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
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
In many applications, we study the time series (or a
random process), which is the sum of the periodic
deterministic function of time and random errors that
distort the periodic signal. It is required to estimate
the length of the period and the periodic component.
It does not assume that the periodic function is
included in any parameter family of functions, such
as finite sums of sines and cosines. It is obvious that
the assumption of occurrence of a periodic function
in parametric family does not meet the
characteristics of the real world, ie, is conditional,
internal mathematical (look for the keys under the
lamp because there is a light, not in the bush, where
lost, because there are dark). For similar reasons, it
is impossible to assume that the distribution function
of the random errors is included in any parameter
family of distributions. In accordance with the new
paradigm of mathematical statistics in this article we
studied the problem of nonparametric estimation
(minimum) length of the period and the periodic
component of the signal. On the basis of natural
variation and scope of indicators is suggested a new
class of nonparametric estimators of the length of the
period and the periodic component in the time
series. Based on the general results of statistics of
objects of non-numeric nature we proved the
consistency of these estimates. From the practical
point of view it is necessary to minimize the
numerical (one parameter - ability length of period
of time) one or more of the 66 functionals, described
in the article
In this work, we investigate the problem of collisions of
particles linked to the singularities of the gravitational
field in the Ricci flow. A system of non-linear parabolic
equations describing the evolution of the axially
symmetric metrics proposed. We consider the metric
having axial symmetry and comprising two singularities
simulating particles of finite mass. There was
numerically investigated the change of the metric in the
collision of particles. Two formulations of the problem
have been considered, one of which scatter particles after
the collision, and the other as a result of the merger of
two particles, a new stable static system, which can be
interpreted as a new particle. The initial and boundary
conditions using the exact solution of the static problem,
so the collision persist particularly metrics caused by the
presence of particles. In numerical experiments
determined that the collision of the particles in the Ricci
flow leads to the formation of gravitational waves,
similar in structure to the waves, registered in the LIGO
experiment. Consequently, we can assume that the
observed gravity waves caused mainly by transients
associated with the change in the metric system. A
model describing the emission of gravitational waves in
the collision of particles in the Ricci flow proposed. The
influence of the parameters of the problem - the speed
and mass of the particles, on the amplitude and intensity
of the emission of gravitational waves was numerically
simulated
The article discusses a model of rocket motor of
electromagnetic type, consisting of a source of
electromagnetic radio frequency oscillations and the
conical cavity, in which electromagnetic waves are
excited. We have created a multi-dimensional
transient numerical model describing the process of
establishing electromagnetic oscillations in the
resonator, taking into account the finite conductivity
of the walls. Separately, the standing waves in the
cavity with conducting walls have been simulated. It
is shown that the oscillations mode in the conducting
resonator different from that in an ideal resonator,
both in a case of steady and unsteady waves. We have
built a dynamic model taking into account the thermal
conductivity and electrical conductivity of the walls,
waves and particles emission and vacuum
polarization. We have also developed a dynamic
model enables to optimize a thrust force on a
considerable number of parameters without the
involvement of the hypotheses about the physics of
the phenomenon. We run the optimization of the
operating parameters of the device, namely by the
excitation frequency, the frequency of the modulating
signal, the magnitude of heat losses of
electromagnetic energy by thermal radiation in the IR
spectrum, the parameters of forced heat transfer and
the temperature dependence of the resistance of the
material of the cavity walls. It is found that the pulse
modulation greatly improves the efficiency of
conversion of electromagnetic energy into thrust. The
mechanism of formation of traction, adjusting the
metrics of space-time, the current contribution of
elementary particles, the Yang-Mills and
electromagnetic fields is proposed. It is shown that
the contribution of the elementary particles in the
thrust force is proportional to the electrical
conductivity of the system multiplied by Abraham
force
Problem having elementary formulation makes us
look for its easier solution. So the combinatorial
method of positive integer’s factorization is an
attempt to do it. The combinatory method possesses
simple algorithm, leading immediately to finding out
all the factorizations and identification of all prime
numbers on any interval of the positive integers.
Prime numbers don’t carry any information except
their own magnitude. Composite numbers, possessing
divisibility properties provide possibility to discover
the law of their distribution. The achievement of this
purpose also completely solves the problem of
finding out the law of prime numbers’ distribution
The general scheme of modern statistical science is
just like this. Mathematical Statistics is a part of
mathematics that studies the statistical structure (it
itself does not give recipes analysis of statistical
data, however, it is developing methods that are
useful for use in theoretical statistics). Theoretical
Statistics - the science dedicated to the models and
methods of analysis of concrete statistical data.
Applied Statistics (in the narrow sense) is devoted to
the statistical techniques of data collection and
processing (it includes the methodology of statistical
methods, the organization of sample surveys, the
development of statistical techniques, the creation
and use of statistical software). Applications of
statistical methods in concrete fields (in economics
and management - Econometrics, in biology -
Biometrics, in chemistry - Chemometrics, in
technical research - Technometric, in geology,
demography, sociology, medicine, history, etc.).
Often positions 2 and 3 together are called Applied
Statistics. Sometimes position 1 is called Theoretical
Statistics. These terminological differences are
related to the fact that the above-described
development of the considered scientific and applied
field not once, not completely and not always
adequately reflected in the minds of experts.
Meanwhile, there are still textbooks of appropriate
level of representation of the mid-twentieth century.
The article analyzes the post-war development of
the national statistics. We have identified five
"growth points": nonparametrics, robustness,
bootstrap, statistics of interval data, and statistics of
non-numeric data. We have discussed content,
development and the basic ideas of statistics of nonnumeric
data. We have given a number of
unresolved problems of theoretical and applied
statistics
We have given a critical analysis of statistical
models and methods for processing text information
in historical records to establish the times when
there were certain events, ie, to build science-based
chronology. There are three main kinds of sources
of knowledge of ancient history: ancient texts, the
remains of material culture and traditions. The
specific date of the extracted by archaeologists
objects in most cases can not be found. The group of
Academician A.T. Fomenko has developed and
applied new statistical methods for analysis of
historical texts (Chronicle), based on the intensive
use of computer technology. Two major scientific
results were: the majority of historical records that
we know now, are duplicated (in particular,
chronicles, describing the so-called "Ancient Rome"
and "Middle Ages", talking about the same events);
the known historical chronicles tell us about real
events, separated from the present time for not more
than 1000 years. It was found that chronicles
describing the history of "ancient times" and
"Middle Ages" and the chronicle of Chinese history
and the history of various European countries do not
talk about different, but about the same events. We
have the attempt of a new dating of historical events
and restoring the true history of human society
based on new data. From the standpoint of statistical
methods of historical records and images of their
fragments – they are special cases of non-numeric
objects of nature. Therefore, developed by the group
of A.T. Fomenko computer-statistical methods are
the part of non-numerical statistics. We have
considered some methods of statistical analysis of
chronicles applied by the group of A.T. Fomenko:
correlation method of maximums; dynasties method;
the method of attenuation frequency; questionnaire
method codes. New chronology allows us to
understand much of the battle of ideas in modern
science and mass consciousness. It becomes clear
the root cause of cautious attitude of the West
towards Russia
The article deals with numerical solutions of MHD
equations describing turbulent flow of a conducting
fluid in a rectangular cavity in the rotating magnetic
field at large values of the magnetic Taylor number,
and Reynolds number. 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. The models based on the properties of
currents of the turbulent environment proposed. A
modification of the continuity equation taking into
account the final magnitude of pressure fluctuations
was considered. It is shown that due to pressure
fluctuation the incompressibility condition can be
violated even for flows with low Mach numbers.
Modification of continuity in the system of NavierStokes
equations by the introduction of turbulent
viscosity allows the regularization of the NavierStokes
equations to solve the problems with rapidly
changing dynamic parameters, for example, in the
case of a conducting fluid flow in a magnetic field
rotating with a high frequency. 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 MHD flow in a rectangular cavity
with rapid change in flow parameters was proposed.
In numerical calculations revealed that under the
influence of a rotating magnetic field in a conducting
fluid there are forces occur, causing unsteady vortex
flow, which is consistent with experimental data. We
have discovered a type of large scale instability of the
turbulent flow, connecting with the secondary flow in
a form of vortices
The article by continues the cycle of their studies
related to the formulation and development of
methods of constructing non-negative solutions of
inverse problems of balance models (in this case,
the model of world trade). Method of constructing
nonnegative solutions of the studied inverse
problems is developed. This technique is based on
the following scheme of the solution. Initially we
convinced of a correct formulation of the direct
problem, then of the solvability of the inverse.
Further, by specified tabular solutions of the direct
problem, a system of algebraic equations
containing the unknown, the estimated parameters
of the studied model is built. Then the inverse
problem reduces to solving the following quadratic
programming, the solution of which is determined
in MS Excel. The theoretical material is
accompanied by solution of specific example,
using statistical data of the Karachay-Cherkess
Republic that shows how actually in practice it is
possible to solve the inverse problem, i.e. to
organize a process of balanced trade of the
Karachay-Cherkess Republic with each of the
subjects of Noth – Caucasion Federal District.
Found the non-negative elements of a matrix, by
which we can judge what proportion of national
income, y, the subject has to spend on the purchase
of goods in the Karachay-Cherkess Republic, to
trade between this pair was balanced. So, the
inverse problem posed in relation to the trading
countries, it is possible to put and solve the
following method and to trade between the
subjects of one country