## Institute of mathematics

734063, Dushanbe, Republic of Tajikistan, 299/1, Aini Street,

tel. (+992 37) 225-80-89

The Institute based on the Department of Mathematics with Computing Centre of the Academy of Sciences of the Republic of Tajikistan was established in 1973. Its first and second director was academician A.D. Juraev Z. D. Usmanov respectively and from 1999 to to-date is prof. Z. H. Rahmonov.

At present the Institute consists of five departments of:

- differential equations;
- theory of functions and functional analysis;
- theory of numbers, algebra and topology;
- mathematical modeling;
- applied mathematics and mechanics.

There are four academician of the AS RT, one corresponding member of the AS RT, 9 doctors and 19 candidates of sciences in the institute.

In the institute following widely known scientific teams in mathematics were formed on:

- the equations of composite type (A. D. Juraev);
- singular differential and integral equations (L. G. Mihaylov, Z. D. Usmanov);
- the spectral theory of differential and pseudo differential operators (K. H. Boymatov, S. A. Iskhokov);
- the qualitative theory of periodic and almost periodic solutions of differential equations (E. M. Muhamadiev, M. I. Ilolov);
- theory of approximation of functions (M. Sh. Shabozov);
- the analytic theory of numbers (Z.H. Rakhmonov);
- computational linguistics (Z. D. Usmanov);
- the theory of nonlinear filtering in low permeable porous media (M. A. Sattarov).

In the institute researches in the following areas are conducted:

- the study of the solvability of boundary problems for differential equations with singular coefficients;
- the study of the solvability of initial-boundary problems for systems of equations with nonlinear diffusion and study of the solubility of the Cauchy and Neumann problems for the classical model of chemo taxis;
- the finding of the exact value of the transversal and quasi-transversal of compact classes of functions in various Banach spaces;
- investigation of the solvability of variational problems for linear and nonlinear differential equations with degeneracy and study of the smoothness of their solutions;
- the study of the solvability of classical boundary problems for multidimensional slightly elliptic systems of differential equations;
- search of topological spaces with given algebraic invariants;
- the study of new signs of the existence and uniqueness of solutions of the Cauchy problem and the study of the phenomenon of bifurcation of periodic and almost periodic solutions of differential equations with complex nonlinearities;
- development of software to ensure optimal water management of trans boundary river basin;
- the study of the structure of the waves of filtration combustion of gases and repression, depending on the thermo physical properties of two phases medium. In the field of theoretical mathematics the new results have been obtained on general and qualitative theory of differential and integral equations, spectral theory, geometry, in general, analytic number theory, function theory and topology.
- the theory of boundary problems for systems of partial differential equations of composite type (the theory of solvability in terms of conjugate problems, index formulas, conditions for normal solvability);
- the solvability of boundary problems for differential equations and systems with singular coefficients;
- the theory of solutions of generalized Cauchy Riemann systems with singular coefficients and the theory of infinitesimal deformations of surfaces of positive curvature with isolated flat point and analytical methods for studying various options for singular generalized Christoffel problem;
- solvability of differential equations with almost periodic coefficients and its analogues for partial differential equations of elliptic type, method of construction of guiding functions for the problems of forced periodic oscillations of nonlinear systems;
- criteria for local controllability of a chaotic dynamical system with the control vector in terms and conditions of the solvability of some classes of initial problems for abstract nonlinear equations;
- a new method of studying of the average values of arithmetic functions such as Chebyshev function over all Dirichlet characters, new estimates for the destiny of zeros of the Riemann zeta function in short rectangles of the critical strip, and a ternary solution of T. Esterman problem with almost equal terms;
- construction of a new method of Green’s functions of parabolic equations and its application in the study of spectral asymptotic of polynomial operator pencils, the theory of separation of differential operators with partial and generalized boundary problems associated with no coercive forms, degenerating on manifolds of different dimensions;
- tauberian method of finding the main part of the asymptotic behavior of multidimensional distribution functions of elliptic operators;
- criteria for determining the optimal parameters for solving linear equations with the approximate coefficients and the establishment of procedures for the convergence of the regularization of ill-posed problems;
- analytical methods for continuation of the holomorphic functions of several complex variables, the solution of problem of approximation from the top by super harmonic functions on compact and open sets of multidimensional space and the theory of complete inerrability of certain differential systems;
- mathematical models for calculation of interval of proper time of an arbitrary process, and natural metrics for a large class of natural processes;
- method of functionalization of parameters for the approximate construction of small oscillations in nonlinear systems;
- theory of the unique solvability of the variational Dirichlet problem for elliptic differential equations with exponential degeneracy on manifolds of different dimensions and degenerate elliptic equations with measurable coefficients in a restricted area;
- exact inequalities between best approximations and modules of continuity of higher orders with calculations of the exact values of the various n-width for some classes of functions defined by these modules of continuity;
- best linear methods of approximation of classes of analytic in the unit circle functions in the weighted Bergmann spaces with computing the exact values for n-width of introduced classes of functions;
- method for solving the extremal problem of finding exact values of the average n-width in the sense of Kolmogorov, Bernstein and Gelfand for the classes of entire functions defined by the modules of continuity on the line and finding the optimal quadrature formulas for the integrals of the first kind on the classes of functions with limited gradient in the metric of space L
_{2}.

Among the most significant achievements in the field of applied mathematics are the followings:

- theory of evolution of collections of any nature with autonomous and no autonomous elements;
- the mathematical models of gradations of liver failure;
- mathematical models of the evaluation of the spectral shapes of shells of gastropor;
- fundamentals of automated morphological analysis of words of the Tajik language;
- mathematical foundation of the theory of filtration in capillary parent media and mathematical methods for solving of nonlinear problems of underground hydrodynamics, hydraulics applying to problems of land reclamation in the irrigation of foothill and intermountain depressions and filler areas;
- mathematical models of a cascade of multi-purpose reservoirs;
- statistical models of turbulent motion of rapid mountain stream at an elevated bottom roughness;
- systems for strengthening banks of mountain rivers, control and prediction of the channel process of mountain rivers in the construction of large reservoirs.

Institute’s staff has obtained 10 patents on inventions. According to the research:

- a variety of options for designing ergonomic keyboards, depending on the ranking of key on labor costs and the volume of the processed text file were offered;
- the standard was developed and the layout proposed on the computer keyboard character Tajik alphabet on UNICODE standard for network technology;
- established the statistical regularities of syllabic structure of the Tajik literary language and on this basis established the synthesizer of the Tajik language through the text;
- set up electronic dictionaries (Tajik-Russian, Russian-Tajik, Tajik-Russian-Tajik, Tajik-English);
- developed a software package for computing the transformation of Tajik Cyrillic texts in to Persian texts in the chart:
- developed a system of forecasting the spread of contamination in Dushanbe due to industrial emissions;
- a long-term program for informatization of the Republic of Tajikistan.

The institute has dissertation councils to award academic degrees of doctor and candidate of physical and mathematical sciences in four specialties. Over the past 10 years in these councils, 10 doctoral dissertations and 41 master’s thesis were defended. During the last 5 years, 8 international conferences were conducted at the institute.

Academician A. D. Juraev for the creation of the theory of boundary problems for systems of differential equations with partial derivatives of composite type was awarded the State Prize named after Abu Ali Ibn Sina.

Academician L. G. Mihaylov for obtaining important scientific results in the theory of differential equations with singular coefficients was awarded the State Prize of Abu Ali Ibn Sina.

Dr. S. A. Iskhokov for obtaining fundamental results on the theory of solvability of variational problems for linear and nonlinear degenerate equations was awarded the prize of academician S. U. Umarov of the Academy of Sciences of the Republic of Tajikistan.

Dr. K. I. Mirozabdugafurov was awarded the prize of I. Somoni of Committee on Youth, sports and tourism under the Government of the Republic of Tajikistan.

The institute maintains close links with V. A. Steklov Mathematical Institute, Moscow State University and collaborates with foreign researches centers and individual scientists from the US, Germany, Chia, Japan, Canada, Iran, Israel, Austria, etc.

The staff of the institute participated in the International Mathematical Congresses in Moscow, Berkeley, Helsinki, Warsaw, Kyoto, Berlin, Barcelona, Beijing, Hyderabad, participated in numerous international conferences, conducted lectures, and joint researches in Russia, USA, Germany, France, Japan, Poland, Slovenia, Italy, Ukraine, Belarus, Kazakhstan, Israel, Iran, Austria.

Institute scientists published 14 monographs abroad. More than 450 scientific articles were translated into English by the American Mathematical Society.