# Reports of the National Academy of Sciencesof Tajikistan

## Доклады Академии наук Республики Таджикистан, том 63, №09-10, 2020 г.

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Article title and abstractPages

Rakhmonov Z.Kh., Nozirov O.O.
543-549
The least Hardy-Littlewood number in an arithmetic progression with a difference equal to a prime power
For - the least Hardy-Littlewood number of the form , lying in the arithmetic progression , , , - prime power an upper bound of the form has been derived.

Key words: Hardy-Littlewood number; complete sums of characters; Chebyshev function.

Rasuli S.
550-556
Time delays in mixed problems for the equation of heat conductivity
In this article, a mixed boundary value problem for the heat equation with a time-delay argument is considered. Using the Fourier method, the problem is reduced to solving a countable system of delayed ordinary differential equations.

Key words: initial function; counting system; boundary conditions; Fourier method; absolute convergence of the Fourier series.

Аbdukarimzoda М.К.
557-563
Optimal weighted quadrature formulas for curvilinear integrals on classes of functions with bounded gradient in the space
The optimal quadrature formula for the class of functions of m variables continuous along the curve is found whose norm of the gradient in the norm of the space is bounded.

Jurakhonov O.A.
564-571
Upper bounds of best approximation by fourier sums over orthogonal polynomials of some classes functions in L(Q)
We evaluate the exact upper bounds for the best approximation by the sums Fourier functions of two variables with respect to orthogonal polynomials in space L:= L(Q), Q:=[a,b]×[c,d] for some classes of functions. The exact values of the -widths series of the specified function classes are calculated.

Key words: orthogonal polynomials; generalized modulus of continuity; generalized shift operator; widths; best aproximation.

Khasanov Yu.Kh.
572-578
On the deviation of uniform almost-periodic functions of two variables from Fourier sums for triangular matrix
In this paper the question of the deviation of uniform almost-periodic functions of two variables from Fourier sums in the sense of Bohr in the case when the Fourier exponents have a single limit point at infinity are investigated. Lower estimates of the value of the deviation of uniform almost-periodic functions from partial Fourier sums for classical triangular matrices are established.

Key words: almost periodic functions; double Fourier series; partial Fourier sums; Fourier exponents; best approximation; classical triangular matrices; trigonometric polynomials.

Ahmedov J.T.
579-585
To theory of periodic and bounded solutions of nonlinear dynamic second-order systems with phase portraits
The work is devoted to questions on the existence of periodic and bounded solutions of second-order nonlinear equations. Based on the results obtained, phase portraits are constructed.

Key words: periodic solution; boundedness; phase portraits; dynamic systems.

Haknazarov K.E.
586-590
On the Fredholm solvability of the variational Dirichlet problem for degenerate elliptic operators of a non-divergent form
We prove an analogue of the Garding inequality for a class of non-divergent form in a bounded domain of the n-dimension euclidian space with power degeneracy along the entire boundary of the domain. Using this inequality we study Fredholm solvability of the variational Dirichlet problem with homogeneous boundary conditions for such operators.

Key words: variational Dirichlet problem; non-divergent form elliptic operator; power degeneration; bounded domain; Fredholm solvability.

591-597
Virginid asteroid-meteoroid complex. II. Near-Earth asteroids associated with the Eta-Virginid meteoroid stream
We present the results of finding of new near-Earth asteroids (NEAs) belonging to the Virginid asteroid-meteoroid complex. On the base of calculation of orbital evolution of a sample of NEAs and determination of theoretical features of related showers a search for observable active showers close to theoretically predicted ones was carried out among published catalogues. As a result, predicted showers of 7 NEAs were identified with the showers produced by the η-Virginid meteoroid stream. Revealed association points to a cometary nature of NEAs that are moving within the stream and may be considered as extinct fragments of a larger comet-progenitor of the Virginid asteroid-meteoroid complex.

Key words: comet; asteroid; extinct comet; meteoroid stream; meteor shower; orbit; radiant.

Shoyoqubov Sh.Sh., Ibragimov A.A.
598-601
The dynamics of the formation of cluster ions in comets
Experimentally, by the method of mass spectral analysis, quantitative data were obtained on the rate of formation of cluster ions from the ice surface with a change temperature of pure ice.

Key words: surface of the comet's nucleus; formation rate; bombardment; vapor condensation; cluster ions.

Mulloev N.U., Fayzieva M.R., Khodiev M.Kh., Lavrik N.L., Tabarov S.T., Khojazoda T.A.
602-609
Manifestation of intermolecular hydrogen bonding in heterocyclic compounds of pyrrol derivatives according to IR spectroscopy and quantum chemical calculations
The formation of H-complexes in solutions of nitrogen-containing heterocyclic compounds of the pyrrole group (pyrrole, indole, carbazole) with a proton-acceptor solvent (acetone) was studied. Based on the results of IR spectroscopic studies and quantum chemical calculations, it has been shown that intermolecular hydrogen bonds of the NH ...O = C type are formed between carbazole and acetone molecules, which leads to a shift in the vibration frequency of NH groups of free molecules towards low frequencies with simultaneous broadening and strengthening of this band. The magnitude of the change in the spectroscopic and energy parameters of the H-complexes depends on the type of structural elements introduced into the pyrrole cycle. It was found that the donor capacity of the compounds under study, which is determined by their equilibrium electronic structure, is low and somewhat increases upon going from pyrrole to its derivatives.

Key words: pyrrole; IR spectrum; electron density functional method; proton exchange capacity; hydrogen bond; induction.

610-617
Evaluation of solar energy and selection of the optimal angle of solar panels for the different cities of Tajikistan
A method of determining the solar radiation on a horizontal surface of the territory of Tajikistan using the NASA meteorological database has been developed. The methodology proposed by Liu and Jordan was used for solar energy comes calculations on an inclined platform. For different cities of Tajikistan, the average annual input of solar energy was estimated. Also, for the selected points, the optimal tilt angles of the solar panels were determined.

Key words: solar radiation; horizontal and inclined surfaces; Tajikistan; NASA; optimal angle of the solar panels; solar potential.

Soliev L., Jumaev M.T., Makhmadov Kh.R., Muzafarova D.Z., Olimdzhonova N.V.
618-625
Phase complex of a mutual system Na,Ca//SO,CO‒HO at 75 and 100°C
The phase complex of the reciprocal system Na,Ca||SO,CO-HO at 75 and 100°C was studied by the translation method. All possible phase equilibria on the geometric images of the studied systems are established. It is shown that the studied system is characterized by the presence of the following number of geometric images, respectively for 75 and 100°C: divariant fields - 8 and 7; monovariant curves - 13 and 11; invariant points - 6 and 5.

Key words: translation method; diagram; sodium; calcium; sulfate; carbonate; phase complex.

Mirzoev D.Kh., Otaev Sh.D., Mirsaidov U.M.
626-630
Comparative analysis of the coagulating ability of aluminum sulphate with mixed aluminum-iron-containing coagulant
In this article a comparative assessment of the coagulating ability of aluminum sulfate with a mixed aluminum-iron-containing coagulant obtained by the interaction of kaolin clays of the Chashma-Sang deposit with hydrochloric acid are presents.

Key words: coagulants; mixed; aluminum-iron-containing; kaolin clay; assessment; hydrochloride acid.

Gaibullaeva Z.H., Asrorov B.I., Bahriddinzoda Sh.B., Nasimov G.T., Sharifov A.
631-636
Regularities of heterogeneous processes of lead recovery from galenite-containing concentrate by gases
The article discusses the main regularities of the heterogeneous process of processing lead-zinc concentrate with reducing water gas. The heterogeneous process depends on such parameters as particle size, method of mixing particles with gases (fixed bed of particles and gas movement through grain voids, turbulent mixing of particles with gas, fluidized bed of solid particles in a gas flow, etc.).

Key words: regularity; mass transfer; concentrate; reduction; water gas; diffusion transfer.

Khaidarov A.M., Eshov B.B., Ganiev I.N., Rakhmonov B.Sh.
637-642
Kinetics of oxidation of alloys of the Pb-Bi system, in solid state
The method of thermogravimetry was used to study the oxidizability kinetics of alloys of the Pb-Bi system in the solid state. The kinetic and energy parameters of the alloy oxidation process have been established.

Key words: Pb-Bi alloys; lead; bismuth; thermogravimetric method; oxidation kinetics; oxidation mechanism; true oxidation rate; activation energy.

Yatimov U.A., Safaraliev N.S., Shilovskikh V.V., Kotlyarov V.A.
643-648
Tsumoit and pilzenite of the Aktash skarn-magnetite deposit (Western Karamazar)
In this article based on the use of modern methods of optical and scanning electron microscopy, and the diffraction of backscattered electrons, describes the rare minerals tsumoite and pilzenite, first discovered in chalcopyrite-pyrite-pyrrhotite ores of the Aktash skarn deposit.

Key words: tsumoite; pilzenite; bismuth tellurides; chalcopyrite-pyrite-pyrrhotite ores; skarns; Aktash deposit; Western Karamazar.