Geodesic
On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces
Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 16-30.

Voir la notice de l'article provenant de la source Math-Net.Ru

The coverings are mostly used in geometry and analysis, and sometimes they are not given explicitly. The problem on defining of covering in concrete situation is substantive. Coverings arose in theory of manifolds, especially in connection with the system of equations. One of powerful methods in this direction is a theorem on implicit functions. In the paper we study these questions in a necessary general form. Such a consideration lead the problems to the basic notions which were studied by classics of mathematics in last two centuries. By him it was analyzed the main points of the theory on behavior of manifolds of less dimensions in manifolds of higher dimensions. Defining of the notion of a curve in the plane is bright example showing how we can establish suitable properties of objects we deal with to get the necessary freedom of actions, does not avoiding simplest generality. Introducing of quadrable curves makes possible to develop an acceptable notion of the integral in the domains on the plane. But this is insufficient for establishing for example, the theorem of Fubini on repeated integrals in that form as in Lebesgue's theory. Here we rest to constraints brought by intersection of manifold with boundary. The useful formulation of this theorem is possible to get only in Lebesgue theory of integration. This is one of multiplicity of questions connected with behavior of manifolds of less dimensions. We show how some notions of the theory must be modified to avoid such difficulties. We establish that the generalization of a notion of "improper" surface integral in some different from the ordinary meaning, makes possible solve the problem in general. In the present work we lead by such method the question on estimating of the number of sheets of covering to some metric relations connected with surface integrals.
Keywords: covering, manifold , system of equation, number of sheets, Jordan domain. 
@article{CHEB_2023_24_5_a1,
     author = {I. Sh. Jabbarov and S. A. Meshaik and M. M. Ismailova},
     title = {On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {16--30},
     publisher = {mathdoc},
     volume = {24},
     number = {5},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a1/}
}
TY  - JOUR
AU  - I. Sh. Jabbarov
AU  - S. A. Meshaik
AU  - M. M. Ismailova
TI  - On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces
JO  - Čebyševskij sbornik
PY  - 2023
SP  - 16
EP  - 30
VL  - 24
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a1/
LA  - en
ID  - CHEB_2023_24_5_a1
ER  - 
%0 Journal Article
%A I. Sh. Jabbarov
%A S. A. Meshaik
%A M. M. Ismailova
%T On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces
%J Čebyševskij sbornik
%D 2023
%P 16-30
%V 24
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a1/
%G en
%F CHEB_2023_24_5_a1
I. Sh. Jabbarov; S. A. Meshaik; M. M. Ismailova. On number of sheets of coverings defined by a system of equations in $n$-dimensional spaces. Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 16-30. http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a1/

[1] Dubrovin B. A., Novikov S. P., Fomenko A. T., Modern geometry, 2-d ed., Nauka, M., 1968, 760 pp. | MR

[2] Dzhabbarov I. Sh., “On an identity of Harmonic Analysis and its applications”, Dokl. AS USSR, 314:5 (1990), 1052–1054 | Zbl

[3] Dzhabbarov I. Sh., “On estimation of trigonometric integrals”, Chebishevskii sbornik, 11:1 (2010), 85–108 | MR

[4] Dzhabbarov I. Sh., “On the Multidimensional Tarry Problem for a Cubic Polynomial”, Mathematical Notes, 107:5 (2020), 15–28 | DOI | MR

[5] Federer H, Geometric measure theory, Nauka, M., 1978, 760 pp.

[6] Fihtengoltz G. M., Differential and integral calculus, v. 1, GIFML, M., 1962, 608 pp.

[7] Grebencha M. K. Novoselov S. I., Course of Mathematical Analysis, v. 2, Vishaya shkola, M., 1961, 560 pp.

[8] Jabbarov I. Sh., “On the structure of some algebraic varieties”, Transactions of NAS of Azerbaijan, issue math., 36:1 (2016), 74–82 | MR

[9] Kurant R., Differential and integral calculus, v. 2, Nauka, M., 1970

[10] Luzin N. N., Theory of functions of real variable, M., 1948, 318 pp. | MR

[11] Natanson I. P., Theory of functions of real variable, Nauka, M., 1974, 480 pp. | MR

[12] Nikolskii S. M., Course of mathematical analysis, v. 1, Nauka, M., 1990

[13] Shilov G. E., Mathematical analysis. Functions of several real variables, Nauka, M., 1972, 622 pp.

[14] Titchmarsh E. C., Theory of functions, Oxford University Press, 1939, 454 pp.

[15] Zorich V. A., Mathematical analysis, v. 2, MCNMO, M., 2002, 788 pp.