The arithmetic properties of the values of hypergeometric function have been studied by various methods since the paper of C. Siegel in 1929. This direction of the theory of Diophantine approximations was studied by such authors as М.~Hata [1]-[2], F.~Amoroso and C.~Viola [3], A.~Heimonen, T.~Matala-aho and K.~V\"{a\"{a}}n\"{a}nen [4]-[5] and other. In recent decades, a number of interesting results in this area have been obtained, many of the previously known estimates for the irrationality measures for values of hypergeometric functions, and other variables have been improved.

Currently one of the widely used approaches in the construction of estimates of the irrationality measure is the use of integral constructions symmetric with respect to replacement of parameters. Symmetrized integrals have been previously used by different authors, for example in the G.~Rhin's article [6], but the most active development of this direction was acquired after the work of V.~,Kh.~Salikhov [7], who received a new estimate for $\ln{3}$ using the symmetrized integral. Subsequently, the symmetry of different types allowed to prove a number of significant results. New estimates for some values of the logarithmic function, the function $\arctg{x}$, and classical constants were obtained (see, for example, [8] -- [18]). In 2014 Q.~Wu and L.~Wang intensified V.~H.~Salikhov's result of the irrationality measure of $\ln{3}$ using common symmetrized polynomials $At-B$, where $t=(x-d)^2$ (see [19]). In the V.~A.~Androsenko's article the idea of symmetry was applied to the integral of Marcovecchio, who previously proved a new estimate for $\ln{2}$ in [21], and it allowed to improve the result for $\pi/3$.

This paper is a continuation of article [22] generalizing results for two types of symmetric integral constructions. The first allows to estimate more effectively the measure of irrationality of numbers of the form $\sqrt{d}\ln{\frac{\sqrt{d}+1}{\sqrt{d}-1}}$ at $d=2^{2k+1}, d=4k+1$ for some $k\in\mathbb N$ (see [22]). It is also possible to obtain estimates of the irrationality measure of numbers $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}},\ k\in\mathbb N$ using this integral. The second considered integral construction makes it possible to estimate the measure of irrationality of some values of the logarithmic function using another type of symmetry, what was discussed in detail in [22]. This integral also allows to estimate the measure of irrationality of values $\frac{1}{\sqrt{k}}\arctg{\frac{1}{\sqrt{k}}}$. A generalization of this case is proposed in this paper.

\textbf{Goal.} The aim of the paper is studying of Russian mathematicians contribution (V.V. Nemytskii, A.N. Tikhonov, A.A. Markov, M.G. Krein, V.L. Shmul'yan, etc.) to the development of the fixed point method for the period from the beginning 1920's until the late 1950's.
Method. The reseach is based on an analysis of the original works of the listed scientists in the context of the worldwide development of nonlinear functional analysis against the backdrop of the achievements of American (J. Birkhoff, O. Kellogg), Polish (S. Banach, S. Mazur, J. Schauder, K. Borsuk, ), Italian (R. Cacciopolli), French (J. Leray) and German (E. Rothe) mathematicians.

\textbf{Result.} The contribution of the Soviet scientists in the field of fixed point method is comparable with that of the rest of the world mathematical community in the period under review. This is confirmed both by the number of proved fixed-point theorems and by their quality. Due to the efforts of the Soviet mathematician M.A. Krasnosel'skii from the mid-1950's a fixed point method became a general method for solving a wide class of problems of a qualitative nature for a nonlinear operators analysis (until this time, the method under discussion was considered only as a tool for proving of the solvability of nonlinear integral or differential equations and their systems abstract analogues).

\textbf{Discussion.} An analysis of the achievements at the area of the fixed-point method in the global context has shown that the development of nonlinear functional analysis (as, indeed, of any other section of mathematics) is a supranational process that is carried out by the efforts of mathematicians from different countries. This process goes beyond any scientific school, no matter how large it may be.

Fibinomial identity is identity that combine Fibonacci numbers and binomial or multinomial coefficients.
In this paper, for obtaining new fibinomial identities we consider determinants and permanents for some families of lower Toeplitz–Hessenberg matrices $H_n=(h_{ij})$,
where $h_{ij}=0$ for all $j>i+1$, $h_{ij}=a_{i-j+1}$, and $a_{i,i+1}=2$, having various translates of the Fibonacci numbers $F_n$ for the nonzero entries.

These determinant and permanent formulas may also be rewritten as identities involving sums of products of Fibonacci numbers and multinomial coefficients.
For example, for $n\geq1$, the following formula holds
$$
\sum_{s_1+2s_2+\cdots+ns_n=n}(-1)^{s_1+\cdots+s_n}{s_1+\cdots+s_n\choose s_1,\ldots, s_n}\left(\frac{F_2}{2}\right)^{s_1}\left(\frac{F_4}{2}\right)^{s_2}\cdots\left(\frac{F_{2n}}{2}\right)^{s_n}=
\frac{1-4^n}{3\cdot 2^n},
$$
where ${s_1+\cdots+s_n\choose s_1,\ldots, s_n}=\frac{(s_1+\cdots+s_n)!}{s_1!\cdots s_n!}$ is multinomial coefficient, and the summation is over non\-negative integers
$s_j$ satisfying Diophantine equation $s_1 +2s_2 +\cdots +ns_n=n$.

Also, we establish connection formulas between Jacobsthal, Pell, Pell-Lucas numbers and Fibonacci numbers using Toeplitz-Hessenberg determinants.

Recently Arestov, Babenko, Deikalova, and Horv\'ath have established a series
of interesting results correspondent to the sharp Nikolskii constant
$\mathcal{L}_\textup{even}(\alpha,p)$ in the weighted inequality
\[
\sup_{x\in [0,\infty)}|f(x)|\le
\mathcal{L}_\textup{even}(\alpha,p)\sigma^{(2\alpha+2)/p}
\biggl(2\int_{0}^{\infty}|f(x)|^{p}x^{2\alpha+1}\,dx\biggr)^{1/p}
\]
for the subspace $\mathcal{E}^{\sigma}\cap
L^{p}(\mathbb{R}_{+},x^{2\alpha+1}\,dx)$ of even entire functions $f$ of
exponential type at most $\sigma>0$, where $1\le p<\infty$ and $\alpha\ge -1/2$.

We prove that, for the same $\alpha$ and $p$
\[
\mathcal{L}_\textup{even}(\alpha,p)=\mathcal{L}(\alpha,p),
\]
where $\mathcal{L}(\alpha,p)$ is the sharp constant in the Nikolskii inequality
\[
\sup_{x\in \mathbb{R}}|f(x)|\le \mathcal{L}(\alpha,p)\sigma^{(2\alpha+2)/p}
\biggl(\int_{\mathbb{R}}|f(x)|^{p}|x|^{2\alpha+1}\,dx\biggr)^{1/p}
\]
for any (not necessary even) functions $f\in
\mathcal{E}_{p,\alpha}^{\sigma}:=\mathcal{E}^{\sigma}\cap
L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$.

Also we give bounds of the normalized Nikolskii constant
\[
\mathcal{L}^{*}(\alpha,p):=
(2^{2\alpha+2}\Gamma(\alpha+1)\Gamma(\alpha+2))^{1/p}\mathcal{L}(\alpha,p),
\]
which are as follows:
\[
\mathcal{L}^{*}(\alpha,p)\le \lceil p/2\rceil^{\frac{2\alpha+2}{p}},\quad p\in
(0,\infty),
\]
and for fixed $p\in [1,\infty)$
\[
\mathcal{L}^{*}(\alpha,p)\ge (p/2)^{\frac{2\alpha+2}{p}\,(1+o(1))},\quad
\alpha\to \infty.
\]
The upper estimate is sharp if and only if $p=2$. In this case,
$\mathcal{L}^{*}(\alpha,2)=1$ for each $\alpha\ge -1/2$.

Our approach relies on the one-dimensional Dunkl harmonic analysis. To prove
the identity $\mathcal{L}_\textup{even}(\alpha,p)=\mathcal{L}(\alpha,p)$ we use
the even positive Dunkl-type generalized translation operator $T^{t}$ such that
is bounded on $L^{p}(\mathbb{R},|t|^{2\alpha+1}\,dt)$ with constant one and
invariant on the subspace $\mathcal{E}_{p,\alpha}^{\sigma}$.

The proof of the upper estimate of the constant $\mathcal{L}^{*}(\alpha,p)$ is
based on estimation of norms of the reproducing kernel for the subspace
$\mathcal{E}_{p,\alpha}^{1}$ and the multiplicative inequality for the
Nikolskii constant. To obtain the lower estimate we consider the normalized
Bessel function $j_{\nu}\in \mathcal{E}_{p,\alpha}^{1}$ of order $\nu\sim
(2\alpha+2)/p$.

For $0<p<\infty$, we investigate the interrelation between the Nikolskii
constant for trigonometric polynomials of order at most $n$
\[
\mathcal{C}(n,p)=\sup_{T_{n}\ne 0}\frac{\|T_{n}\|_{\infty}}{\|T_{n}\|_{p}}
\]
and the Nikolskii constant for entire functions of exponential type at most~$1$
\[
\mathcal{L}(p)=\sup_{f\ne 0}\frac{\|f\|_{\infty}}{\|f\|_{p}}.
\]

Recently E.~Levin and D.~Lubinsky have proved that
\[
\mathcal{C}(n,p)=\mathcal{L}(p)n^{1/p}(1+o(1)),\quad n\to \infty.
\]
M.~Ganzburg and S.~Tikhonov have extend this result on the case of
Nikolskii--Bernstein constants.

We prove inequalities
\[
n^{1/p}\mathcal{L}(p)\le \mathcal{C}(n,p)\le (n+\lceil
p^{-1}\rceil)^{1/p}\mathcal{L}(p),\quad n\in \mathbb{Z}_{+},\quad 0<p<\infty,
\]
which improve the result of Levin and Lubinsky. The proof follows our old
approach based on properties of the integral Fejer kernel. Using this approach
we proved earlier estimates for $p=1$
\[
n\mathcal{L}(1)\le \mathcal{C}(n,1)\le (n+1)\mathcal{L}(1).
\]

Using such inequalities, we can estimate the constant $\mathcal{L}(p)$ solving
approximately $\mathcal{C}(n,p)$ for large $n$. To do this we use recent
results of V.~Arestov and M.~Deikalova, who expressed the Nikolskii constant
$\mathcal{C}(n,p)$ using the algebraic polynomial $\rho_{n}$ that deviates
least from zero in the space $L^{p}$ on the segment $[-1,1]$ with the weight
$(1-t)v(t)$, where $v(t)=(1-t^{2})^{-1/2}$ is the Chebyshev weight. As
consequence, we refine estimates of the Nikolskii constant $\mathcal{L}(1)$ and
find~that
\[
1.081<2\pi \mathcal{L}(1)<1.082.
\]
To compare previous estimates were $1.081<2\pi \mathcal{L}(1)<1.098$.

Continuous non-negative positive definite functions satisfy the following
property:
\[
\int_{-R}^{R}f(x)\,dx\le C(R)\int_{-1}^{1}f(x)\,dx,\quad R\ge 1,
\tag{$*$}
\]
where the smallest positive constant $C(R)$ does not depend on $f$. For $R=2$,
this property is well known as the doubling condition at zero. These
inequalities have applications in number theory.

In the one-dimensional case, the inequality ($*$) was studied by B.F.~Logan
(1988), as well as recently by A.~Efimov, M.~Ga\'al, and Sz.~R\'ev\'esz (2017).
It has been proven that $2R-1\le C(R)\le 2R+1$ for $R=2,3,\ldots$, whence it
follows that $C(R)\sim 2R$. The question of exact constants is still open.

A multidimensional version of the inequality ($*$) for the Euclidean space
$\mathbb{R}^{n}$ was investigated by D.V.~Gorbachev and S.Yu.~Tikhonov (2018).
In particular, it was proved that for continuous positive definite functions
$f\colon \mathbb{R}^{n}\to \mathbb{R}_{+}$
\[
\int_{|x|\le R}f(x)\,dx\le c_{n}R^{n}\int_{|x|\le 1}f(x)\,dx,
\]
where $c_{n}\le 2^{n}n\ln n\,(1+o(1))(1+R^{-1})^{n}$ при $n\to \infty$. For
radial functions, we obtain the one-dimensional weight inequality
\[
\int_{0}^{R}f(x)x^{n-1}\,dx\le c_{n}R^{n}\int_{0}^{1}f(x)x^{n-1}\,dx,\quad n\in
\mathbb{N}.
\]

We study the following natural weight generalization of such inequalities:
\[
\int_{0}^{R}f(x)x^{2\alpha+1}\,dx\le
C_{\alpha}(R)\int_{0}^{1}f(x)x^{2\alpha+1}\,dx,\quad \alpha\ge -1/2,
\]
where $f\colon \mathbb{R}_{+}\to \mathbb{R}_{+}$ is an even positive definite
function with respect to the weight $x^{2\alpha+1}$. This concept has been
introduced by B.M.~Levitan (1951) and means that for arbitrary
$x_{1},\ldots,x_{N}\in \mathbb{R}_{+}$ matrix
$(T_{\alpha}^{x_i}f(x_j))_{i,j=1}^{N}$ is semidefinite. Here $T_{\alpha}^{t}$
is the Bessel--Gegenbauer generalized translation. Levitan proved an analogue
of the classical Bochner theorem for such functions according to which $f$ has
the nonnegative Hankel transform (in the measure sense).

We prove that for every $\alpha\ge -1/2$
\[
c_{1}(\alpha)R^{2\alpha+2}\le C_{\alpha}(R)\le c_{2}(\alpha)R^{2\alpha+2},\quad
R\ge 1.
\]
The lower bound is trivially achieved on the function $f(x)=1$. To prove the
upper bound we apply lower estimates of the sums
$\sum_{k=1}^{m}a_{k}T^{x_{k}}\chi(x)$, where $\chi$ is the characteristic
function of the segment $[0,1]$, and also we use properties of the Bessel
convolution.

We propose a high precision method of finding of potential for multi-atomic
quantum-mechanical tasks in real space. The method is based on dividing of electron
density and potential of a multi-atomic system into two parts. The
first part of density is found as a sum of spherical atomic
densities; the second part is a variation of density generated by
interatomic interaction. The first part of potential is formed by
the first part of density and may be calculated correctly using
simple integrals. The second part of potential is found through a
Poisson equation from the second part of density. To provide a
high precision we divided a work space into Voronoy's polyhedrons
and found the boundary conditions by means of a multi-pole
distribution of potentials formed by local densities concentrated
in these polyhedrons. Then we used double-grid approach, and fast
Fourier transformations as initial functions for iterative
solution of the Poisson's equation. We estimated accuracy of the
offered method and carried out test calculations which showed that
this method gives the accuracy several times better than accuracy
of the fast Fourier transformation.

A survey of results obtained within the project 0AAAA-A16-116070810025-5 and the recent joint project with Indian algebraists S.Chakrabarti, S. Gangopahyay, S. Pal and also with Russian participants V.T. Markov, A.E. Pankratiev.

The aim of projects is a study of algebraic properties of finite polynomially complete quasigroups, the problem of their recognition from its Latin square and constructions of polynomially complete quasigroups of sufficiently large order.
We are also interested in poly nomially complete quasigroups with no subquasigroups.
There are found sufficient conditions of polynomial completeness of a quasigroups $Q$ in terms of a group $G(Q)$. For example it suffices if $G(Q)$ acts doubly transitive in $Q$.
There is found a behaviour of $G(Q)$ under isotopies.

It is shown that any finite quasigroup can be embedded into a polynomial complete one. The results are applied for securing an information.

The paper continues consideration of a new class of the Dirichlet --- Zeta function of monoids of natural numbers. The main task solved in this paper is to construct a monoid of natural numbers for which the Zeta function of this monoid has a given abscissa of absolute convergence.

Previously, the author solved a similar problem of constructing a set of natural numbers for which the corresponding Zeta function has a given abscissa of absolute convergence.

To solve the problem for the Zeta function of the monoid of natural numbers there are certain difficulties associated with the need to build a sequence of primes that meet certain requirements for the growth of terms.

The notion $\sigma$"=sequences $\mathbb{P}_\sigma$ of primes was introduced, whose terms satisfy the inequality $n^\sigma\le p_n<(n+1)^\sigma.$

With the help of a theorem of Ingham with a cubic growth of Prime numbers was able to build a $\sigma$"=a sequence of primes for any $\sigma\ge3$. For the corresponding Zeta function of a monoid generated by a given $\sigma$ " =sequence of primes, the abscissa of absolute convergence is $\frac{1}{\sigma}$. Thus, with the help of Ingam's theorem it was possible to solve the problem for the abscissa values of absolute convergence from 0 to $\frac{1}{3}$. For such monoids it is possible to obtain an asymptotic formula for the Prime number distribution function $\pi_{\mathbb{P}_\sigma}(x)$: $\pi_{\mathbb{P}_\sigma}(x)=x^{\frac{1}{\sigma}}+\theta(x)$, where $-2<\theta(x)<-1$.

To prove the existence of a monoid of natural numbers, for whose Zeta function the abscissa value of absolute convergence is from $\frac{1}{3}$ to 1, it was necessary to use Rosser's Prime number theorem. For this purpose, the concept $\sigma$"=sequences of the second kind was introduced.

In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

The paper examines the Linux process tree data structure, which arises from the hierarchical scheme of processes generation in Unix-like operating systems. The purpose of study is to highlight the properties of the Linux process trees, which allow to conclude the applicable methods for analyzing such trees, in aim to solve the checkpoint-restore problem of executable environments in Unix-like operating systems.

The inverse discrete problem of restoring chains of system calls that generate a certain tree of processes is formulated, as well as a number of restrictions on the form of the system call and an assertion about the existence of a solution that concludes the correctness of the input.

Combinatorial estimation of total trees number, which are provided by fork system call, is presented, and correction is noted for the discernibility of identifiers. The feasibility of indexing by nodes is substantiated, due to the formation of non-root identifiers of the symmetric group. Thus, the functional equivalence of automorphic trees with permutations of non-root identifiers is proved. A combinatorial explosion of the functionally different trees number is shown by the procedure of adding a new system call. In view of the above estimations, a conclusion about the ineffectiveness of process trees restoring by bruteforce or direct search is drawn. The idea of constructing restoring mathematical formalisms that take into account the structure of the problem is proposed.

Next, the inheritance property of attributes in the process trees is examined, which allows to localize a required attribute when checking the applicability of a system call rule, thus reducing the number of checks. The segmentation property of the Linux process trees is provided. On the basis of the above properties, the conclusion is formulated on the goal of constructing a solution of restoring the syscall chains, which constructing a certain process tree, on the basis of the theory of formal languages and grammars, using formalisms of the class of mildly-context-sensitive. The alternative methods of solution are reviewed too.

In the article it is proved that a given real number $N>N_0(\varepsilon)$ can be approached by the sum of squares of three primes by a distance not exceeding $H = N^{217/768 + \varepsilon}$ and can be approached by the sum of four squares of primes by a distance no greater than $H = N^{1519/9216 + \varepsilon}$, where $\varepsilon$ is an arbitrary positive number.

These results were obtained using the density technique developed by Yu.V. Linnik in the 1940s. The density technique is based on applying explicit formulas expressing sums over prime numbers with sums over nontrivial zeros of the Riemann zeta function and using density theorems that estimate the number of nontrivial zeros of the zeta function lying in the critical strip such that their real part is greater than some $\sigma$, $1> \sigma \geq 1/2$.

The results obtained in this paper are based on the application of modern density theorems obtained by A. Ivich. In addition, the proof used the theorem of Baker, Harman, and Pintz: one can approach a given real number $N>N_0(\varepsilon)$ by a prime number by a distance no more than $H = N^{21/40 + \varepsilon}$. Also, the following result obtained by the author is used: one can approach a given real number $N>N_0(\varepsilon)$ by the sum of squares of two prime numbers by a distance no greater than $H = N^{31/64 + \varepsilon}$.

Functional gradient pyromaterials are found
wide application in the creation of various diagnostic instruments. For
correct calculation of devices using pyroeffect, you need knowledge
material characteristics. In the case of inhomogeneous pre-stressed
bodies, direct measurements of material characteristics are impossible, since they
represent some functions of the coordinates. Finding characteristics
heterogeneous pyromaterials is possible only on the basis of apparatus
coefficient inverse problems of thermoelectroelasticity (KOZT), which
practically not developed. The paper presents the formulation of the inverse problem
thermoelectroelasticity for prestressed
functional gradient rod. For this, based on the approach,
proposed by Guzem AN, the equations of thermoelectroelasticity for
prestressed rod. The problem is dimensioned.
A weak formulation of the direct problem of thermoelectroelasticity is obtained. Based
weak formulation and the linearization method, the operator equations for
solution of the inverse problem on the basis of the iterative process. During the course of
Iteration process correction to recoverable characteristics
thermoelectroelastic rod were determined from the solution of integral equations
Fredholm of the first kind. A direct problem was solved on the basis of the method of reduction to
system of Fredholm integral equations of the second kind in transformants in
Laplace and the use of the treatment procedure implemented in accordance with
The theory of residues A series of computational experiments on
The restoration of characteristics, the change of which has an essential
influence on additional information. In computational experiments
one of the characteristics of a thermoelectroelastic rod under
known others. Practical recommendations for choosing the most
informative time intervals for measuring the input information.
It was found that the appearance of initial stresses significantly affects
results of reconstruction of the characteristics of the rod.

In paper the problem of diffraction of a plane sound wave by a
homogeneous elastic sphere with radially non-uniform elastic coating
located near a plane. It is necessary that the body is placed in an
ideal fluid, the spreading flat surface is absolutely rigid and
absolutely soft, heterogeneity laws of a coating material are
described by continuous functions.

The problem is replaced by a problem of diffraction on two bodies.
According to a method of imaginary radiants the dividing boundary of
mediums is substituted by with mirrorly mapped imaginary sphere
which is situated in the field of two plane waves. The analytical
solution of the problem of diffraction of a plane sound wave by two
identical homogeneous elastic spheres with radially non-uniform
coatings situated in an ideal unlimited fluid is received. For
solution of the problem the addition theorem for spherical wave
functions is used. Analytic expressions In the form of decomposition
on spherical functions are obtained which describe the wave fields
in the containing medium and the homogeneous elastic bodies. The
boundary-value problem for the system of ordinary differential
equations of the second order is constructed for determination of
the displacement fields in non-uniform coatings. On the basis of
solution of problem of diffraction a plane wave by two bodies the
diffraction problem for case of scattering of second plane wave is
received. By summation of results of solutions of two diffraction
problems the analytical solution of the problem of diffraction of a
plane sound wave by a elastic sphere with coating located near a
plane is received.

By means of an continuous-non-uniform elastic coatings it is
possible to change effectively scattering performances of bodies in
determinate directions if to pick up corresponding the inhomogeneity
laws for mechanical parametres of a coating.

For the early {\it Pythagoreans}, in perfect agreement with their
philosophical-mathematical thought, given segments $s$ and $t$ there
was a segment $u$ contained exactly $n$ times in $s$ and $m$ times
in $t$, for some suitable integers $n$ and $m$. In the sequel, the
{\it Pythagorean} system is been put in crisis by their own
discovery of the incommensurability of the {\it side} and {\it
diagonal} of a {\it regular pentagon}. This fundamental historical
discovery, glory of the {\it Pythagorean School}, did however ``{\it
forget}'' the research phase that preceded their achievement. This
phase, started with numerous attempts, all failed, to find the
desired common measure and culminated with the very famous odd even
argument, is precisely the object of our ``{\it creative
interpretation}'' of the {\it Pythagorean} research that we present
in this paper: the link between the {\it Pythagorean identity}
$b(b+a)-a^2=0$ concerning the {\it side} $b$ and the {\it diagonal}
$a$ of a {\it regular pentagon} and the {\it Cassini identity}
$F_{i}F_{i+2}-F_{i+1}^2=(-1)^{i}$, concerning three consecutive {\it
Fibonacci numbers}, is very strong. Moreover, the two just mentioned
equations were ``{\it almost simultaneously}'' discovered by the
{\it Pythagorean School} and consequently {\it Fibonacci numbers}
and {\it Cassini identity} are of {\it Pythagorean origin}. There
are no historical documents (so rare for that period!) concerning
our audacious thesis, but we present solid mathematical arguments
that support it. These arguments provide in any case a new (and
natural!) characterization of the Fibonacci numbers, until now
absent in literature