##### Work number - M 3 AWARDED

__Authors :__*Sevost’yanov E.A., Salimov R.R., Kolomoitsev Yu.S.*

The series is presentedbythe Institute of Applied Mathematicsand Mechanics

**The goal of the work****:**The main aim of the work includes:1) an investigation of metric, differential, boundary, and other properties of mappings satisfying modular-capacity inequalities of arbitrary order; 2) an investigation of local and boundary properties for homeomorphisms of the Orlicz-Sobolev classes of finite distortion, an obtaining of normality and compactness theorems; 3) an investigation of non-classical methods of approximation of functions,moduli of smoothness and *K*-functionals of fractional order by methods of Fourier analysis; 4) anobtaining of new sufficient conditions for Fourier multipliers.

**The****scientific****novelty****. **It is obtained a series offundamental theorems about topological and metric properties of subclasses of mappings of finite distortion, which are widegeneralization of quasiconformal mappings.It is established a number ofimportant properties of mappings of Orlicz-Sobolev classes, which were used to prove new theorems on the existence of solutions of quasilinear degenerate Beltrami equations.It is obtained some fundamentalproperties of moduli of smoothness and K-functionals of fractional order. It is investigated a number properties of non-classical methods of approximation of functions. It isproved newsufficient conditions for Fourier multipliers that strengthen some known conditions and essentially extend the classes of functions which can be study by methods of Fourier analysis.

**The practical****significance of the results****.** Theworkismainlytheoretical. However, its results can be applied in theoretical investigations in hydrodynamics and gas dynamics, wherethere is inhomogeneity of the medium and singularity.

The results ofthe work published in one monograph, 28 articles (including 13 works in foreign journals). Work of the authors cited in 39 scientific journals, the overall index of citing publications is 55 citations (according the Scopus databases), mean value of h-indexing equals to 2,33 (E.A. Sevost’yanov – 4, R.R. Salimov – 3, Yu.S. Kolomoitsev – 0).

**Total number of publications is 146 works****:**E.A. Sevost'yanov – 64 (including 62 articles in printed journals and one book), R.R. Salimov – 55 (including 54 articles in printed journals and one book), Y.S. Kolomoitsev – 27.