Speciality:
01.01.04 (Geometry and topology)
Birth date:
26.08.1937
Website: https://www.chronos.msu.ru/lab-kaf/Usmanov/us-person.html Keywords: partial differential equations of elliptic type; general theory of elliptic systems; generalized Cauchy–Riemann systems with a singular point and a singular line; integral representations of solutions; general theory; boundary value problems differential geometry in Euclidean space; theory of surfaces; intrinsic geometry of surfaces; bending of surfaces; infinitesimal bendings of surfaces with an isolated flat point and a conic point; generalized Christoffel problem application of mathematical methods in different sciences; in sciences on space and time; modelling of an intrinsic time of a process.
UDC: 51-7, 51-71, 513.7, 513.73, 513.736.4, 514.752.43, 514.752.434, 514.752.435, 517.944, 517.956.222, 517.956.223 MSC: 35с15, 35j45, 35j55, 35j70, 53a05, 53b, 53c24
Subject:
The theory of generalized Cauchy–Riemann systems with a point of singularity of the 1-st order in coefficients, that is an immedate extension of the classical apparatus of the Vekua theory of generalized analytic functions, was developed. On this basis local and global problems in the theory of deformation of surfaces with a flat point as well as their local rigidity and non-rigidity were investigated. Together with the disciples theories of generalized Cauchy–Riemann systems with a point singularity of order greater than one in coefficients (with H. Najmiddinov) and with the 1-st order singularity on a circle (with A. Abdushukurov) were worked out. A certain progress was attained in solution of the extended Christoffel problem in searching convex surfaces with a given sum of conventional radii of curvature determined in a convex surface with an isolated flat point (with A. Khakimov). For an extensive class of processes described by linear and partial differential equations natural metrics similar to the Minkowskii space-time one were constructed. In particular, it enabled to create the natural metrics of the gravitational field, processes of string oscillation, heat conductivity and extension, and some others.
Main publications:
Z. D. Usmanov, Generalized Cauchy–Riemann systems with a singular point, Pitman Monographs and Surveys in Pure and Applied Mathematics, 85, Longman, Harlow, 1997, 222 p., ISSN 0269-3666, ISBN 0-582-29280-8
