of National Academy of Sciences
of Tajikistan

Reports of the National Academy of Sciences
of Tajikistan

Доклады Национальной академии наук Таджикистана, том 64, №03-04, 2021 г.

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Rakhmonov Z.Kh., Rahmonov D.D.
Short exponential sums with primes in a small neighborhood of the center of major arcs
For a exponential sums of the form in the neighborhood of the center of major arcs when , , the asymptotic formula is proved.

Key words: short exponential sum with primes; major arcs; density theorem; Dirichlet L-function.

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Khayrulloev Sh.A.
Distance between adjacent zeros of the derivative of the j-th order of the Hardy function
In this paper, a new lower bound for the length of an interval of the critical line containing a zero of odd order of the j-th order derivative of the Hardy function is obtained.

Key words: Hardy function; exponential pair; the Riemann zeta function; critical line; trigonometric sum.

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Zulfonov Sh.M.
Application of the Laplace-Carson transform to the solution of the initial-boundary value problem for integro differential equations
In this paper, we investigate the question of the application of operational calculations in two variables to the solution of integro-differential equations in partial derivatives.

Key words: Laplace-Carson transform; initial-boundary value problem; integro-differential equation; Heaviside function.

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Shabozov M.Sh., Quraishi A.S.
On exact upper bounds of root mean square approximations some classes of periodic differentiable functions in space L
In the article, the exact values of the upper bounds of the best mean non-quadratic approximations of some classes of periodic of rendered functions by trigonometric polynomials, characterizing generalized modulus of continuity generated by a specific generalized shift operator in the space L:=[0,2π] are found.

Key words: best root mean square approximation functions; upper bounds; generalized modulus of continuity; generalization operator displacement.

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Yusupov G.A., Shabozova A.A.
Average squared polynomial approximation of periodic differentiated functions
In this paper, we obtain the exact constants in the Jackson - Stechkin inequalities and for the function in which the sequential derivatives are belong to the space the behavior of the value of the best joint approximations on the class is studied.

Key words: periodic function; best polynomial approximation; exact constant; Jackson-Stechkin inequalities; modulus of continuity; weight function; n-widths.

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Raimzoda F.
Exact inequalities containing best approximations and high order difference norms in L
The exact inequalities between the value of the best polynomial approximation of a periodic differentiable function in L and integrals containing the norms of the difference of higher orders of functions are found, and some applications of the results are given.

Key words: exact inequalities; best approximation; trigonometric polynomial; m-th order difference; joint approximation.

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Pirov R., Jabborov M.A.
On the study of some systems of partial differential equations with two unknown functions in spase
This paper considers a single class of redefined systems of first-order partial differential equations with two unknown functions in space. Explicit compatibility conditions are found, and the existence and uniqueness theorems of solutions are proved.

Key words: the condition of jointness; the system of total differential; overrides the system; the diversity of solutions.

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Kokhirova G.I., Ivanova О.V., Rakhmatullaeva F.Dzh., Borysenko S., Agnihotri V.K., Buriev A.M.
The disintegrations of comet Atlas by observations in Tajikistan and India
Long-period comet C/2019Y4 (Atlas) was discovered in the end of 2019, it has passed the orbits perihelion in 31 May 2010. Observations of the Hubble space telescope have registered a break-up of the comet nucleus into several fragments in the end of March 2020. To investigate the event observations of comet Atlas were carried out in the Cepheid observatory of India, International astronomical observatory Sanglokh and Hissar astronomical observatory of the Institute of astrophysics, NAST, during March-April 2020.The decreasing of the apparent and absolute brightness of comet in VRI bands is shown that confirms the nucleus disintegration occurred at this period. The estimate of the speed of the fragments scattering is several m/s, at such speeds the disintegration of the nucleus can be caused by a powerful ejection of gases, which led to the destruction of the mechanical bonds of the conglomerates of the cometary nucleus. The coordinates of comet were determined, the orbit was calculated, and it is shown that the nucleus break-up did not affect the stability of the orbit of the main component of the cometary nucleus.

Key words: comet; photometry; light curve; astrometry; nucleus; coordinates; orbit.

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Rahimi F., Pirov B.G.
Heisenberg magnets with spin S=3/2 in multidimensional space
This work presents a common problem in the study of ferromagnets with spin S = 3/2. In the semiclassical approach to the system the results obtained has given a complete description of magnetic materials and contribute to the study of the dynamic, kinetic and thermodynamic properties of magnetic systems and obtain their nonlinear localized solutions.

Key words: quantum; ferromagnets; coherent states.

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Abdulloev Kh.O., Solihov D.K., Rakhmonov S.S.
Solution of the nonlinear Schrödinger equation within the easy-axis Heisenberg model and the behavior of spin wave packets
In this work, the semi classical behavior of initial packets of spin waves of a more general form is investigated and conditions are found under which, depending on the initial state, packets either collapse or inflate.

Key words: spin waves; easy-axis; self-localization; wave packets.

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Tuichiev Sh., Rashidov D., Tabarov S.Kh., Sodikov F.Kh.
Influence of nanographenes on structure and some properties of polymers
The influence of nanografens on the structure, mechanical and thermal properties of LDPE and PMMA composites was studied by X-ray diffraction, mechanical and thermal tests. It is shown that the introduction of nanoparticles into polymers leads to a decrease in the mobility of the chain molecules of the matrix and, accordingly, to the change in the physical state of the amorphous regions and their transition from the highly elastic state to the glassy state.

Key words: polymer; nanographene; composite; structure; properties.

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Mulloyev N.U., Majidov N.A., Lavrik N.L.
Quantum-chemical calculation of the structure of imidazole dimers
This article is devoted to the topical issue of identifying the mechanisms of complex organic reactions using quantum-chemical calculations, which make it possible to calculate most of the characteristic parameters of medicinal compounds. According to the authors, this method is widely used in the interpretation of the obtained analytical and experimental results. The authors calculated the molecular structure of the hydrogen-bonded complex (dimer) of imidazole by the method of quantum-chemical calculation using the electron density functional B3LYP / 6-31G using the Gaussian software package. The work revealed the amphoteric properties of imidazole, since imidazole is both a donor and an acceptor of a proton with the possible formation of hydrogen bonds. The calculations performed by the authors can help in the interpretation of the IR spectra of heterocyclic azole compounds and in the calculation of the vibrational spectra of these compounds.

Key words: quantum-chemical calculations; imidazole; medicinal compounds; dimers; molecular structure; heterocyclic compounds; amphoteric properties.

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Nizomov J.N.
Boundary integral equations three-dimensional problems of elasticity theory
The article presents issues related to the mathematical modeling of three-dimensional problems of the theory of elasticity by the method of boundary integral equations. On the basis of the theorem on the reciprocity of works, an equation is obtained, which, upon passing to the limit to the boundary of the domain, allows one to obtain boundary integral equations of external and internal problems of mechanics of a deformable solid.

Key words: boundary equations; reciprocity theorem; external problem; internal problem; additional state; initial state; Einstein's rule; stress tensor; 2nd rank tensor.

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Mirsaidov U.M., Barotov B.B., Boboev K.O., Mirsaidzoda I., Eshov J.N.
Uranium concentrates separation from brine, drainage and process waters
The article presents the physicochemical bases of uranium concentrates separation from the Sasyk-Kul lake brine and from the Tajikistan technical and drainage waters. The basic technological scheme for uranium separation from uranium-containing waters has been proposed.

Key words: brine; uranium-containing waters; uranium concentrate; drainage and mine waters.

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Sharifov A., Gaibullaeva Z.H., Nematzoda D.S., Goziev Z.A.
Corrosion of hydraulic concrete and causes of its flow
The article is devoted to the corrosion processes of hydraulic concrete with clarifying the reasons for their flow. The concept of corrosion of concrete is given and its varieties are indicated. The composition of cements for the resistance of clinker minerals in aggressive environments was analyzed. It is indicated that the hydration of hydration of cement minerals hydrates is the most easily soluble in water, the Ca (OН) calcium hydration and on its content depends on the corrosion resistance of the hydrosilicates and other compounds of the concrete cement stone in the aggressive medium. The process of pore formation and formation of a structure of cement stone, contracting of clinker minerals during hydration was analyzed. The dependence of the coefficient of filtration of water through the structure of concrete from the parameters of its water absorption is given. The characteristics of aggressive medium for hydrotechnical concrete are indicated.

Key words: hydraulic concrete; concrete corrosion; aggressive medium; cement stone structure; mineral contracting; water filtration coefficient; strength; durability.

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Fayziev F.A., Yatimov S.B., Fayziev A.R.
On the formation of polymetal deposits of South-Western Karamazar (on the example of Konsoy and Takeli groups)
The article discusses the genetic features of polymetallic deposits of South-West Karamazor. Regardless of whether the mineralization is located in the skarn, it is found under hydrothermal conditions.

Key words: genesis; polymetallic deposits; Konsoy field; Takeli field; hydrothermal process; mineralization; rock.