# Research

## Department of Algebra

The basic scientific directions of chair are connected with researches in such areas as the categorical and homological topology, the model theory and universal algebra, the group theory, the ring theory, the module theory, algebraic methods of information security, the methodology of mathematics teaching.

Professor Skurikhin E.E. is a supervisor of the direction "Categorical and homological topology". Under this direction they research in such areas as algebraic topology, homological algebra, homological theory of dimension, the Sheaf theory on Grotendics sites, the theory of categorical topological spaces, of posets, of frames, of topological and uniform spaces, the theory of categories, the theory of spaces Chu. By the results of the work in this direction there are the thesis for a Associate Professor's degree and Ph.D.

Professor Stepanova A.A. is a supervisor of the direction "Model theory and universal algebra". Under this direction they receive new original results in model theory of S-sets and S-posets, they solve some problems of universal algebra, connected with varieties of groupoids, semigroups, quasigroups. They prove a Mustafin - Poizat hypothesis of description of monoids over which theories of all S-sets are stable. By the results of the work in this direction there are the thesis for a Associate Professor's degree and Ph.D.

## Department of Mathematical Analysis

Department of Mathematical Analisys explores two scientific areas at the same time. One of them is analysis of differential equation systems. This scientific direction is represented by the scientific work of Professor Shepeleva R.P. and Associate Professor Durnov G.G. The second scientific direction is represented by the scientific work of Kalmykov C.I., Kim V.Y., Prilepkina E.G. in the field of geometric function theory and potential theory. These works are closely related with researches pursued by Dubinin V.N., Shlyk V.A., Dymchenko Y.V. at the Department of Function Analysis within Far Eastern Mathematical high school. Great attention is paid to solving extremal problems of geometric function theory, developing of methods of symmetrization and techniques of generalized condensers, obtaining inequalities for polynomials and rational functions.

## Department of Mathematical Physics and Computer Simulation

Principal scientific research conducted at the Department of Mathematical Physics and Computer Simulation is closely coupled with research in mathematical methods in economics, computeraized tomography, hydrodynamics, underwater acoustics and mathematical ecology.

Associate Professor Kovtanyuk A.E. and Professor Prokhorov I.V. have experience in computeraized tomography. Research is pointed towards development of new computeraized tomography methods based on usage of Radiative transfer equation. Particularly behavior of solution of inverse problems for radiative transfer equation is developed. Algorithms of diagnostics of scattering and absorbing mediums are developed.

Professor Alekseev G.V. is at the head of research in extremal problems in hydrodynamics and hydroacoustics within the bounds of which problems of hydrophysical field optimization are being developed. Particularly structure of solution set for optimal control problems in viscous medium with nonlinear restrictions is declared. Algorithms of hydrodynamical and hydroacoustical field optimization are developed.

Professor Abakumov A.I. is at the head of research in mathematical simulation of ecological problems within the bounds of which primal and inverse problems in mathematical ecology and models of environmental pollution with waste products including radioactive waste are being developed. The most attention is paid for researh of oceanic pollution problems.

Associate Professor Han I.S. is at the head of research in mathematical methods in economics. Principal scientific research is closely coupled with problems in synergetic economics, fractal structure and cellular automata of nature and society.

There were defended a Associate Professoral dissertation and four candidate's dissertations of late five years. Graduate school works intensively.

Web application: Database of radiological properties of materials valuable for x-ray diagnostics.

## Department of Function Theory and Functional Analysis

Principal scientific research conducted at the Department of Function Theory and Functional Analysis is closely coupled with research in symmetrization, major components, weighting function spaces and extremal problems in geometrical function theory. On the basis of this research new scientific school is founded and being developed. Professor Dubinin V.N., Professor Frolov N.N. and Professor Shlyk V.A. become leaders of this school. Within the bounds of this school new original symmetrization transformations are got and many important and difficult extremal problems coupled with international directions of research in geometrical function theory are solved. Some of these problems were raised earlier and it was impossible to solve them in a certain way in the course of many years. Among these problems there are covering vertical segments problem of Heiman, harmonious measure problem of Gonchar, extreme distance problem of Vuorinen, spatial analog of minimal capacity problem of Gonchar, intractable problems. This research is considered as new perspective direction in function theory.

There were defended three Associate Professoral and four candidate's dissertations of late five years. Graduate school works intensively.

## Department of Software

Scientific research conducted at the Department of Software is closely coupled with research conducted at the laboratory of intelligence systems of IACP FEB RAS.

Principal scientific directions:

- Ontologies and knowledge bases - theory, application and estimation.
- Implements for software development, based on methods of artificial intelligence - models, methods, implementation.
- Applied intelligence systems - theory, implementation, usage experience.

Principal results:

- Mathematical theory of subject field ontologies, subject field level-based models and structural properties of ontologies.
- Models of ontologies and knowledge bases for a number of subject field.
- Method of inductive generation of medical diagnostics knowledge.
- Multipurpose computer knowledge bank - WEB-system based on ontologies and intended for solving problems coupled with information gathering, formalization, storing, management and processing.
- Tools for design and automatic generation of user interface for different platforms with support of different types of dialogs.
- Productional systems of parallel programming.
- Specialized computer banks of medical diagnostics knowledge, chemical knowledge, mathematical knowledge and program modification knowledge.

Professor Artemeva I.L., Professor Gribova V.V. and Professor Knyazeva M.A. have defended a Associate Professoral dissertation of late three years.

There are also conducted following research at the Department of Software:

- quality metrology and estimation of software products and processes
- creating of intelligence systems of management
- application and development of e-learning technologies