of National Academy of Sciences
of Tajikistan

Reports of the National Academy of Sciences
of Tajikistan

Доклады Национальной академии наук Таджикистана, том 64, №05-06, 2021 г.

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S.A.Iskhokov , B.A.Rakhmonov
About isomorphism of some fractional order Sobolev type spaces
In the paper, we first study the interpolation properties of one type of normed spaces of differentiable functions in the entire Euclidean space with a power-law weight, and then, using an elliptic operator with degeneration, an isomorphism of some scale of such spaces is constructed.

Key words: Sobolev-type space of fractional order; interpolation of functional spaces; elliptic operator; power degeneration; an isomorphism of functional spaces.

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M.Sh.Shabozov , A.S.Quraishi
On the joint approximation of functions and its derivatives in the average on the real axis by algebraic polynoma with the weight of Chebyshev-Hermit
Sharp Jackson-Stechkin-type inequalities are obtained on the sets , in which the values of the best joint polynomial approximations are estimated from above by the generalized modulus of continuity of the m-th order.

Key words: joint approximation; Jackson-Stechkin type inequalities; upper bound; generic module continuity; K-functional.

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A.G.Aydarmamadov , N.U.Qadamshoev
On approximation of analytic functions in weight Bergmans space
In the weighted Bergman B space for classes of analytic in the unit circle function defined by generalized modules continuity of higher order -th derivatives on argument some inequalities between best approximation and medium meaning of modules continuity high order are praved.

Key words: analytical function; the generalized modulus of continuity of m-th order; best approximation; weight Bergman's space; complex algebraic polynomial; n-widths.

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A.A.Shabozova , G.Farzom
The best joint approximation of some classes of periodic functions determined by the module of continuity of the m-th order in L
In this paper, one extremal problem the best approximation of a function and their sequential derivatives for a class of functions given by the averaged value of the modulus of of continuity of the -th order of the highest derivative of the -th order fÎL and bounded from above by a given majorant are considered.

Key words: complex-valued polynomial; modulus of continuity; joint approximation; upper bounds; best joint polynomial approximation.

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Upper bounds of the best mean-square approximation of function of many variables by triangular Fourier sums over arbitrary orthogonal systems
In this article, the exact upper bounds of the best mean-square approximations of functions of many variables by partial Fourier sums over arbitrary orthogonal systems of functions on some classes of functions of many variables, characterized by a generalized modulus of continuity generated by the generalized shift operator are found.

Key words: Fourier series; orthogonal system; shift operator; generalized modulus of continuity; "triangular sums".

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L.N.Rajabova , F.M.Akhmadov
On some cases of two-dimensional integral equations of Volterra type with boundary singular and strongly-singular lines
Some cases of a two-dimensional integral equation of Volterra type with a singular and strongly - singular lines are studied, in which, depending on the roots of the characteristic equation and the sign of the parameters of the equation, explicit representations of the variety of solutions in terms of arbitrary functions are obtained.

Key words: two-dimensional integral equations; special line; strong-special line; characteristic equation.

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G.I.Kokhirova , F.Dzh.Rakhmatullaeva , S.A.Borysenko
Results of observations of comet P/2019 LD2 in the Sanglokh observatory
A short-period comet P/2019 LD2 (Atlas) was discovered in June 2019. The object was originally classified as a Trojan asteroid, but later included in the Jupiter family of comets. The origin of the object has not been established for certain and therefore the study of the comet continues to be of particular scientific interest. The optical observations of the comet were carried out in August 2020 at the Sanglokh International Astronomical Observatory (IAOS) of the Institute of Astrophysics of the National Academy of Sciences of Tajikistan. The apparent and absolute brightness of the comet in the R filter is determined, the dust production parameter and the upper limit of the radius of the nucleus are estimated. The distribution of brightness along the tail is shown and the structure of the dust tail is revealed. Photometric data indicate that during the monitoring period, the comet was in a state of normal cometary activity, associated mainly with the recent passage of perihelion.

Key words: comet; photometry; light; dust production; radius; isophote; Finson-Probstein diagram.

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D.Rashidov , T.Kh.Salikhov , S.Kh.Tabarov , A.Mirzo , D.M.Sharifov , F.Kh.Sodikov
Thermaldiffusivity of fullerene-filled polyethylene
Temperature changes in thickness, density, and coefficients of thermal shrinkage or thermal expansion of fullerene-containing samples of low density polyethylene-Сhas been studied by dilatometers and thermography methods. It is shown that the coefficient of thermaldiffusivity of the composite samples of LDPE-С decreases with increasing concentration of С, which is associated with an increase in phonon scattering on crystallites and the nanoparticles of fullerene С.

Key words: polymer; nanocomposite; fullerene; phonons; thermaldiffusivity.

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N.S.Beknazarova , J.O.Shoalifov
Complex formation in the system Hg (II) - 1-phenyl-2,3-dimethylpyrazoline-5-thion - 0.1 mol / l HNO - 298 K
The process of complexation of mercury (II) ions in a medium of 0.1 mol / L HNO at a temperature of 298 K was studied by potentiometry using a redox electrode based on 1-phenyl-2,3-dimethylpyrazoline-5-thione and its oxidized form. It is shown that titration of the Hg (II)-1-phenyl-2,3-dimethylpyrazoline-5-thione - 0.1 mol /L HNO - 298 K system leads to a regular increase in the potential of the indicator electrode. It was found that the process of complexation proceeds stepwise with the formation of four complex forms of compositions: [HgL(HO)], [HgL(HO)], [HgL(HO)] and [HgL] For each complex particle, their stability constants are determined by the Bjerrum method.

Key words: complexation; 1-phenyl-2; 3-dimethylpyrazoline-5-thione; mercury(II); redox electrode; stability constant; equilibrium concentration.

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A.S.Kurbonov , M.M.Tagoev , R.Akramzoda , A.P.Tagaev , U.M.Mirsaidov
Comparative assessment of thermodynamic characteristics of borosilicate ore decomposition with mineral acids and acetic acid
In the paper a comparative thermodynamic analysis of the possibility of a number of chemical reactions during the processing of Tajikistan borosilicate ores with nitric, sulfuric, hydrochloric, orthophosphoric and acetic acids are presents. The analysis showed that the probabilities of the occurring reactions of minerals of borosilicate ores with nitric acid are more preferable in comparison with other acids.

Key words: borosilicate ores; decomposition; thermodynamic analysis; mineral acids; acetic acid.

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M.M.Mamadvafoev , D.A.Davlatshoev
To the petrochemistry of basaltoids and alkaline basaltoids of the Kafandara ore field (Eastern Gissar)
The article contains the most complete data on the petrochemistry of basaltoid and alkaline-basaltoid subvolcanic dikes, which are products of the final taphrogenic magmatism of the Kafandara ore field and the southern Tien Shan.

Key words: petrochemistry; taphrogenic; subvolcanic; basaltoid and alkaline-basaltoid dikes; Kafandar ore field; East Gissar; South Tien Shan.

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N.S.Saidov , D.S.Sattarov
Results of the introduction of hoseed plants in the North American area the Central Botanical Garden of Tajikistan for 85 years
In this article the results of the introduction of conifers in the North American section of the Central Botanical Garden of Tajikistan over the past 85 years of activity are presents. It has been ascertained that over the past 30 years, the biodiversity of this site has decreased by 16 species, which belong to 1 family and 2 genera, and for this period more than 40 taxa have been preserved on the site. This number is dominated by representatives of the cypress family - Cupressaceae Bartl., which is represented by 5 genera and 28 species.

Key words: introduction; garden; North American site; conifers; family; genus.

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The content of nitrogen and protein in green biomass and in grain tritikalе
In this article the results of the analysis of the content of nitrogen and protein in the leaves, stem, spike, as well as in the mature grain of the triticale samples in different phases of development are presents. The study of nitrogen and protein in vegetative organs in the process of vegetation of triticale plants revealed a clear relationship - the increased nitrogen concentration in the leaf stem mass in D-83, T-25 and Vose-3 contributes to an increased protein content in the grain.

Key words: triticale; wheat; rye; leaf; stem; spike; nitrogen content; protein.

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К.Aliev , N.S.Dilovarova , N.Kh.Norkulov , M.Kh.Shukurova , Z.Kh.Norkulova
Organ specificity of pro- and antioxidant system of potato in vitro and ex vitro
In this article the organ-specificity of the antioxidant defense system of potato plants ex vitro has been established. The main mechanism of the plant resistance against to upcoming stress mainly by the concentration of antioxidants in the root system is determined. It is supposed that the root system has a high resistance potential in contrast to leaves. Moreover, the activity of the studied enzymes was higher in roots than in leaves.

Key words: potato; reactive oxygen species; lipid peroxidation; enzyme; hydrogen peroxide.