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.

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.

Biography

1954–1958 — Student of Moscow State University, Faculty of Mathematics and Mechanics; 1959–1962 — Post-graduate, Department of Mechanics, Moscow State University; 1966 — Ph.D. thesis was defended; 1973 — D.Sci. thesis was defended; 1983 — Professor of Math. (VAK USSR). Author of more than 180 papers in theoretical and applied mathematics including 4 monographs.

1976 — Corresponding member, Tajik Academy of Sciences, Dushanbe; 1981 — Full member, Tajik ACademy of Sciences, Dushanbe; 1997 — Professor, Moscow Power Engineering Institute, Voljski Department; 1999 — Professor, Department of Informatics, Tajik Technological University, Dushanbe; Head of Department "Natural metrics of processes", Vertual Institute of interdisciplinary study of time, Moscow State University.

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

A. L. Goncharov, S. B. Klimentov, Z. D. Usmanov, “An analogue of Liouville's theorem for a class of systems of Cauchy-Riemann type with singular coefficients”, Vladikavkaz. Mat. Zh., 7:4 (2005), 7–16

1999

2.

Z. D. Usmanov, “Relationship between manifolds of solutions to generic and model generalized Cauchy–Riemann systems with a singular point”, Mat. Zametki, 66:2 (1999), 302–307; Math. Notes, 66:2 (1999), 240–244

1996

3.

Z. D. Usmanov, “The variety of solutions of the singular generalized Cauchy–Riemann System”, Mat. Zametki, 59:2 (1996), 278–283; Math. Notes, 59:2 (1996), 196–200

4.

Z. D. Usmanov, “On Efimov surfaces that are rigid 'in the small'”, Mat. Sb., 187:6 (1996), 119–130; Sb. Math., 187:6 (1996), 903–915

1977

5.

Z. D. Usmanov, “The problem of local rigidity in the class $C^\infty$”, Mat. Zametki, 22:5 (1977), 643–648; Math. Notes, 22:5 (1977), 850–853

1973

6.

N. V. Efimov, Z. D. Usmanov, “Infinitesimal bending of a surface with a point of flatness”, Dokl. Akad. Nauk SSSR, 208:1 (1973), 28–31

7.

Z. D. Usmanov, “A certain class of generalized Cauchy-Riemann systems with a singular point”, Sibirsk. Mat. Zh., 14:5 (1973), 1076–1087; Siberian Math. J., 14:5 (1973), 751–758

1972

8.

Z. D. Usmanov, “An effective solution of a certain two-dimensional integral equation with a homogeneous kernel, and its applications”, Differ. Uravn., 8:12 (1972), 2267–2270

9.

Z. D. Usmanov, “On a problem concerning the deformation of a surface with a flat point”, Mat. Sb. (N.S.), 89(131):1(9) (1972), 61–82; Math. USSR-Sb., 18:1 (1972), 61–81

1971

10.

Z. D. Usmanov, “A boundary value problem for generalized analytic functions with a fixed singular point”, Dokl. Akad. Nauk SSSR, 197:2 (1971), 288–291

1970

11.

Z. D. Usmanov, “On infinitesimal deformations of surfaces of positive curvature with an isolated flat point”, Mat. Sb. (N.S.), 83(125):4(12) (1970), 596–615; Math. USSR-Sb., 12:4 (1970), 595–614

1967

12.

A. I. Achil'diev, Z. D. Usmanov, “Rigidity of a surface with a point of flattening”, Mat. Sb. (N.S.), 73(115):1 (1967), 89–96; Math. USSR-Sb., 2:1 (1967), 77–83

1966

13.

L. G. Mikhailov, Z. D. Usmanov, “Infinitesimal bendings of surfaces of revolution with positive curvature having a conic or a parabolic point at the pole”, Dokl. Akad. Nauk SSSR, 166:4 (1966), 791–794

1992

14.

V. A. Il'in, S. M. Nikol'skii, B. V. Boyarsky, Z. D. Usmanov, “Abdukhamid Dzhuraevich Dzhuraev (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 47:3(285) (1992), 187–190; Russian Math. Surveys, 47:3 (1992), 201–204

1988

15.

S. M. Nikol'skii, L. D. Kudryavtsev, Z. D. Usmanov, È. M. Muhamadiev, “Leonid Grigor'evich Mikhailov (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 43:2(260) (1988), 171–172; Russian Math. Surveys, 43:2 (1988), 205–207