The article is devoted to the doctor of physico-mathematical Sciences, academician of the
Academy of Sciences of the Republic of Tajikistan, one of the foremost experts in the field
of number theory, Zarullo Husenovich Rakhmonov in connection with his 60–year anniversary.
Provides a brief biography, the main stages of development of his scientific career. We give
the review of results of Z. H. Rakhmonov on following problems: on the distribution of the
Goldbach’s and Hardy–Littlewood’s numbers in short arithmetical progressions, on the problem
of mean values of the Chebyshev’s function and the problem of the Riemann zeta-function zeros
belonging to short rectangular in the critical strip, to estimations short trigonometric sums over
primes and on the Goldbach’s problem with almost equals summands, on the Selberg’s problem
concerning to the Riemann’s zeta-function zeros lying on short intevals of the critical line. In
conclusion, the author presents a list of main scientific publications Z. H. Rakhmonov

Let X be enough big real number and $k\geq2$ be a natural number, M be a set of natural numbers n not exceeding X, which cannot be written as a sum of prime and fixed degree a prime, $E_k (X)=card M.$ In present paper is proved theorem.

Theorem. For it is enough greater $X-$equitable estimation $ E_k (X)\ll X^{\gamma},$ where
$$ \gamma<\left\{
\begin{array}{lll}
1-(17612,983k^2 (\ln k+6,5452))^{-1}, & \text{при} & 2\leq k\leq 205,\\[1mm]
1-(68k^3 (2\ln k+\ln\ln k+2,8))^{-1}, & \text{при} & k>205,\\[1mm]
1-(137k^3 \ln k)^{-1}, & \text{при} & k>e^{628}.
\end{array}\right.
$$}

In particular from this theorems follows that estimation $$\gamma<1-(137k^3 \ln k)^{-1},$$ got by V. A. Plaksin for it is enough greater k,
remains to be equitable under $\ln k>628$.

In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I. M. Vinogradov's method. Here we obtain for $$n=r=2$$ lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for $$P\to\infty$$ for the number of solutions of the following system of Diophantine equations
$$
\sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}\dots x_{r,j}^{t_r}=0,\quad 0\leq t_1,\dots, t_r\leq n,
$$
where $$n\geq 2,r\geq 1, k$$ are natural numbers, moreover an each variable $$x_{i,j}$$ can take
all integer values from 1 to $$P\geq 1.$$

An evaluation of irrationality measure for various transcendental numbers is one of the field in diophantine approximation theory.
Starting with the works of Е. Borel at the end of 19th century, were developed both general methods of evaluation
for classes of some functions values and specialized approaches for estimating peculiar numbers.
Diverse methods particularly were practiced for the investigating of arithmetic properties of the function $$\arctg x$$ values.

For getting evaluation on irrationality measure of $$\arctg x $$ values many authors regarded them as particular case of Gauss hypergeometric function.
One of the first such kind of papers was the article of M. Huttner 1987, who proved a generalized theorem
about estimation on irrationality measure of the Gauss hypergeometric function values $$F_2^1\left(1,\frac{1}{k},1+\frac{1}{k}|\varepsilon x^k\right), k\in\mathbb N, k\geq 2, \varepsilon=\pm 1.$$
A big role in progress of theme have been played by works of A. Heimonen, T. Matala"=aho, K. V\"{a}\"{a}n\"{a}nen,
in which was also constructed a method for evaluation on irrationality measure of the Gauss hypergeometric function values of the form $$F_2^1\left(1,\frac{1}{2},1+\frac{1}{2}|z\right), k\in \mathbb N, k\geq 2,$$
including $$F_2^1\left(1,\frac{1}{2},\frac{3}{2}|-z^2\right)=\frac {1}{z}\arctg z.$$ The approach considered by them had used approximation of the Gauss hypergeometric function by Jacobi type polynomials and
gave a lot of concrete results.

Last decades for evaluation of various numbers were broadly spreading methods, which used symmetric on some changes of variable integrals.
Originally, integral qualitatively using the property of symmetry was applied by V.Kh.Salikhov, who used it to got the new estimate for $$\ln 3.$$
A little later V. Kh. Salikhov had applied similar symmetrized complex integral for obtaining new evaluation of $$\pi.$$
In that work he put to use classical equality $$\frac{\pi}{4}=\arctg \frac{1}{2}+\arctg \frac{1}{3}.$$
The same method, i.e. complex symmetrized integral was used by E. B. Tomashevskaya, who had estimated values of $$\arctg \frac{1}{n}, n\in\mathbb N, n>2$$
and some of previous results for such numbers were improved by her. Later on E. B. Tomashevskaya had elaborated analogical integral for estimation of $$\arctg\frac{1}{2},$$
which one had allowed to prove the best result until now $$\mu(\arctg \frac{1}{2})\leq 11.7116....$$

In 2014 K. Wu and L. Vang improved the result of V. Kh. Salikhov for $$\ln 3,$$ applying a new type integral construction, which also had used a property of symmetry.
In present paper we took the idea of K. Wu and L. Vang and applied it to the integral of E. B. Tomashevskaya. It allowed us to improve arithmetic properties of integral
and obtain better result for extent of irrationality $$\arctg\frac{1}{2}.$$

The article considers linear ordinary differential equations of the second order with variable
coefficients (initial equations). Along with each initial equation the same equation is considered
only with constant coefficients (accompanying equation). It is shown that the general solution
of the initial equation is represented in the integral form through the general solution of the
accompanying equation and the fundamental solution of the original equation. The fundamental
solution is the perturbation method in the form of an infinite rows. Research is carried out on
the convergence of rows. As a concrete example of the application of the developed methodology
is considered the Chebyshev equation.

The paper discusses a set of states of Linux processes as data structure, which is used in the
task of process-tree reconstruction in Unix-like operating systems. The purpose of the study
is to analyze dependencies in such structure, to introduce the natural order of dependencies,
to propose the class of such reconstruction structure as upper complete semilattice. Following
from the technical properties of the applied problem attributes’ hierarchy allow to introduce
additional restrictions on the minimum upper bounds in such semilattice.
Constraints are formally described as suitable pre-closure and closure operators. The
constraints implies the necessary condition for the correctness of the process tree. Based on
the properties of points returned by the proposed operators and system call execution scheme,
a sufficient condition for correctness is given. The introduced conditions form the criterion for
process-tree correctness, which can be useful in such tasks as generating tests for checkpointrestore in Unix-like operating systems, anomalies detection, increasing portability and reliability
of software. Dependency schemes between attributes that impose particular constraints on the
reconstructing set are also shown. Opened questions are also highlighted and further steps are
suggested.

The main difficulty one has to deal with while investigating arithmetic nature of the values
of the generalized hypergeometric functions with irrational parameters consists in the fact that
the least common denominator of several first coefficients of the corresponding power series
increases too fast with the growth of their number. The last circumstance makes it impossible
to apply known in the theory of transcendental numbers Siegel’s method for carrying out the
above mentioned investigation. The application of this method implies usage of pigeon-hole
principle for the construction of a functional linear approximating form. This construction
is the first step in a long and complicated reasoning that leads ultimately to the required
arithmetic result. The attempts to apply pigeon-hole principle in case of functions with irrational
parameters encounters insurmountable obstacles because of the aforementioned fast growth of
the least common denominator of the coefficients of the corresponding Taylor series. Owing to
this difficulty one usually applies effective construction of the linear approximating form (or a
system of such forms in case of simultaneous approximations) for the functions with irrational
parameters. The effectively constructed form contains polynomials with algebraic coefficients
and it is necessary for further reasoning to obtain a satisfactory upper estimate of the modulus
of the least common denominator of these coefficients. The known estimates of this type should
be in some cases improved. This improvement is carried out by means of the theory of divisibility
in quadratic fields. Some facts concerning the distribution of the prime numbers in arithmetic
progression are also made use of.
In the present paper we consider one of the versions of effective construction of the
simultaneous approximations for the hypergeometric function of the general type and its
derivatives. The least common denominator of the coefficients of the polynomials included
in these approximations is estimated subsequently by means of the improved variant of the
corresponding lemma. All this makes it possible to obtain a new result concerning the arithmetic
values of the aforesaid function at a nonzero point of small modulus from some imaginary
quadratic field.

The work focuses on obtaining coercive estimates and separability theorems of second-order nonlinear differential operators. Based on the obtained coercive estimates, the coercive solvability of the second-order nonlinear differential equations in the space $$L_{2,\rho}(R^n)$$ is investigated. For the first time the problem of the differential operators separability was dealt with by the English mathematicians V.N.Everitt and M. Girz. They studied in details the separability of the Sturm-Liouville operator and its degrees. Further development of this theory belongs to K.H.Boimatov, M.Otelbayev and their students. The main part of the published works on this theory applies to linear operators. There are only individual works that consider nonlinear differential operators, which are a weak nonlinear perturbations of linear operators. The case where the operator under study is strictly nonlinear, that is, it cannot be represented as a weak perturbation of the linear operator, is considered only in some individual separate works. The results obtained in this work also refer to this insufficiently studied case. The paper examined the coercive properties of a second-order nonlinear differential operator in the Hilbert space $$L_{2,\rho}(R^n)$$
$$
L[u]=-\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum_{j=1}^n b_{j}(x)\frac{\partial u}{\partial x_j}+V(x,u)u(x),
$$
and on the basis of coercive estimates, its separability in this space has been proved. The operator under study is not a weak perturbation of the linear operator, i.e. is strictly nonlinear. Based on obtained coercive estimates and separability, solvability of nonlinear differential equation in the space $$L_{2,\rho}(R^n)$$ is investigated.

The article investigates the full utility of economic activity in a finite period of time in the
case when the utility function admits with high accuracy the approximation of a linear function.
The estimation of the best approximation of the utility function on the segment with a given
ratio of the ends of the linear function is given.

We study the full utility of economic activity in a certain model, when the investment in
the production of an economic resource is given as an exponent and the utility function —
logarithm. We prove the existence and uniqueness of the optimal exponent and find the interval
in which the optimal exponent is contained.

Over the past decades, the theory of functional differential inclusions, primarily, the delayed
functional differential inclusion, has received significant development. Scientists from different
countries conduct research in the theory of initial-boundary value problems for various classes of
differential, integro-differential and functional differential inclusions in partial derivatives with
integer and fractional orders of derivatives.
The present work is devoted to fractional functional-differential and integro-differential
inclusions of Hale type, which occupy an intermediate place between functional-differential
inclusions with delay and inclusions of a neutral type. Sufficient conditions for the existence of
weak solutions of inclusions of Hale type with fractional order of the derivative are established.
The methods of fractional integro-differential calculus and the theory of fixed points of
multivalued mappings are the basis of this study. It is known that the dynamics of economic,
social, and ecological macrosystems is a multi-valued dynamic process, and fractional differential
and integro-differential inclusions are natural models of macrosystem dynamics. Such inclusions
are also used to describe some physical and mechanical systems with hysteresis. At the end of
the paper, an example illustrates abstract results.

The problem of the periodicity of functional continued fractions of elements of a hyperelliptic field is closely related to the problem of finding and constructing fundamental S-units of a hyperelliptic field and the torsion problem in the Jacobian of the corresponding hyperelliptic curve. For elliptic curves over a field of rational numbers, the torsion problem was solved by B. Mazur in 1978. For hyperelliptic curves of genus 2 and higher over the field of rational numbers, the above three problems remain open.
The theory of functional continued fractions has become a powerful arithmetic tool for studying these problems. In addition, tasks arising in the theory of functional continued fractions have their own interest. Sometimes these tasks have analogues in the numerical case, but tasks that are significantly different from the numerical case are especially interesting. One such problem is the problem of estimating from above the lengths of periods of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers.
In this article, we find upper bounds on the period lengths for key elements of a hyperelliptic field over a field of rational numbers. In the case when the hyperelliptic field is defined by an odd degree polynomial, the period length of the elements under consideration is either infinite or does not exceed twice the degree of the fundamental S-unit. A more interesting and complicated case is when a hyperelliptic field is defined by a polynomial of even degree.
In 2019, V. P. Platonov and G. V. Fedorov for hyperelliptic fields
$$L = \mathbb{Q}(x)(\sqrt{f}), \deg f = 2g + 2,$$ found the exact interval values
$$s \in \mathbb{Z}$$ such that continued fractions of elements of the form
$$\sqrt{f}/h^s \in L \setminus \mathbb{Q}(x)$$ are periodic.
Using this result in this article, we find exact upper bounds on the period lengths of functional continued fractions of elements of a hyperelliptic field over a field of rational numbers, depending only on the genus of the hyperelliptic field and the order of the torsion group of the Jacobian of the corresponding hyperelliptic curve.

The paper is consists from two parts. In first part summarizes the
review of findings on best approximation of periodic functions by
trigonometric polynomials in Hilbert space $$L_{2}:=L_{2}[0,2\pi].$$
The sharp inequalities between the best approximation and averaged
with given weights modulus of continuity of mth order values rth
derivatives of functions and analogues for some modified modulus of
continuity presented.

In second part, some new sharp Jackson-Stechkin type inequalities
for characteristics of smoothness studied by K. V. Runovski and more
detail by S. B. Vakarchuk and V. I. Zabutnaya are proposed. The sharp
result on joint approximation of function and successive derivatives
for some classes of functions defined by modulus of smoothness
obtained.

The definition of saturation condition was formulated at the end of the last century.
Saturation condition has become useful in study of infinite groups. A description of various
classes of infinite groups with various variants of saturating sets was obtained. In particular,
it was found that periodic groups with a saturating set consisting of finite simple non-Abelian
groups of Lie type, under the condition that ranks of groups in saturation set are bounded in
the aggregate, are precisely locally finite groups of Lie type over a suitable locally finite field.
A natural step in further research was the rejection of the periodicity condition for the group
under study, and the rejection of the structure of the saturating set as a set consisting of finite
simple non-Abelian groups of Lie type with ranks bounded in the aggregate. In this paper,
we consider mixed Shunkov groups (i.e., groups that contain both elements of finite order and
elements of infinite order).
It is well known that the Shunkov group does not have to have a periodic part (i.e., the set of
elements of finite order in the Shunkov group is not necessarily a group). As a saturating set, we
consider the set of full linear groups of degree 2 over finite fields of even characteristic. The lack
of analogues of known results V. D. Mazurova on periodic groups with Abelian centralizers of
involutions for a long time did not allow us to establish the structures of the Shunkov group with
the saturation set mentioned above. In this paper, this difficulty was overcome. It is proved that
a Shunkov group saturated with full linear groups of degree 2 is locally finite and isomorphic
to a full linear group of degree 2 over a suitable locally finite field of characteristic 2.

This paper consists of 3 sections. In the first section, we will give a brief introduction to the ''Feigin's homomorphisms'' and will see how they will help us to prove our main and fundamental theorems related to quantum Serre relations and screening operators.

In the second section, we will introduce Local integral of motions as the space of invariants of nilpotent
part of quantum affine Lie algebras and will find two and three-point invariants in the case of $U_q(\hat{sl_2}) $ by using Volkov's scheme.

In the third section, we will introduce lattice Virasoro algebras as the space of invariants of Borel part $U_q(B_{+})$ of $U_q(g)$ for simple Lie algebra $g$ and will find the set of generators of Lattice Virasoro algebra connected to $sl_2$ and $U_q(sl_2)$

And as a new result, we found the set of some generators of lattice Virasoro algebra.

The development of modern industry puts forward a responsible and complex task of
protecting the population, service personnel and the environment from accidents. The analysis
of possible deviations from normal operating conditions in these industries and a thorough
study of the possible development of various emergency situations that lead to dynamic
effects on structures and finding conditions for the destruction of structural elements is of
paramount importance. The article proposes a mathematical method for finding the conditions
of destruction of structural elements by dynamic loading. To solve dynamic problems, a
variational approach is used, based on the construction of a functional for calculating the power
of elastic deformation taking into account the power of inertia forces, in the context of using
modern software systems based on the finite element method. As an example, the problem of
computer modeling of the dynamic load located above the center of the reinforced concrete slab,
which allows to determine the stress-strain state of the simplest elements of building structures
of plates, is considered. All calculations were performed in ANSYSLS-DYNA environment. The
results are obtained in the form of graphs of strain rates and stress fields. The obtained results
are compared with the analytical solution of a similar problem presented in the work of G.T.
Volodin.

The paper describes the historical aspects of the study of light and electrical phenomena
that contribute to the emergence of the laser and the development of laser technology. The
principle of operation of the laser is presented, the main types and characteristics of lasers are
listed. The dependence of the radiation power on the laser wavelength is shown. It is told about
various fields of application of lasers. The list of modern scientific literature with technological
parameters of laser processing of various materials is given.

The article is devoted to the distinguished mathematician A.Ja.KHintchine in connection
with his 125–year anniversary and 60-year of the his death.
We give the comment of results of A. Ja. Khintchine in the number theory.

The authors set themselves the task: to describe the main stages of life of the scientist and
teacher of Tula state pedagogical University. L. N. Tolstoy Vladislav Ivanovich Rybakov and
give a brief analysis of his scientific work, which had a significant impact on the development
of functional analysis.
Special attention is paid to the research of V. I. Rybakov on the theory of measure and
integral.
Under his leadership, conducted scientific work of individual students, who later became
candidates of physical and mathematical Sciences. Vladislav Ivanovich Rybakov obtained deep,
meaningful scientific results. For example, "the classical theorem of Rybakov" can be read in
books and articles published in the international mathematical press.
Vladislav Ivanovich Rybakova was actively engaged in scientific activities until his death.
The article presents the results obtained by V. I. Rybakov in different periods of his life.