NATIONAL ACADEMY OF SCIENCES OF TAJIKISTAN

Institute of Mathematics named after A. Juraev

National Academy of Sciences of Tajikistan

 

The Institute of Mathematics named after A. Juraev of the National Academy of Sciences of Tajikistan was organized in 1973 on the basis of the Department of Mathematics with the Computing Center of the Academy of Sciences of the Tajik SSR. Its first director-organizer was Academician A. D. Juraev (1973-1987), in the following years - Academician Z. D. Usmanov (1987-1999) and from 1999 to February 2024 the Institute was headed by Academician of the National Academy of Sciences Rakhmonov Z. Kh. From March 2024 to the present, the Institute is headed by Candidate of Physical and Mathematical Sciences Rahimzoda Alisher Orzu. The emergence of mathematical science in Tajikistan was facilitated by two important events of the Government of the Republic: the organization in the 1950s of training young specialists in mathematics at the Tajik State University and the formation in 1957 at the Academy of Sciences on the initiative of the President of the Academy of Sciences of the Republic of Tajikistan S. Umarov of the Department of Physics and Mathematics, in which the formation of a scientific research mathematical team began. The department was headed by L.Sh. Khodjaev (1957-1959), L.G. Mikhailov (1959-1962) and A.J. Juraev (1962-1964)

During this period, the mathematics sector of this Department began to be staffed with the first graduates-mathematicians of the Physics and Mathematics Department of the Tajik State University, as well as other higher educational institutions, Samarkand University, etc. At the initiative of Academician S. Umarov, a broad program was launched to train highly qualified personnel in the mathematics sector and, in particular, in the leading mathematical centers of the 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 half of the 60s, 2 doctors and 14 candidates of physical and mathematical sciences were already working in the mathematical team. Since the beginning of the 21st century, during the period of independence of the Republic of Tajikistan, authoritative scientific schools in mathematics have been formed at the Institute, the achievements of which are briefly characterized by the following indicators:

 

1. In analytical number theory (headed by Academician of the National Academy of Sciences of Tajikistan Z.Kh.Rakhmonov).

• Estimates of the average values ​​of the Chebyshev function were obtained for all Dirichlet characters of a given modulus and for all primitive Dirichlet characters whose modulus does not exceed a given value (1999-2020), which are more accurate in comparison with the well-known estimates of G. Montgomery (USA, 1974) and R.Vaughan (Britain, 1975).

• Non-trivial estimates of the sums of the values ​​of the non-principal Dirichlet character by a composite modulus in a sequence of shifted prime numbers were obtained (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).

• Estimates of short cubic trigonometric sums with prime numbers and short cubic trigonometric sums with the Möbius function in small arcs were found (2015-2016), which are an improvement on the well-known estimates of A.V. Kumchev (USA, 2012) and W. Yao (China, 2015).

• A complete theory of short trigonometric sums of G. Weyl was constructed, with the help of which previously unsolved additive problems of Esterman and Waring with almost equal terms were solved (1999-2018).

• Final results were obtained in the theory of zeros of the Riemann, Hardy and Davenport-Heilbronn functions in short intervals of the critical line (2006-2019), which are an improvement on the known results previously belonging to A. Selberg (Norway), A.A. Karatsuba (Russia), J. Moser (Czech Republic).

 

2. On the spectral theory of differential and pseudo-differential operators (headed by Academician of the Russian Academy of Sciences K.Kh. Boymatov and Corresponding Member of the National Academy of Sciences of Russia S.A. Iskhokov).

• The spectral asymptotics of some degenerate elliptic operators of higher order with nonsmooth coefficients in an unbounded domain is investigated, and the influence of the coefficients of the operators under study on the main part of their spectral asymptotics is studied.

• The method of "perturbation by a singular potential" is developed, which allows solving the Gasimov-Kostyuchenko spectral problem for some classes of partial differential operators in unbounded domains.

• An analogue of the Gårding inequality for degenerate elliptic operators in an arbitrary domain is proved for the first time, which 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 for some classes of degenerate elliptic operators are obtained, and the dependence of the smoothness of the solution of this problem on the smoothness of the coefficients of the operator under study is studied. The obtained results generalize the corresponding results of professors S.M. Nikolsky, L.D. Kudryavtsev, P.I. Lizorkin, N.V. Miroshin and others.

• New embedding theorems for some weighted spaces of differentiable functions of several variables are proved, which are applied in the theory of solvability of the variational Dirichlet problem.

• A Tauberian method of spectral asymptotics of elliptic operators with non-smooth coefficients is developed, which was previously considered one of the difficult problems of the spectral theory of differential operators.

• A theory of solvability of variational problems for elliptic operators associated with non-coercive forms is constructed and many results known in the case of coercive forms are generalized.

• A modern method for studying the solvability of initial-boundary value problems for multidimensional systems of differential equations of composite type is developed.

• Summability in the sense of Abel-Lidskii of the system of root vector functions of some classes of non-self-adjoint elliptic operators with degeneracy is proved.

• New separability theorems are proved for some strictly non-linear differential operators, which are used in the theory of solvability of boundary value problems for non-linear differential equations.

• A new method for constructing an inverse operator for non-self-adjoint elliptic operators of high order in the entire space, generated by non-coercive sesquilinear integro-differential forms, is developed.

3. On computer linguistics (headed by Academician of the National Academy of Sciences Z.J. Usmanov).

• A standard for Tajik graphics was proposed 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 the Decree of the Government of the Republic of Tajikistan dated August 2, 2004, No. 330.

• A computer Tajik-Persian converter of graphic writing systems was created, allowing automatic conversion of Tajik language texts into texts in Persian graphics;

• Computer synthesis of Tajik speech based on text was implemented.

• An automatic TajSpell-2.0 system was developed to check the spelling of the Tajik language in the MS Office 2010-2019 office application package.

• Computer Tajik-Russian, Russian-Tajik, Tajik-English, English-Tajik dictionaries have been created.

• A fundamentally new type of the so-called computer gamma classifier has been developed, with the help of which a wide variety of practical problems have been solved, such as identification of authors of text fragments, automatic recognition of plagiarism, borrowings, identification of homogeneity of texts and the original and its translation, determination of the language of works and much more. 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 academician of the National Academy of Sciences of the Russian Federation M.Sh.Shabozov):

• Exact constants were found in Jackson-Stechkin inequalities between the best approximations of periodic, complex, entire functions and averaged values ​​of the moduli of continuity of derivatives of higher orders of functions. • Exact Kolmogorov-type inequalities for differentiable periodic functions of two variables are found, in which successive partial derivatives are estimated from above by means of the product of the norms of the function itself and the norms of the greatest derivative of the function.

• Similar Kolmogorov-type inequalities are found for complex functions of two variables, analytic in the bicircle.

• The best linear methods for approximating functions analytic in the disk in the weighted Bergman space are found. Optimal subspaces are indicated that realize exact values ​​of the diameters of classes of functions defined by moduli of continuity and smoothness.

• The obtained results provide the possibility of solving problems of recovery and coding of some classes of functions analytic in the unit disk, belonging to the weighted Bergman space;

• The extremal problem of finding the best cubature formulas for classes of functions of several variables, defined by various modifications of moduli of continuity, depending on the metrics of the space, is solved. The obtained results are a generalization of the known results of N.P. Korneichuk and V.F. Babenko.

 

5. On the theory of initial-boundary value problems for partial differential equations (headed by Academician of the National Academy of Sciences of the Russian Federation M. Ilolov).

• Theorems of existence and uniqueness of solutions of evolutionary equations with fractional derivatives in a Banach space were proved and conditions of maximum regularity were found.

• A quasilinear parabolic equation obtained from the parabolic-elliptic Keller-Siegel system is reduced to a linear differential equation with partial derivatives and variable coefficients.

• A simpler Keller-Siegel system, which is widely used in mathematical biology, is investigated. A quasilinear equation is obtained from the Keller-Siegel system, which is difficult to solve in the original formulation. However, using the Hopf-Cole transform, it is possible to write out a relationship linking the quasilinear equation and the linear equation with variable coefficients.

• Criteria for local controllability of a chaotic dynamic system are found, that is, conditions are 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 was studied; in the case of the Neumann problem, difference schemes were proposed, the stability of which was proven by the sweep method.

• Dispersion equations were obtained and the boundaries of oscillatory and exponential instability of the gas combustion front in adiabatic and non-adiabatic modes were determined.

6. On the theory of differential equations with singular coefficients (headed by Academician of the Russian Academy of Sciences L.G. Mikhailov and Doctor of Physical and Mathematical Sciences G. Dzhangibekov).

• the theory of differential equations with singular coefficients;

• methods for studying overdetermined systems of differential equations and the theory of generalized analytic functions of many complex variables;

• the theory of a special class of singular integral equations with homogeneous kernels;

• theories of generalized Cauchy-Riemann systems with a polar singularity of the 1st and higher first order in the coefficients;

• development of the theory of two-dimensional singular integral operators over a bounded domain;

• basic boundary value problems for general elliptic systems of differential equations on a plane.

 

Currently, the institute includes 5 departments:

 

1. The department was formed in 1999 in connection with the transfer of Academician Z.Kh. Rakhmonov to work at the institute. The employees of the department conduct scientific research in all the main areas of analytical number theory and the most striking achievements of the department were previously noted within the framework of the achievements of the scientific school of Z.Kh. Rakhmonov.

Scientific research in the department is carried out in the following areas:

• the behavior of short trigonometric sums of G. Weyl in large arcs and their estimation in small arcs;

• the Esterman problem with almost equal terms for powers of prime numbers;

• the Waring-Goldbach problem with almost equal terms;

• the problem of zeros of a linear combination of Dirichlet series that do not have an Euler product;

• the distribution of residues, non-residues and indices by a composite modulus in a sequence of values ​​of a square polynomial whose argument takes on prime numbers;

• the distribution of fractional parts of values ​​of a polynomial whose argument runs through prime numbers from short intervals;

Scientific research of Doctor of Physical and Mathematical Sciences U.Kh.Karimov is devoted to the problems of algebraic topology and he has obtained the following notable results:

• the Bestvina-Edwards problem on the existence of a non-contractible compact set whose superstructure is contractible has been solved;

• a theorem on the existence of a two-dimensional simply connected cell-like Peano continuum whose two-dimensional homotopy group is non-zero has been proved;

• a theorem on the existence of a cohomology manifold that is not homologically locally connected has been proved;

• a non-contractible compact set has been constructed, all of whose homology and homotopy groups are trivial.

In the last 10 years, the department has trained 2 doctors and 12 candidates of sciences, who work at the Institute and universities of the republic. Currently, the department employs specialists in number theory, Academician Z.Kh.Rakhmonov, candidates of physical and mathematical sciences D.M.Fozilova, A.S. Aminov, specialist in the field of quasigroup algebra, Corresponding Member of the National Academy of Sciences of the Russian Federation A.Kh.Tabari, Doctor of Physical and Mathematical Sciences U.Kh.Karimov and Doctor of Physical and Mathematical Sciences Khairulloev Sh.A. Previously, the department employed specialists in number theory, candidates of physical and mathematical sciences V.M.Panov, Sh.K.Boboyorov, K.I.Mirzoabdugafurov, S.N.Ismatov, Z.Kamaridinova. 2. The Department of Function Theory and Functional Analysis was formed in 2005 as a result of the merger of two departments - "Function Theory" and "Functional Analysis". Until 2006, the department was headed by Academician of the Tatarstan Academy of Sciences K.Kh.Boimatov, from 2006 to 2013 - Academician of the Tatarstan Academy of Sciences M.Shabozov. From 2013 to 2024, the department was headed by Doctor of Physical and Mathematical Sciences O.Kh.Karimov. Since April 2024, the department has been headed by Corresponding Member of the National Academy of Sciences of the Russian Federation, Doctor of Physical and Mathematical Sciences, Professor S.A.Iskhokov. The Department of "Theory of Functions" was created in 1970 from among the employees who worked in various divisions of the institute: M.K.Sattarov, O.Shabozov, M.Shirinbekov, who conducted research in the relevant areas of function theory, and later I.Shokamolov and A.M.Abdushukurov joined them. The head of the department was Candidate of Physical and Mathematical Sciences M.Sh.Shirinbekov. In the department, scientific research was mainly carried out on the analytical continuation of holomorphic functions of many complex variables, approximation of functions of a real variable by positive operators, and solving the problem of approximation from above by superharmonic functions on compacts and open sets of multidimensional space. From 2000 to 2005, the department was headed by Corresponding Member of the National Academy of Sciences of the Russian Federation S.A. Iskhokov.

The Department of Functional Analysis was organized in 1986. The organizer and permanent head of the department was Academician K.Kh. Boymatov, a major specialist in the field of functional analysis and differential equations. In different years, S.A. Iskhokov, A. Sharifov, M.Z. Zamonov, S. Ashurov, V. Fayziev worked in the department.

The most important scientific results of the department's employees are listed above within the framework of the achievements of the scientific schools of K.Kh. Boymatov, S.A. Iskhokov and M. Shabozov. The presented results obtained in the school of K.Kh.Boimatov, S.A.Iskhokov are improvements of the corresponding results in the theory of separability of differential operators (V.N.Everitt, M.Girtz, M.Otelbaev, etc.), in the spectral theory of differential and pseudodifferential operators (A.G.Kostyuchenko, M.Gasymov, Ya.T.Sultanaev, A.N.Kozhevnikov, G.V.Rosenblyum, etc.) and in the theory of weighted function spaces (S.M.Nikol'skii, P.I.Lizorkin, N.V.Miroshin, etc.).

At present, the department conducts scientific research on:

• separability of differential operators;

• spectral asymptotics of differential and pseudodifferential operators;

• construction of the resolvent of differential operators of elliptic type;

• convergence of functional series in systems of root vector functions of differential operators;

• extremal problems of approximation theory of functions.

 

Over the past 10 years, the department has trained 1 doctor and 5 candidates of science within the scientific school of K.Kh.Boimatov and S.A.Iskhokov, and 1 doctor and 12 candidates of science within the school of M.Shabozov.

 

Currently, the department employs Corresponding Member of NAST S.A.Iskhokov, Academician of NAST M.Sh.Shabozov (part-time), Candidate of Physical and Mathematical Sciences A.Sharifzoda (part-time), research associates Z.Dzh.Khakimova and P.M.Fozilova.

3. The Department of Differential Equations was formed in 2005 as a result of the merger of two departments - "Partial Differential Equations" and "Equations of Mathematical Physics". Until 2017, the department was headed by Academician of the Academy of Sciences of the Republic of Tatarstan L.G.Mikhailov. From 2017 to 2022, the head of the department was Corresponding Member of the National Academy of Sciences of the Russian Federation, Doctor of Physical and Mathematical Sciences, Professor S.A.Iskhokov. From 2022 to the present, the department is headed by Candidate of Physical and Mathematical Sciences B.A.Rakhmonov. The Department of Partial Differential Equations was created in 1963 on the basis of employees working in the differential equations sector of the Department of Physics and Mathematics under the Presidium of the Academy of Sciences of the Tajik SSR. The organizer and first head of the Department was Academician A.D.Dzhuraev. In different years, D.M. Murtazaev, R. Abdurakhmonov, A. Kaziyev, A. Sanginov, D.M. Kazidzhanova, S.B. Boboev, S. Bayzayev, I. Sirazhdinov, M. Nurubloev, Kh. Radzhabov and F.N. Gafurova worked in the Department. Scientific research of the Department staff under the supervision of A.D. Dzhuraev was devoted, first of all, to equations with partial derivatives. The Department was the first to develop a theory of boundary value problems for systems of differential equations with partial derivatives in bounded domains on a plane, possessing at each point of the domain both real and imaginary characteristics. For such systems, A.D. Dzhuraev was the first to formulate natural boundary value problems and develop methods for their study based on the use of singular integral equations on two-dimensional bounded multiply connected domains with a boundary, and then applied them to study the natural formulations of boundary value problems he found for general elliptic systems on a plane. He managed to prove that in addition to the Dirichlet and Neumann problems, there is another natural boundary value problem (problem A), which is Fredholm in an arbitrary bounded multiply connected domain for an elliptic system, regardless of whether it is strongly elliptic or not. On this basis, he managed to construct a theory of mixed (initial-boundary value) problems for non-stationary systems of partial differential equations that do not belong to the classical types.

The Department's staff also constructed and developed:

• the theory of boundary value problems in the theory of functions and elliptic systems; • methods for studying systems of multidimensional singular integral equations on manifolds with boundaries in the class of systems that have applications in geometry;

• solvability theory for systems of elliptic equations that degenerate on the boundary;

• apparatus of multidimensional complex analysis for studying overdetermined systems of equations that arise in complex differential geometry;

• modified solvability theory for boundary value problems for singular elliptic systems;

• finding ways to apply degenerate problems of mathematical physics that could not be solved by standard methods.

 

The Department of Mathematical Physics Equations was created in 1971 within the structure of the Department of Physics and Mathematics under the Presidium of the Academy of Sciences of the Tajik SSR. The organizer and first head of the department was Academician L.G. Mikhailov. In different years, the department employed B.M. Bilman, A.I. Achildiev, Z.D. Usmanov, N.R. Radjabov, G. Nazirov, A. Muminov, G. Dzhangibekov, A. Mukhsinov, M. Tursunov. The main achievements of the department's staff are the development of

• the theory of differential equations with singular coefficients;

• methods for studying overdetermined systems of differential equations and the theory of generalized analytic functions of several complex variables;

• the theory of a special class of singular integral equations with homogeneous kernels;

• the theory of generalized Cauchy-Riemann systems with a polar singularity of the 1st and higher first order in the coefficients.

Currently, the department conducts research on:

• the theory of generalized boundary value problems for degenerate elliptic equations;

• the application of function theory methods in the theory of boundary value problems for partial differential equations;

• the theory of initial-boundary value problems for linear and nonlinear equations of electrodynamics;

• the theory of periodic and almost-periodic solutions of differential equations;

• the theory of elliptic systems on the plane.

In the Department of Differential Equations during the years of independence, under the supervision of Academician L.G. Mikhailov, 4 doctoral and 6 candidate dissertations were defended.

Currently, the following work in the department: Corresponding Member of the National Academy of Sciences of Tajikistan I.Kurbanov (part-time), Doctor of Physical and Mathematical Sciences, Professor G.Dzhangibekov, Candidate of Physical and Mathematical Sciences B.A.Rakhmonov and Researcher M.S.Sharifzoda.

4. The Department of Applied Mathematics and Mechanics was formed in 2005 on the basis of the Department of Mechanics and the School of Academician M.I. Ilolov on the Theory of Initial-Boundary Value Problems for Partial Differential Equations, headed by Academician M.I. Ilolov (2005-2013). From 2013 to 2019, the department was headed by PhD in Physics and Mathematics P.B. Sadriddinov.

Since 2005, the main scientific directions of the Department of Applied Mathematics and Mechanics are:

• dynamics of chaos in multi-frequency systems of differential equations;

• localization of solutions to the modified Keller-Segel system of nonlinear diffusion;

• study of a stationary wave and temperature of filtration combustion of gases;

• hydrodynamic studies of turbulent flows in open channels and the possibilities of taking into account their structure for hydraulic calculations. The most notable achievements of the department staff in the first two areas are presented within the framework of scientific achievements of the M.I. Ilolov school. The achievements of the department staff in the other two areas are as follows:

• a formula for the maximum combustion temperature was obtained for the first time, additionally taking into account the coefficient of heat removal into the surrounding space;

• an analytical solution of the main equations of hydromechanics was obtained for modeling the propagation of flood waves in mountain rivers, which makes it possible to determine the dependence of the parameters of a stable channel on the average flow velocity;

• the dependences of the wave velocity, equilibrium temperature, characteristic size of the combustion zone and the diffusion coefficient of the missing component on the gas injection velocity were studied

• difference equations were compiled for a two-temperature model of filtration combustion of gases in the adiabatic regime, which make it possible to find discrete values ​​​​of the temperature of the porous medium, the gas mixture and the concentration of the missing component. Convergence conditions and approximation errors were found.

• the properties of three-dimensional turbulent flows in pipes and channels of a longitudinally homogeneous flow in the presence of transverse flows with closed streamlines were studied;

• analytical solutions for two types of elliptical tubes were obtained taking into account the shape parameters, which agree with the results obtained by direct numerical modeling methods (scientists from Russia, Israel, 2007, 2009).

Currently, the department employs specialists in applied mathematics, Academician M. Ilolov, D. N. Guljonov, and specialists in mechanics, Doctor of Physical and Mathematical Sciences M. Kabilov, Candidates of Physical and Mathematical Sciences N. N. Stepanova, P. B. Sadriddinov. In different years, Candidates of Physical and Mathematical Sciences H. Kuchakshoev and M. Shabolov worked in the department.

Over the past 10 years, the department has trained 4 candidates of science.

The department has close scientific ties with the Institute of Mathematics of the National Academy of Sciences of Ukraine, the Institute of Mechanics of Moscow State University named after M. V. Lomonosov, and the Institute of Oceanography of the Russian Academy of Sciences.

5. The Department of Mathematical Modeling was formed in the 1980s; until 2021, the head of the department was Academician Z.D. Usmanov. Since 2022, the department has been headed by PhD in Physics and Mathematics F.Z. Rakhmonov. Among its employees at various times were doctors of science S.T. Navruzov, I.Dzh. Nurov, M.K. Yunusi, V.I. Borzdyko and candidates of science M.A. Ismailov, T.I. Khaitov, H. Nazhmiddinov, R.I. Sadulloev, A. Gafforov, P.A. Pulatov, M. Ganiev, H.T. Maksudov, M.Ch. Yusupov, K.Kh. Zakhidov, Yu. Gorelov, U. Khaitova, M. Saksanov, M. Umarov, Z. Sanginov and others.

Among the scientific achievements of this department are the development of mathematical models:

• various phenomenological phenomena in physics, mechanics, ecology, which are described by complex nonlinearities and systems of differential equations containing complex nonlinearities;

• non-stationary heat and mass transfer processes during transportation of products through turbo pipelines taking into account changes in the phase state of the environment;

• stochastic analogues of the Pontryagin maximum principle for control systems described by stochastic differential equations;

• evolution of collection material of arbitrary nature;

• dynamics of the tugai-desert ecosystem of the Tigrovaya Balka Nature Reserve (jointly with the Department of Nature Protection and Rational Use of Natural Resources of the Academy of Sciences of the Tajik SSR);

• dynamics of cotton bush growth;

• dynamics of the number of insects in a cotton field;

• determination of the optimal scale of application of chemical and biological means of protecting cotton crops from pests (jointly with the Institute of Zoology and Parasitology of the Academy of Sciences of the Tajik SSR);

• water supply of the cascade of reservoirs on the Vakhsh River and water distribution in the Vakhsh Valley;

• siltation of the Nurek Reservoir and reformation of the side of the mountain reservoir under the influence of abrasion;

• distribution of water resources of transboundary rivers of Central Asia;

• development of economy of Tajikistan in conditions of market relations.

Since mid-1990s, extensive development of research in the department on automation of information processing in Tajik language has been received, in particular

• automatic morphological analysis of words of Tajik language has been developed;

• ergonomic layout of English, Russian and Tajik letters on computer keyboard and mobile phone keyboard has been proposed;

• statistical portrait of the most general Tajik sentence has been received;

• alphabetical coding of words of natural languages ​​has been proposed, on the basis of which automatic definition of variety of word-form anagrams in individual works and creative works of various writers and poets has been carried out;

• mathematical model of perception by brain of meaning of anagrammatically distorted texts of natural languages ​​has been proposed;

• new type of classifier of discrete random variables has been proposed, allowing to recognize the author by the fragment of text, which is an order of magnitude smaller than required for other classifiers.

Over the past years, the department has completed significant work on the implementation of the results of scientific research in the economy of the Republic:

• an automated system for distributing steamed cocoons to cocoon-winding machines for the Dushanbe silk-winding factory was developed and implemented;

• an automated subsystem for accounting for the supply of components was developed for the Tajiktekstilmash plant;

• the “Efir” subsystem was implemented for the Hydrometeorological Service Department for the prompt determination of air pollution by emissions from industrial enterprises;

• mathematical foundations were developed for optimizing the enrichment process of the extractant in the technological chain of counter-current extraction with the implementation of the results in the practical extraction of sea buckthorn oil from cake;

• mathematical foundations were developed for the automatic design of slotted grooves of winding drums for the Tajiktekstilmash plant;

• the developed driver for the layout of Tajik letters on a computer keyboard and instructions for its installation for use in everyday work were implemented through the Ministry of Communications of the Republic of Tajikistan. The Department implemented a broad program for training young scientific personnel in the central scientific institutions of the country, which were, first of all, the Computing Center of the USSR Academy of Sciences, the Institute of Advanced Training of the USSR Academy of Sciences, Moscow State University named after M.V. Lomonosov, the Institute of Cybernetics of the Academy of Sciences of the Ukrainian SSR, and the All-Russian Research Institute of PAS. Over the past time, the Department has essentially played the main role in providing the Republic with highly qualified specialists in computer science. Currently, more than 40 candidates and 4 doctors of science in computer science, who have undergone scientific training through the Department, make up the bulk of Tajikistan's leading specialists in this field.

Over the past 10 years, the Department has trained one doctor of science and 8 candidates of science.

Among the employees 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. In the last 10 years, the employees of the Institute have received 10 author's certificates.

The Institute plays a major role in training highly qualified personnel in mathematics, mechanics and computer science in Tajikistan. Until 2017, the Institute had dissertation councils for awarding the academic degrees of Doctor and Candidate of Physical and Mathematical Sciences under 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 were trained, representatives of both metropolitan universities and regions of the republic.

Since January 2018, the Dissertation Council 6D.KOA-037 of the Higher Attestation Commission under the President of the Republic of Tajikistan has been functioning to defend dissertations for the degree of Doctor of Philosophy (PhD), Doctor in the specialty 6D060100 - mathematics in the following specialties:

• 01.01.01 - 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 sciences, who successfully work at the Institute and in all universities of the republic. Also during this period, 65 highly qualified specialists in mathematics and computer science were trained through the Master's program and 59 through the postgraduate program. Currently, 14 people are studying at the Institute in the Master's program and 6 people in the doctoral program (PhD).

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

The following scientists from the Institute were awarded the State Prize of the Republic of Tajikistan named after Abu Ali ibn Sino:

• Academician A. Juraev - 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. Dzh. Usmanov - for the creation of the theory of generalized Cauchy-Riemann systems with a singular point and their applications in geometry in general;

 

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

• Academician Z. Dzh. Usmanov for the development of information technologies in the Republic of Tajikistan;

• Corresponding Member of the National Academy of Sciences of Tajikistan S. A. Iskhokov for obtaining fundamental results in 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 in the theory of separability of nonlinear differential operators and its applications.

 

The institute has held 18 international conferences over the past 10 years. These conferences have contributed to the establishment of creative contacts, the organization of joint research and, in general, have had a positive impact on the overall level of research in mathematics, mechanics and computer science in the republic.

 

For a series of scientific papers published in renowned 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 staff have published 37 monographs, 2037 scientific articles, of which 847 were published 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 the branch of Moscow State University named after M.V. Lomonosov in Dushanbe. As part of cooperation, the Institute's staff gives lectures, supervises students' term papers and theses, supervises postgraduate students' work, holds joint scientific seminars, and conducts joint scientific research.

The Institute maintains scientific ties with the Steklov Mathematical Institute, the Computing Center of the Russian Academy of Sciences, the Institute of Market Problems of the Russian Academy of Sciences, the Lomonosov Moscow State University, the Institute of Mathematics of the National Academy of Ukraine, the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, the Institute of Applied Mathematics and Automation of the Kabardino-Balkarian Center of the Russian Academy of Sciences, the Ammosov North-Eastern Federal University, the Tolstoy Tula Pedagogical University, and the Termez State University (Uzbekistan).