Institute of Mathematics named after A. Dzhuraev National Academy of Sciences Tajikistan was organized in 1973 based on the Department of Mathematics with computing center of the Taj Academy of Sciences. SSR. His first the organizing director was academician A.D. Dzhuraev (1973-1987), in subsequent years - academician Z.D. Usmanov (1987-1999) and since 1999 until february 2024, academician  NAST  Z. Kh. Rakhmonov headed the Institute. From March 2024 to the present, candidate of physical and mathematical sciences Rakhimzoda A.O. heads the institute.

Rahimzoda Alisher Orzu,

director of the institute, candidate of physical and mathematical sciences

Institute of Mathematics named after A. Juraeva conducts her activities in accordance with the laws of the Republic of Tajikistan “On science and state scientific and technical policy", "About the Academy of Sciences of the Republic Tajikistan", "Strategies of the Republic of Tajikistan in the field of science and technologies for 2007-2015”, charter of the Institute. The main scientific task Institute is implementation fundamental scientific research, also scientific research of practical importance and preparation highly qualified specialists in mathematics,

mechanics and computer science.

Karimov Olimjon Khudoiberdievich

deputy director for science and education, doctor of physical and mathematical sciences, laureate of the NAST prize named after S.U. Umarov

The emergence of mathematical science in Tajikistan was facilitated by two important events of the Government of the Republic:  organization in the 50s training of young mathematicians in the Tajik State University and education in 1957 at the Academy of Sciences initiative of the President of the Academy of Sciences of the Republic of Tajikistan S. Umarov department physics and mathematics, in which the formation of scientific research began Vat's mathematical team. During this period, the mathematics sector this department began to be staffed with the first mathematician graduates faculty of physics and mathematics of the Tajik State University, as well as other higher educational institutions - Moscow State University. M.V. Lomonosov, Samarkand University, etc. On the initiative of academician S. Umarov there was a broad training program for highly qualified personnel has been launched in mathematics sector and, in particular, in leading mathematics centers Soviet Union - in Moscow, Novosibirsk, Voronezh, Kyiv and Leningrad. The first fruits of this work appeared already in the early 60s, and in the second in the mid-1960s, the mathematical team already included 2 doctors and 14 candidates of physical and mathematical sciences.

Nazrubloev Nasrulo Nurubloevich

Scientific secretary of the institute, candidate of physical and

mathematical sciences

At the Institute from the beginning of the 21st century during the period of independence of the Republic Tajikistan has formed authoritative scientific schools in mathematics, whose achievements are briefly characterized by the following indicators:

1. On analytical number theory (headed by academician of NAST Z.Kh.Rakhmonov).

• Received estimates of the average values ​​of the Chebyshev function were obtained for all Dirichlet characters of a given module and over all primitive Dirichlet characters whose modulus does not exceed a given value (1999-2020), which are more accurate compared to the known ones estimates by G. Montgomery (USA, 1974) and R. Vaughn (Britain, 1975).
• Received non-trivial estimates for sums of sums of a non-principal nature are obtained. Direction of a composite module in a sequence of shifted primes numbers (2013-2018). Previously, the best estimates belonged to I.M. Vinogradov, A.A. Karatsuba (Russia), K. Gong (China), J.B. Friedlander (Canada) and I.E. Shparlinsky (Australia).
• Found estimates for the execution of cubic trigonometric sums with prime numbers and perform cubic trigonometric sums with Möbius decrease in small arcs (2015-2016), which are improvement of the known estimates of A.V. Kumchev (USA, 2012) and U. Yao (China, 2015).
• Built a complete theory of short trigonometric sums by G. Weyl was constructed with with the help of which previously unsolved additive problems were solved Esterman and Waring with almost equal terms (1999-2018).
• Received the final results in the theories of zeros of Riemann functions have been obtained, Hardy and Davenport-Heilbronn at critical intervals direct (2006-2019), which are improvements to the well-known results, early achievements of A. Selberg (Norway), A.A.Karatsube (Russia), J.Mozer (Czech Republic).

2. On spectral theories of differential and pseudodifferential operators (leaders: NAST Academician K.Kh. Boymatov and Corresponding Member of NAST S.A. Iskhokov).

• Researched the spectral asymptotic behavior of some degenerate elliptic operators of higher order with no smooth coefficients in an unlimited region and studied the influence coefficients of the operators under study to the main part of them spectral asymptotic.
• A method of “perturbation by singular potential” has been developed, which allows us to solve the Gasimov-Kostyuchenko spectral problem for some classes of differential operators with factors derivatives in unlimited areas.
• For the first time, an analogue of Gording's inequality has been proven for degenerate elliptic operators in an arbitrary domain, who played an important role in the study of the solvability of generalized boundary value problems

for degenerate elliptic equations.

• New conditions for the solvability of the variational Dirichlet problem are obtained for some classes of degenerate elliptic operators and dependence of the smoothness of the solution to this problem on the smoothness coefficients of the operator under study. Results summarize the relevant results of Professors S.M. Nikolsky, L.D. Kudryavtsev, P.I. Lizorkina, N.V. Miroshin and others.
• New investment conclusions have been proven for some western territories differentiable functions, many functions that are included in theoretical solutions of the variational Dirichlet problem.
• Developed Tauber method of spectral asymptotics of elliptic operators with no smooth coefficients, which was previously considered one of the difficult problems of the spectral theory of differential operators.
• Built a theory of solvability of variational problems for elliptic operators associated with non-coercive forms and generalized many results known in the case of coercive forms.
• A modern method has been developed for studying the solvability of initial-boundary value problems for multidimensional systems of differential equations of composite type.
• The summability in the Abel-Lidsky sense of the system of root vector functions of certain classes of non-self-adjoint elliptic operators with degeneracy is proved.
• New derivations of separability are proven for some strictly nonlinear differential operators, which are used in the theory of solvability of boundary value problems for nonlinear differential equations.
• A new method has been developed for constructing the inverse operator itself for high-order non-conjugate elliptic operators in the entire space generated from non-coercive sesquilinear integro-differential forms.

3. On computer linguistics (headed by NAST academician Z.J. Usmanov).

• Has been proposed a standard of Tajik graphics for use in network technology; the development was accepted by the Moscow representative office of MICROSOFT and included in the WINDOWS editor. Approved as a standard by Decree of the Government of the Republic of Tajikistan dated August 2, 2004 No. 330.
• Has been created computer Tajik-Persian converter of graphic systems allows  automatically convert Tajik language texts into texts in Persian graphics;
• Has been implemented computer synthesis of Tajik speech based on text.
• Has been developed an automatic system TajSpell-2.0 to check the spelling of the Tajik language in the MS Office 2010-2019 office suite.
• Created computer Tajik-Russian, Russian-Tajik, Tajik-English, English-Tajik dictionaries.
• A fundamentally new type of so-called computer gamma classifier has been developed, with the help of which a wide variety of practical problems have been solved and are being solved, such as identifying the authors of text fragments, automatically recognizing plagiarism, borrowings, identifying the homogeneity of texts and the original and its translation, determining the language of works, and much more other. The developed classifier turned out to be quite competitive with the most popular classifiers in world practice, such as support vector machines and neural networks.

4. On the theory of approximation of functions (headed by NAST Academician M.Sh. Shabozov):

• Founded exact constants in inequalities type Jackson-Stechkin   between the best approximations of periodic, complex, entire functions and averaged values ​​of the modulus of continuity of derivatives of higher orders of functions. 
• Founded exact burners of Kolmogorov type for differentiable periodic functions of two procedures, in which subsequent partial derivatives of the connections arise from above, according to the norms of the function itself and the norms of the largest derivative of the function.
• Similar inequalities of Kolmogorov type were found for complex functions of two variables that are analytic in the bicircle.
• Found the best linear methods for approximating functions analytic in a circle in a weighted Bergman space. Optimal subspaces are indicated that realize the exact values ​​of the diameters of classes of functions specified by the moduli of continuity and smoothness.
• The results obtained provide the opportunity to solve problems of reconstruction and coding of certain classes of functions that are analytic in the unit disk and belong to the weighted Bergman space;
• The extremal problem of finding the best cubature formulas for classes of functions of many functions, defined by various modifications of moduli of continuity, structured from metric spaces, has been solved. The results obtained represent a generalization of the known results of N.P. Korneychuk and V.F. Babenko.

5. On the theory of initial-boundary value problems for partial differential equations (supervisor: NAST academician M.I. Ilolov).

• Existence and uniqueness theorems for solutions of evolution equations with fractional derivatives in a Banach space are proven and conditions for maximum regularity are found.
• A quasilinear parabolic equation derived from the parabolic-elliptic Keller-Siegel system is reduced to a linear differential equation with partial derivatives and with variable coefficients.
• A simpler Keller-Siegel system, which is widely used in mathematical biology, has been studied. From the Keller-Siegel system, a quasilinear equation is obtained, which is difficult to solve as a result staging. However, the use of the Hopf-Cole transformation allows us to write out relations connecting a quasilinear equation and a linear equation with variable coefficients.
• Have been found criteria for the local controllability of a chaotic dynamic system , that is, conditions have been specified for the control vector so that, depending on the parameter, the given dynamic system has an invariant torus.
• The solvability of initial boundary value problems for the Keller-Segel chemotaxis model with a nonlinear diffusion term is studied; in the case of the Neumann problem, difference schemes are proposed, the stability of which is proven by the sweep method.
• Obtained dispersion equations  and the boundaries of the oscillatory and exponential instability of the gas combustion front in adiabatic and nonadiabatic regimes are determined.

Among the staff of the institute there are 3 full members of the NAST, 3 corresponding members of the NAS of Tajikistan, 10 doctors and 13 candidates of science. Over the past 10 years, the Institute’s employees have received 10 copyright certificates.

The Institute plays a main role in training highly qualified personnel in the field of mathematics, mechanics and computer science in Tajikistan. Until 2017, the institute operated dissertation councils for awarding scientific degrees of doctor and candidate of physical and mathematical sciences at the Higher Attestation Commission of the Russian Federation in 4 specialties:

01.01.01 - real, complex and functional analysis;

01.01.02-differential equations, dynamic systems and optimal control;

01.01.06-mathematical logic, algebra and number theory;

05.13.18 - mathematical modeling, numerical methods and software packages.

During this period, 10 doctors of physical and mathematical sciences and 77 candidates of physical and mathematical sciences, representatives of both capital universities and regions of the republic, were trained.

Since January 2018, the dissertation council 6D.KOA-037 of the Higher Attestation Commission under the President of the Republic of Tajikistan began to function for the defense of dissertations for the degree of Doctor of Philosophy (PhD), doctor in the specialty 6D060100 - mathematics in the following specialties:

• • real, complex and functional analysis;
• 01.01.06 - mathematical logic, algebra and number theory.

Over the past 10 years, the institute has trained 8 doctors and more than 54 candidates of science, who successfully work at the institute and in all universities

republics. Also during this period, 65 highly qualified specialists in mathematics and computer science were trained through master's programs and through postgraduate studies. Currently, 30 people are studying at the institute for master's degree and 9 people are studying for doctoral studies (PhD).

         Over the past 10 years, the institute has held 17 international conferences. These conferences contributed to the establishment of creative contacts, the establishment of joint research and, in general, had a beneficial impact on the general level of research in mathematics, mechanics and computer science in the republic.

Among the scientists of the institute, the following were awarded the State Prize of the Republic of Tajikistan named after Abuali ibn Sino:

• academician A. Dzhuraev - for the creation of the theory of boundary value problems for systems of partial differential equations of composite type;
• academician L.G. Mikhailov - for his great contribution to the development of mathematical science in Tajikistan;
• academician Z.J. Usmanov - for the creation of the theory of generalized Cauchy-Riemann systems with a singular point and their applications in geometry in general;

Prize of the National Academy of Sciences of Tajikistan named after Academician S.U. Umarova:

• academician Z.J. Usmanov for the development of information technologies in the Republic of Tajikistan;
• Corresponding Member of NAST S.A. Iskhokov for obtaining fundamental results on the theory of solvability of variational problems for linear and nonlinear degenerate differential equations;
• doctor of Physical and Mathematical Sciences O.Kh. Karimov for obtaining fundamental results on the theory of separability of nonlinear differential operators and its applications.

For a series of scientific works published in well-known academic journals in Germany, academician M.Sh. Shabozov was awarded the international prize “Springer Top Author - 2015” in 2015.

Over the 30 years of independence, the Institute’s employees have published 37 monographs, 2037 scientific articles, of which 847 in republican publications, 511 in CIS publications, 165 in foreign publications, and 547 abstracts.

The Institute coordinates research work in the field of mathematics, mechanics and computer science in research institutes and universities of the republic.

The Institute maintains scientific contacts with TNU, Russian-Tajik (Slavic), Pedagogical, Technical, Technological, Dangara universities and a branch of Moscow State University. M.V. Lomonosov in Dushanbe. As part of cooperation, the institute’s employees give lectures, supervise coursework and students' theses, supervise the work of graduate students, conduct joint scientific seminars, and conduct joint scientific research.

The Institute maintains scientific relations with the Mathematical Institute named after. V.A. Steklov, Computing Center of the Russian Academy of Sciences, Institute of Market Problems of the Russian Academy of Sciences, Moscow State University. M.V. Lomonosov, Institute of Mathematics of the National Academy of Ukraine, Institute of Mathematics named after. S.L. Sobolev SB RAS, Institute of Applied Mathematics and Automation of the Kabardino-Balkarian Center of the Russian Academy of Sciences, North-Eastern Federal University named after. M.K. Ammosov, Tula Pedagogical University named after. L.N. Tolstoy  and with Termez State University (Uzbekistan).

Within the framework of cooperation, joint scientific research is carried out, scientific information is exchanged, scientific research in promising scientific areas of modern mathematics, mechanics and computer science is clarified.

The Institute maintains scientific relations with foreign scientific centers and individual scientists from the USA, Britain, Germany, France, Greece, Poland, China, Japan, Slovenia, Croatia, Israel, Iran, Pakistan, etc.

Academician Rakhmonov Z.Kh. is a member of the editorial boards of the international scientific mathematical journals “Chebyshev Collection” (Russian Federation) and “Journal of Number Theory” (USA), included in the list of peer-reviewed journals Scopus and Web of Sciences.

Academician Usmanov Z.J. was a member of the editorial board of the journal from the list of the Higher Attestation Commission of the Russian Federation “Bulletin of Samara University”, the international “Central Asian Journal of Mathematics” and a reviewer of the journal “ComplexVariables”, published in the USA.

Corresponding Member of the NAS of Tajikistan Ishokov S.A. is a member of the American Mathematical Society and a regular reviewer of the journal “Mathematical Reviews”, published in the USA.

The Institute's scientific research topics are relevant and promising for further development. Evidence of this is the constant invitations of a number of leading researchers to major international scientific centers to conduct research, give lectures, participate in international scientific conferences and publish scientific papers abroad.

Currently, the institute includes 5 departments: