of National Academy of Sciences
of Tajikistan

Reports of the National Academy of Sciences
of Tajikistan

Доклады Национальной академии наук Таджикистана, том 65, №03-04, 2022 г.

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M.Sh.Shabozov , Z.А.Parvonaeva
About Jackson-type inequalities in L[-1,1] and solutions of some extremal problems
In the space L[-1,1] an exact Jackson-type inequality between E(f), the best approximation of a continuous function by the subspace - of algebraic polynomials of degree (-1) the generalized modulus of continuity of the first order ω(f ,t) of the derivative f Î L[-1,1] and introduced by A.A.Abilov was found. A similar inequality between E(f) and the -functional Petre is proved.

Key words: Jackson-type inequality; best mean-square approximation; Legendre polynomials; modulus of continuity.

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Yu.Kh.Khasanov , M.M.Makhamadieva
About the summation of Fourier series by Voronoi-Nerlund method
A measurable and Lebesgue integrable function f(x) belonging to the space L(1 ≤ p ≤ ∞) is considered. The question of the behavior of the Linear operator, or (W,p) - Voronoi averages, which are formed using partial sums of order k of the Fourier series of the function f(x), is investigated. A statement is proved in which, for each n>1, an upper bound of the value p(f;x;W) is established through a sequence of the best approximations of the function in a uniform metric.

Key words: periodic function; Fourier series; linear operator; Voronoi averages; regularity of summation method; trigonometric polynomials; Fourier coefficients; uniform metric.

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S.A.Iskhokov , K.E.Khaknazarov
On the finite dimensionality of the kernel of some non-divergece form degenerate elliptic operators
We prove an inequality which contains the norm of a non-divergence form elliptic operator in a bounded domain of the -dimensional Euclidean space that degenerates along the entire boundary of the domain. Then, with the help of this inequality, the finite-dimensionality of the kernel of the operators under investigation is proved. In contrast to previously published papers, the coefficients of the operators under investigation are not differentiable, and therefore they cannot be reduced to divergence form.

Key words: non-divergence form elliptic operator; power degeneration; bounded domain; a finite-dimensional kernel; closeness of a range of operator.

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I.Kurbonov , U.Bakhtovar
E-problem of electrodynamics for an isotropic suspension-linear and nonlinear medium with memory
The problems of propagation of electromagnetic waves in piecewise- inhomogeneous nonlinear media are considered. Substitutions of boundary nonlinear integro-differential problems are given to obtain systems of partial differential equations. Their approximate time-periodic solutions are sought by the method of equivalent linearization. A generalization of the Fresnel formulas of the E-problem of electrodynamics for media with memory properties is obtained. It is shown that for non-conducting media without memory . The is, the angles of incidence, reflection and refraction are equal.

Key words: piecewise linear medium; inhomogeneous medium; memory; linear and nonlinear medium; E-problems of electrodynamics.

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Sh.B.Khalilov , B.N.Rushanov
On the theory of boundary problems for multidimensional Petrovsky elliptic systems of equations of the second order
In the article, for one elliptic system of four second-order equations depending on four unknown functions, the Neumann problem in a half-space is considered. Using the Fourier transform method, it is proved that the problem posed is uniquely solvable and this solution is found in an explicit form.

Key words: elliptic system; Neumann problem; Fourier transform; strongly elliptic systems; homotopy classification; characteristic determinant; matrix differential operator.

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G.Kh.Khujanazarova , I.Z.Khonov
Basic boundary-value problems for elliptic systems of differential equations of the fourth order with displacement coefficients
The work has established effective necessary and sufficient Noetherianness conditions for elliptic systems of differential fourth-order equations with discontinuous coefficients and the formula for calculating the index is obtained.

Key words: basic boundary value problems; elliptic systems; singular integral operator; symbol; index; operator impatience.

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Thermal ignition of a mixture of gas in an inert porous medium
A two-temperature model of thermal ignition of a mixture of gases in an inert porous medium is investigated in the adiabatic formulation. An approximate solution of the problem is obtained.

Key words: wave; combustion; mixture; velocity; ignition; model; porous medium.

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In this article a possible connection between the activity of the Sun and the absolute brightness of comets of the families of Saturn and Jupiter are considered. The dependence of the brightness of some comets on the activity of the Sun has been revealed. It is shown that for comets of the Jupiter family this dependence is stronger than for comets of the Saturn family. This is due to the lack of observational data on comets of the Saturn family due to rarer encounters with the Sun compared to comets of the Jupiter family. Rare observations do not allow a full-fledged analysis.

Key words: comet; photometric parameter; absolute brightness; correlation; solar activity.

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A.M.Olimov , M.Subhoni , Tomoyuki Yamamoto , F.Rahimi
Effect of pressure on the geometric and elastic properties of CsPbX (X = Cl, Br, I)
In this paper the results of studying the dependence of unit cell parameters, elasticity constants, bulk moduli of elasticity, and other elasticity parameters of CsPbX (X = Cl, Br, I) perovskites on pressure in the range from 0 to 30 GPa with a step of 5 GPa, in the framework of the density functional theory are presents. It is determined that for all structures, the values of the bulk modulus, the average speed of sound, the transverse and longitudinal speeds of sound, as well as the Debye temperature increase with increasing pressure.

Key words: geometric properties; density functional theory; bond length; elasticity constants; average sound speed; transverse and longitudinal sound speeds; shear modulus; Debye temperature.

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S.Odinaev , D.A.Abdurasulov , A.A.Abdurasulov
Molecular statistical study of the orientation elastic properties of nematic liquid crystals
The orientational elastic properties of nematic liquid crystals are investigated on the basis of the locally equilibrium statistical distribution function generalized for the description of liquid systems with arbitrary shapes of molecules. Analytical expressions are obtained for the coefficients of orientation elasticity during deformation of transverse bending ( ) longitudinal bending ( ) and torsion ( ) In the framework of the approximation of the separability of the intermolecular interaction potential into radial and angular parts, using the Lennard-Jones and Mayer-Zaupe potentials as its radial and orientation parts, the dependence of the orientation elasticity coefficients ( ) is investigated from temperature, pressure and density for NLC (PAA).

Key words: statistical distribution function; nematic liquid crystals; orientation elasticity; radial and orientation order.

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An exact solution of the coupled equations in the theory of stimulated brillouin scattering using Riemann method
In this article we have obtained an exact solution of the coupled equations in the theory of stimulated Riemann scattering of electromagnetic waves in the field of a two-dimensionally localized pump wave for arbitrary scattering angles by the Riemann method. The spatial variation of the amplitudes of the interacting waves is studied for various values of the pump field and the sizes of its localization region.

Key words: stimulated Riemann scattering; stimulated Brillouin scattering; ion-acoustic wave.

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Таҷзияи кинетикии маъданҳои урандори Тоҷикистон
Дар мақола тавсифи кинетикии таҷзияи маъданҳои урандори конҳои «Тоҷикистони Марказӣ», «Тоҷикистони Ғарбӣ», «Тоҷикистони Шарқӣ» ва «Октябрское» бо кислотаи сулфат оварда шудааст.

Key words: кинетика; маъдани урандор; таҷзия; кислота; энергияи фаъолкунӣ.

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Phase equilibrium in the In-Te system
In this article the results of a study of the temperaturedependence of the speed of propagation of ultrasound in exfoliating melts of the In-Te system are presents. In the In-Te system, anomalies in the polytherms ϑs. It was found that these anomalies increase gradually as the concentration of melts approaches the critical one. The lines of monovariant liquid-liquid equilibrium and critical parameters have been refined.

Key words: temperature; phase; system; indium; tellurium; ultrasonic method; concentration.

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T.M.Umarovа , S.S.Holova , D.S.Kuchakshoev
Synthesis and physicochemical properties of Al-Te alloys alloyed by neodymium
The analysis of the available literature data on the aluminum-tellurium phase diagram, shows the results of studies of X-ray phase and metallographic analyzes of alloys of the Al-Te-Nd system and the prospect of their application are briefly presented.

Key words: state diagram; aluminum alloy; neodymium; aluminum telluride; diffractogram; microstructure.

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Modification of bitumen with polymer additives using secondary raw materials and by-products of the industry
In this article the effect of polymer additives from dynistyrene thermoplastic elastomer (DST), vegetable oil tar, gossypol resin (GS) and the residue from the regeneration of used lubricating oils (ORSM) on the physical and mechanical properties of bitumen are studied. It is shown that the use of the above-mentioned additives makes it possible to improve the physical and mechanical properties of bitumen.

Key words: polymer; oils; additive; gossypol resin; properties.

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F.M.Toshov , Sh.R.Samikhov , Sh.A.Kurbonov , N.Kh.Oyzoda , Kh.A.Abdullaev
Action of phosphate fertilizer, produced from phosphorites of the Rivat deposit on biological and economical tratts of midle staple cotton
In the study found that by adding superphosphate to the soil with a P0 content of 13-18% obtained from Rivat deposits, it positively affection on biological, economical and technological qualities of cotton fiber. It was found that as a result of the application of phosphate fertilizers obtained from Rivat deposits, increases the main stem (3/1 times), the total leaf surface area per plants (3.2 times), the number of formed fruit organs (3.7 times), total biological yield (2.7 times) with comparison to control plants of cotton.

Key words: cotton; phosphate rock raw materials; superphosphate; morpophysioloqical patterns; distribution of assimilates; biological and economic harvest; technological properties of cotton fiber.

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T.Sh.Avezov , A.Ergashev
Effect of seed treatment with physiologically active substances on the activity of antioxidant enzymes in wheat leaves under soil dry conditions
It was shown that the treatment of seeds before sowing with antioxidants- a-tocopherol (α-TOK), ascorbic acid and Zn-hydroxyethyl phenolic acid (Zn-ODEPK) under soil drought conditions increases the drought resistance of plants of soft and durum wheat varieties and affects the activity of antioxidant enzymes in the leaves.

Key words: wheat; antioxidants; soil drought; enzymes activity.

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A.K.Sayfudinov , M.A.Babazhanova
The kinetic behavior robotfactory multienzyme complex of Calvin cycle in extracts from leaves of cotton leaves
The results of kinetic studies robotfactory reactions in a multienzyme complex of Calvin cycle in extracts from cotton leaves of variety 108-F. The dependence robotfactory activity multienzyme complex of reaction time, protein concentration and ribozo-5-phosphate in the reaction medium. It is established that robotfactory refers to allosteric enzymes. The sigmoid form of the curve of the reaction rate dependence on the protein and substrate concentration indicates a positive cooperative interaction between the active centers of the enzyme subunits.

Key words: cotton; robotfactory; reaction; rate; kinetics; multienzyme complex; Calvin cycle.

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B.I.Ganiev , S.A.Avezov
Hyperbilirubinamia in the development of encephalopathy in liver cirrhosis
The importance of hyperbilirubinemia in the development of hepatic encephalopathy (HE) in patients with liver cirrhosis (LC) was studied. 27 patients of the viral etiology LC were examined (18 - chronic hepatitis B, 9 - chronic hepatitis C). The content of bilirubin in blood serum in patients with LC varies depending on the severity of the pathological process. In univariate analysis, the following measures were significantly (p≤0.05) associated with poor prognosis: HE, hyperbilirubinemia, hypercreatinemia, hypoalbuminemia, and reduced prothrombin time. Our studies confirm that bilirubin levels increase with irreversible impaired liver function. Increasing bilirubin over time helps reduce the likelihood of survival. On the base, the bilirubin content of patients with LC can be divided with a low or high risk of death from liver failure with the development of HE.

Key words: cirrhosis of liver; hepatic encephalopathy; hyperbilirubinemia; survival.