of National Academy of Sciences
of Tajikistan

Reports of the National Academy of Sciences
of Tajikistan

Доклады Национальной академии наук Таджикистана, том 64, №11-12, 2021 г.

Article title and abstractPages
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Mean value of short cubic exponential G.Weyl sums with prime numbers
For the integral of the eighth degree of the modulus of a short cubic exponential sum of G.Weyl with prime numbers, the correct estimate is found in order. The result obtained is a generalization of the Hua Lo-ken theorem for short cubic exponential sums of G. Weyl with prime numbers.

Key words: short exponential sum with primes; diophantine equation; exponential integral.

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A.A.Sobirov ,
Short linear exponential sums with primes in a neighborhood of the center of major arcs
At , in neighborhood , of the center of major arcs, using the second moment of Dirichlet L-series, an asymptotic formula of the form is proved.

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

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S.I.Isroilov , S.M.Aslonov
Algorithmic solvability of the problem of inequality for one class of linearly ordered semigroups
The article studies the conditions for the algorithmic solvability of the inequality problem for one class of naturally linearly ordered semigroups. For the class of Archimedean finitely semigroups - the conditions of the algorithmic solvability of the natural order problem are defined. Using Archimedean semigroups, the problem of the algorithmic solvability of this problem for the class of finitely defined linearly ordered semigroups was solved.

Key words: semigroups; predicat; problem; algorithmic; decidability; definitely; finitely approximable; linear order.

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G.A.Yusupov , M.A.Mavlonazarov
Exact values of the widths of some functional classes in L
In this paper, exact inequalities between the best polynomial approximations and the averaged values of the generalized moduli of continuity of functions in the norm of the space L are found. For some classes of functions defined by the indicated generalized moduli of continuity, the exact values of the series of -widths are calculated.

Key words: best polynomial approximation; generalized modulus of continuity; Fourier series; n-widths.

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On the best joint polynomial approximation of functions and their derivatives in Bergman space
In the article, we solve some extremal problems associated with the best joint polynomial approximation of a function and their derivatives in the Bergman space.A number of theorems are obtained in which the exact value of the upper bounds of the best joint approximations on the classes of functions is found.

Key words: best joint approximation; analytic function; modulus of continuity; extremal problems.

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Of the stability of a stationary solution of a nonlinear system of differential equations
In this article the question of the asymptotic stability of the stationary solution of a system of three nonlinear differential equations describing the interaction of the primary processes of photosynthesis with the system of metabolic reactions in autotrophic systems is considered.

Key words: nonlinear differential equation; stationary solution; stability; asymptotic stability; Jacobi matrix.

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Effect of temperature dependence of transfer coefficient on the characteristics of the combustion wave of a methane-air mixture in an inert porous medium
In this article the effect of the temperature dependence of the transfer coefficient on the characteristics of the stationary wave of filtration combustion methane-air mixture is investigated. The characteristics of the wave are compared taking into account the temperature dependence of the transfer coefficients and without it. It is revealed that for a poor methane-air mixture, the wave velocities are relatively higher at the temperature dependence of the transfer coefficients than at their constantly values. For a stoichiometric mixture, the effect is expressed in the form of a change in the mode of propagation of combustion wave from counter to passing direction.

Key words: combustion wave; combustion velocity; burning temperature; methane -air mixture; heat capacity; thermal conductivity; diffusion coefficient.

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Definition of uniformity of originals and translation of text
A model collection of texts in Tajik and Russian is considered, composed of 12 text by 3 authors. Each text is associated with a digital portrait characterizing the distribution of the frequency of the letter unigrams in the texts. The effectiveness of applying the γ-classifier for identifying uniformity of originals and translations of text is established.

Key words: Russian language; Tajik language; text; frequency; classifier; identification.

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G.I.Kokhirova , F.Dzh.Rakhmatullaeva , M.Husarik , О.V.Ivanova , S.A.Borysenko
Quasisynchronous observations of asteroid (6478) Gault in Tajikistan and Slovakia
In this paper the results of quasi-synchronous optical observations of the asteroid (6478) Gault, carried out on the Zeiss-1000 telescope of the Sanglokh Observatory Institute of Astrophysics NAST and the 1.3-m and 0.61-m telescopes of the Skalnato Pleso observatory of the Astronomical Institute of the Slovak Academy of Sciences in August-October 2020 are presented. The object episodically shows signs of cometary activity and therefore is classified as an active asteroid in the Main Belt. The apparent and absolute brightness of the asteroid in the BVR bands was determined, based on the color indices the mineralogical composition of the asteroid was suggested. According to our estimates, the average diameter of the asteroid is 2.8 km. During the observation period the activity of the asteroid was not detected.

Key words: asteroid; photometry; brightness; diameter.

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F.M.Shokiriyon , A.A.Ibragimov
Influence of precession and proper motion of stars on the accuracy of cometocentric coordinates
In this paper, the cometocentric coordinates of the points of the plasma tail of comet C/2019 Y4 (ATLAS) taking into account the influence of precession and the proper motion of stars are determined. The calculated relative errors, show an error of less than 10%.

Key words: precession; cometocentric coordinate; stellar proper motion; accuracy.

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U.J.Rakhimova , M.Kh.Egamov
Modifications of the structure of polymer-liquid crystal films for the production of polarizers
Changes in the boundary conditions, orientation structure and anisotropy of light transmission of polymer-liquid crystal composites modified with a surfactant were studied, depending on the composition of the composite and the degree of stretching of the samples. The optical characteristics of the films (degrees of polarization, transmission and absorption, optical density and dichroism) of the film are determined with a perpendicular and parallel arrangement of the electric vector of light radiation incident on the sample and the orientation axis of the film. It was established that the polarization characteristics of uniaxially elongated films will be higher the higher the degree of orientation of macromolecules, and therefore the higher the dichroic ratio.

Key words: polymers; liquid crystals; anisotropy of light transmission; uniaxial deformation; absorption and transmission spectra; polarizer.

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S.S.Bobosaidzoda , A.S.Sodatdinova , S.M.Safarmamadzoda , N.N.Niyozov
Stability of silver complexes with 2-mercaptoimidazoles
The pH-metric titration method was used to study the acid-base equilibria of 2-mercaptoimidazole (2MI) in aqueous solutions at 298 and 308 K. It is established that 2MI belongs to very weak bases and its pKa decreases with increasing temperature. By the method of potentiometric titration, it is shown that silver (I) with 2MI in aqueous solutions with an ionic strength of 0.1-0.2 mol/l created by NaClO forms three complex forms in the range of 273-318 K. The stability constants of the complexes are calculated.

Key words: silver; 2-mercaptoimidazole; thermodynamics; stability constants; complex formation.

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S.G.Yormamadova , I.U.Kamolova , U.Radzhabov
Complex formation of iron(II) with mercasolil at 298 K and ionic force 1.0 mol/l
The process of complexation of iron (II) with mercosalyl at a temperature of 298 K and an ion layer of 1.0 mol/l by an oxo-redmetric method was studied. The composition, the regions of dominance of coordination compounds of zinc with mercosalyl, were determined by the method of oxedmetry, and the constants of formation were determined using the oxidation function.

Key words: iron; mercosalyl; complexation; constant stability; ionic strength.

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M.B.Sharipova , M.B.Ikrami , O.S.Karimov , B.A.Solieva
Enzymes of flour from sprouted wheat
The article considers the results of studying the amylolytic enzymes of flour from sprouted wheat grain. It is shown that the content of amylases in this flour significantly exceeds that in flour from non-sprouted wheat. The effect of the pH of the medium and the storage time on the state of amylases is also shown.

Key words: bread; wheat germ; flour; amylases; pH of the medium; food products.

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N.B.Odinaeva , I.N.Ganiev , Z.R.Obidov , M.J.Isobaev , F.R.Safarova
High-temperature oxidation of Zn+0.5%Al alloy, doped with indium, in the firm condition
In this paper the thermo gravimetrical method interaction of Zn+0.5%Al alloy, doped with indium, with air oxygen in an interval to temperature 473-623 K in the firm condition are investigated. Kinetic parameters of process of high-temperature oxidation of alloys are defined. Showed that additives of an alloying component slightly increase oxidability of alloy Zn+0.5%Al.

Key words: Zn+0.5%Al alloy; doped; indium; thermo gravimetrical a method; high-temperature oxidation; speed of oxidation; energy of activation.

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Standardization of the ointment of bromine thiadiazole pyrimidine 1%
In this paper the results of standardization of the ointment of bromine thiadiazole pyrimidine 1% are presented. Threefold extraction with water for 10 minutes provides almost complete extraction of 2-bromo-7-methyl-5-oxo-5H-1,3,4-thiadiazolo[3,2-a]pyrimidine from the ointment base. The method of quantitative determination of the active substance in the ointment meets the necessary requirements for linearity, repeatability, convergence and correctness.

Key words: ointment; 2-bromo-7-methyl-5-oxo-5H-1,3,4-thiadiazolo[3,2-a]pyrimidine; standardization.