Доклады Национальной академии наук Таджикистана, том 64, №0304, 2021 г. 
File  Article title and abstract  Pages 


Download  Rakhmonov Z.Kh., Rahmonov D.D.  121128 
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 Lfunction.  

Download  Khayrulloev Sh.A.  129134 
Distance between adjacent zeros of the derivative of the jth 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 jth order derivative of the Hardy function is obtained. 

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

Download  Zulfonov Sh.M.  135141 
Application of the LaplaceCarson transform to the solution of the initialboundary 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 integrodifferential equations in partial derivatives. 

Key words: LaplaceCarson transform; initialboundary value problem; integrodifferential equation; Heaviside function.  

Download  Shabozov M.Sh., Quraishi A.S.  142149 
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 nonquadratic 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.  

Download  Yusupov G.A., Shabozova A.A.  150155 
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; JacksonStechkin inequalities; modulus of continuity; weight function; nwidths.  

Download  Raimzoda F.  156164 
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; mth order difference; joint approximation.  

Download  Pirov R., Jabborov M.A.  165170 
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 firstorder 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.  

Download  Kokhirova G.I., Ivanova О.V., Rakhmatullaeva F.Dzh., Borysenko S., Agnihotri V.K., Buriev A.M.  171182 
The disintegrations of comet Atlas by observations
in Tajikistan and India 
Longperiod 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 breakup 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 MarchApril 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 breakup 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.  

Download  Rahimi F., Pirov B.G.  183192 
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.  

Download  Abdulloev Kh.O., Solihov D.K., Rakhmonov S.S.  193197 
Solution of the nonlinear Schrödinger equation within
the easyaxis 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; easyaxis; selflocalization; wave packets.  

Download  Tuichiev Sh., Rashidov D., Tabarov S.Kh., Sodikov F.Kh.  198201 
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 Xray 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.  

Download  Mulloyev N.U., Majidov N.A., Lavrik N.L.  202206 
Quantumchemical calculation of the structure of imidazole dimers 
This article is devoted to the topical issue of identifying the mechanisms of complex organic reactions using quantumchemical 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 hydrogenbonded complex (dimer) of imidazole by the method of quantumchemical calculation using the electron density functional B3LYP / 631G 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: quantumchemical calculations; imidazole; medicinal compounds; dimers; molecular structure; heterocyclic compounds; amphoteric properties.  

Download  Nizomov J.N.  207218 
Boundary integral equations threedimensional
problems of elasticity theory 
The article presents issues related to the mathematical modeling of threedimensional 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.  

Download  Mirsaidov U.M., Barotov B.B., Boboev K.O., Mirsaidzoda I., Eshov J.N.  219223 
Uranium concentrates separation from brine, drainage and process waters 
The article presents the physicochemical bases of uranium concentrates separation from the SasykKul lake brine and from the Tajikistan technical and drainage waters. The basic technological scheme for uranium separation from uraniumcontaining waters has been proposed. 

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

Download  Sharifov A., Gaibullaeva Z.H., Nematzoda D.S., Goziev Z.A.  224231 
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.  

Download  Fayziev F.A., Yatimov S.B., Fayziev A.R.  232238 
On the formation of polymetal deposits of SouthWestern Karamazar (on the example of Konsoy and Takeli groups) 
The article discusses the genetic features of polymetallic deposits of SouthWest 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.  
