The Euler function is very important in number theory
and in Mathematics, however, the range of its values in
the natural numbers has not been written off. The
greatest value of the Euler function reaches on Prime
numbers, furthermore, it is multiplicative. The value of
the Euler function is closely associated with the values
of the Moebius function and the function values of the
sum of the divisors of the given natural number. The
Euler function is linked with systems of public key
encryption. The individual values of the Euler function
behave irregularly because of the irregular distribution
of primes in the natural numbers. This tract is
illustrated in the article with charts; summatory
function for the Euler function and its average value
are more predictable. We prove the formula of
Martinga and, based on it, we study the approximation
accuracy of the average value of the Euler function
with corresponding quadratic polynomial. There is a
new feature associated with the average value of the
Euler function and calculate intervals of its values. We
also introduce the concept of density values of the
Euler function and calculate its value on the interval of
the natural numbers. It can be noted that the results of
the behavior of the Euler function are followed by the
results in the behavior of functions of sums of divisors
of natural numbers and vice versa. We have also given
the results of A.Z.Valfish and A.N.Saltykov on this
subject
The review of resolvability of the beginning-boundary problem describing dispersion of an impurity in turbulent atmosphere, correctness of mathematical models, describing impurity dispersion in atmosphere and Koshi’s problem, the first right problem, the third right problem is given in the article
The technique for computing of the turbulent diffusion coefficient vertical component in the context of a mathematical model of admixture dispersion in the surface layer is proposed
Adequate and effective assessment of the efficiency, effectiveness and the quality of scientific activities of specific scientists and research teams is crucial for any information society and a society based on knowledge. The solution to this problem is the subject of scientometrics and its purpose. The current stage of development scientometrics differs greatly from his previous appearance in the open as well as paid on-line access to huge amount of detailed data on a large number of indicators on individual authors and on scientific organizations and universities. The world has well-known bibliographic databases: Web of Science, Scopus, Astrophysics Data System, PubMed, MathSciNet, zbMATH, Chemical Abstracts, Springer, Agris, or GeoRef. In Russia, it is primarily the Russian scientific citing index (RSCI). RSCI is a national information-analytical system, accumulating more than 9 million publications of Russian scientists, as well as the information about citation of these publications from more than 6,000 Russian journals. There is too much information; it is so-called "Big data". But the problem is how to make sense of these large data, more precisely, to identify the meaning of scientometric indicators) and thus to convert them into great information ("great information"), and then apply this information to achieve the objective of scientometrics, i.e. to transform it into a lot of knowledge ("great knowledge") about the specific scientists and research teams. The solution to this problem is creating a "Scientific smart metering system" based on the use of the automated system-cognitive analysis and its software tools – an intellectual system called "Eidos". The article provides a numerical example of the creation and application of Scientometric intelligent measurement system based on a small amount of real scientific data that are publicly available using free on-line access to the RSCI
In this article, there is a numerical method of solving
the problem of self-organization of the labor
resources. The problem deals with finding
probabilities of hiring and the layoffs of specialists
from the sectors of the labor market. A mathematical
model of labor resources dynamics is used to solve
this problem. The initial problem is incorrect,
because number of equations of the descriptive
system is less than number of unknown variables. A
special algorithm is designed for guaranteed finding
the normal solution in finite number of iterations.
The algorithm is separated into two key stages.
Initially, unconditional normal solution of the
problem is found by applying the modified method of
Gauss for underdetermined systems. Later, this
solution is projected in the subspace of permissible
values. After that, the normal solution of the problem
with consideration of non-negativity of the desired
values is being found by using the gradient projection
method. The proposed algorithm has been
successfully used to develop application in
programming environment C++. This application is
focused on solving of the problem of selforganization
of the labor resources. Comparative
analysis of speed of the application and add-ins MS
Excel "Solver" showed that the same problem is
solved much faster in the application designed by the
author than in a table processor MS Excel when
using the add-in "Solver". This demonstrates the high
efficiency of the proposed method
We have proposed the general scheme for studying the stability of the conclusions obtained by mathematical methods and models regarding tolerance deviations of the original data and background models. The concrete problems of stability are discussed: towards a change of data, its size and distributions, to allowable transformations measurement scales, to the temporal characteristics (time of start of the project, the planning horizon). Reducing the uncertainty can be conducted by changing the type of data, i.e. with the aid of the transition to non-numerical data.
The models of concrete management processes of industrial organizations are considered on the examples of stability characteristics of investment projects to change the discount factors and in models of inventory management to change in the coefficients of the model and batch size production
A model for minority carrier mobility in polysilicon emitter contacts is developed. It is based on the effect of the segregation of electrically active dopants to polysilicon grain boundaries and the thermionic emission - diffusion theory of the hole current. An analytical equation is derived which allows to calculate hole mobility in polysilicon emitter contacts and its dependence on dopant concentration and polysilicon grain size
The investigated and correct boundary value problem for mixed hyperbolic-parabolic equation of second order in a bounded domain is posed and studied in this work. Boundary conditions are of a classical nature. On line of type changes, which is also the line of the parabolic degeneracy for hyperbolic equations considered in the lower half-plane, a continuous bonding condition for the function itself and the breaking condition for the trace of the derivative is given. The main result is the proof of its unique solvability in the required class of functions. In particular, based on the properties of the operators of fractional integro-differentiation and in view of the ratio of the first boundary value problem for the heat equation, the question of the solvability of the original problem was equivalently reduced to the problem of solvability of the corresponding integral equation of the Voltaire second kind. In the hyperbolic part of the region, the question of solvability of the problem has also been reduced to the problem of solvability of the integral equation Voltaire second kind. The properties of the hypergeometric function of Gauss, as well as classical methods of integral equations were used. Thus it is proved the uniqueness and the existence of classical solution to the initial problem
In the article we have derived mathematical models of non-stationary transport binary electrolyte in EMS (electromembrane systems: electrodialysis apparatus, electromembrane cell, etc.) for the galvanostatic mode. To be specific, as EMS viewed channel of desalting of EDA (electrodialysis apparatus) and EMS with RMD (rotating membrane disk). We present a formula expressing the intensity of the electric field through the current density and concentration. Also, we have received the differential equation for the current density. The fundamental point here is derived new equation for the unknown vector function of current density of the initial system of equations of Nernst-Planck. In addition, the article shows the output equation for the current density in three dimensions; we have proposed various methods for solving the equation of the current density and the boundary conditions for the current density. The proposed mathematical models of transport binary electrolyte are easy to be generalized to an arbitrary electrolyte. However, the corresponding equations are cumbersome. It should be also noted that the boundary conditions can be varied and depend on the purpose of a particular study in this regard, in this work are just the equation having the general form
There is a 2D mathematical model of ion transport
binary salt with the main conjugate effects of
concentration polarization in the overlimiting current
mode: the bulk charge and the dissociation/
recombination of water, gravity and electroconvection
and Joule heating the solution in the form of a
boundary value problem for systems of differential
equations with partial derivatives in the article. This
system is presented in a form convenient for numerical
solution. We describe the necessary boundary
conditions. This article presents a theoretical study of
the interaction of forced, gravitational and
electroconvection, the dissociation / recombination of
water molecules, and Joule heating of the solution and
heat transport through membranes. We have
constructed a mathematical model of two-dimensional
non-stationary ion transport binary salt in a smooth
rectangular channel desalting electrodialysis device
using equations Nernst-Planck-Poisson, heat
conduction and Navier-Stokes equations and the
natural boundary conditions. For numerical solution
we use the finite element method, with the splitting of
task at each new time layer into three subtasks:
electrochemical, thermal conductivity, hydrodynamic.
Such approach to the development of numerical
methods is the original and can solve arising in
modeling boundary-value problems for a nonlinear system of partial differential equations