis of
symmetry of the system. 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. Using the properties of relativistic potentials
we have justified transition from the relativistic motion
of the particles to the dynamic equations in the classic
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
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
This article focuses on the mathematical modeling of
evaluation of financial and economic activities of a
company and on definition (based on this model) of
such balance settings (line F1 and F2) which would
make financial-economic indicators of the activities of
the organization optimal, and the total cumulative
score was the maximum. The knowledge and the use
of the optimal parameters of the balance will allow the
managers to plan strategy for the future development
of the company. The article analyzes the dependencies
of each of the 15 basic indicators (profitability,
turnover, financial stability, liquidity and solvency) of
financial and economic activity of the organization on
the balance parameters. The optimal values of the
parameters of the balance and the main indicators of
financial and economic activities of the organization
have been found. We have also built a mathematical
model of optimal control of financial and economic
indicators in the form of a problem of mathematical
programming. For example, for the company called "Nika" it is shown the possibility of improving
estimation of financial and economic condition of the
organization. Knowledge of the optimal parameters of
the balance will allow the managers to plan strategy
for the future development of the organization. To
solve this problem we have used the method of
generalized reduced gradient implemented in Excel,
with which there was found a maximum of the
objective function for the article restrictions. The
article describes the analysis algorithm of the
optimization problem. A common assessment was
carried out in stages, based on the calculation
algorithm of sequentially improved target functions
This article is a continuation of the previous works of
the authors [The influence of reaction dissociation /
recombination of molecules of water on transportation
of electrolyte 1:1 in the membrane systems in the
diffusion layer. Part 1. Mathematical model //
Scientific journal of Kuban State Agrarian University,
2016. No. 07(121) and The influence of the reaction of
dissociation / recombination of molecules of water on
transportation of electrolyte 1: 1 in membrane systems
in the diffusion layer. Part 2. Asymptotic analysis //
Scientific journal of Kuban State Agrarian University,
2016. – №08(122)] and devoted to assessing the
possibility of gravitational convection due to the
recombination of hydrogen and hydroxyl ions. The
article presents the solution of a boundary-value
problem, which is a mathematical model of
electrodiffusion for the four types of ions at the same
time (two ions of salts and hydrogen and hydroxyl
ions) in the diffusion layer in electro-membrane
systems with ideal selective membrane, with the heat
transfer equation and the Navier-Stokes equation. The
article shows the possibility of the emergence of
gravitational convection due to the exothermic reaction
of recombination of water molecules in the depth of
the solution. The article considered the reaction of
recombination of hydrogen ions and hydroxyl,
although the main results can be applied, after appropriate modifications, and to amfolit-containing
solutions, such as wine, juices, dairy products,
microbiological processing of biomass (amino acids,
anions of polybasic carboxylic acids), municipal
effluent (anions of phosphoric acid), etc.
This article discusses an economic game called
"The struggle for markets". We have generated a
mathematical model of quantum realization of this
game. For clarity, the algorithms are derived for
soft and hard quantum games for assessing the
impact of the degree of entanglement to work and
the result of the algorithm. There are step-by-step
instructions for the sequence of actions and
operations to create a quantum model of the game.
The aim is to assess the influence of the degree of
entanglement on work algorithms. Also, we
investigate the influence of quantum entanglement
on the win for two or more players. The article
gives a comparison with classical results
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