All articles - Mathematics

Subjects of research: Finite dimensional complex Zinbiel algebras, filiform Leibniz algebras.
Purpose of work: To investigate n-dimensional complex filiform Zinbiel algebras, to examine the structural theory of Zinbiel algebras.
Methods of research: In this work methods of structural constants, classification methods, gradation methods and the methods of invariant theory are used.
The results obtained and their novelty: The main results of the work are the following:
- criteria of isomorphism of filiform Leibniz algebras class, natural gradation of which are Lie algebras, is obtained;
- the classification of four-dimensional complex Zinbiel algebras is obtained;
- zero-filiform and filiform complex Zinbiel algebras are described. Based on this description, the derivations of such algebras are investigated. Moreover, the description was extended to the class of complex naturally graded quasi-filiform Zinbiel algebras;
- some properties of characteristic sequence for the Zinbiel algebras are obtained. Furthermore, the classifications of n-dimensional complex Zinbiel algebras with nilindex n-2 with characteristic sequences (n-3, 3) and (n-3, 1, 1, 1) are obtained.
Practical value: The results of the dissertation are of theoretical character.
Degree of embed and economic effectivity: It can be used at reading of special courses.
Field of application: The main scientific results and methods presented in the work can be used in research of other algebras and superalgebras, in the theory of categories, in the study of algebras with various types of gradation, in calculation of cohomological and homological groups.

Subject for inquiry: functions of positions of mechanisms, connections equations of these mechanisms, singular positions of mechanisms and their algorithm of computation, Newton’s polyhedron.
Aim of the inquiry: description of connections equations of mechanisms with the help of a system of nonlinear algebraic equations. Classification singular points of the position function of mechanisms. Construction of algorthm for computation of singularities of the position function of mechanisms. Investigation of singularities fiflink mcchnisms, plane mechanism with three degrees of freedom and plane fourlink with hydrosilindrs.
Methods of inquiry: in the work methods of computational mathematics, linear algebra and exponential geometry, and algorithms of finding of singularities of curves arc applied.
The results achieved and their novelty: classification of singularities of position function of mechanisms which arc expressed by algebraic curves is obtained. The algorithm for computation of singular positions of position functions of mechanisms is constructed. Local presentations of position function of plane mechanisms with two and three degrees of freedom arc found.
Practical value: the results of the dissertation have scientific-applied character.
Sphere of usage: the results of the present dissertation work may be used in the further development of the theory of singularities of algebraic curves, in problems which appear in investigations and design of mechanisms, in creation automatic and semiautomatic robots and in other theoretical and practical problems.

Subjects of the inquiry: the study heat and mechanical processes in electronic charge with provision for anisotropies mechanical and heat features of the electronic charges.
Aim of the inquiry: the development algorithm decisions initial-marginal problems to anisotropic theories to bounce and the numerical realization of the marginal problems with reference to calculation element radio electronics equipments (REE).
Method of inquiry: it is used new recurrence-operators method of the decision of the linear differential equations and their systems with provision for anisotropies, dissipation and heat conductivity.
The results achieved and their novelty: Constructed the new decisions of the modified equations of heat conductivity will built in two variants, equations unbound theories heat elasticity, equations to Lame, complemented member, taking into account viscosity and dissipation energy, as well as is for the first time solved problems of the fluctuation pivotal and flat design REE in new wave production.
Practical value: the got results of the analytical decision of the boundary problems allows on the one hand to value accuracy of the numerical methods of the decision of these problems, but will on the other hand get the more reliable results, for their account when designing REE.
Degree of embed and economic effectivity: the results of the work arc introduced in the training processes and can be enclosed in automated system of the provision to stability and quality of the equipment that allows in total to reduce the expenses under automatic designing REE.
Sphere of usage: the got results can be using not only when designing REE, but also in the other branch of the technology, where is researched heat and mechanical processes in anisotropic building design, in shipbuilding, machine building and others.

Subjects of research: quasilincar parabolic equations with the gradient nonlinearity, describing nonlinear processes heat conductivity.
Purpose of work: the research of qualitative properties solutions of nonlinear mathematical models, in homogeneous and heterogeneous environment, when coefficient heat conductivity depends on a gradient of temperature, in view of absorption or source.
Methods of research: algorithm of nonlinear splitting, technique of comparison of the solutions, iterative numerical methods, method of variable directions and proracc method.
The results obtained and their novelty: for the quasilincar parabolic equations with the gradient nonlinearity of the second order develops of asymptotic theory based on a method of nonlinear splitting. The estimations of the decisions problem of Cauchy by algorithm of nonlinear splitting for the nonlinear heat conductivity equation with strong absorption divergent and non divergent types arc received. Asymptotic behavior of the solution in a critical ease is investigated. The conditions of occurrence of the unlimited solutions for the equation with a nonlinear source in heterogeneous media are received. Basing on the received estimations of the solutions and fronts, the computing experiments with use MathCad is carried out.
Practical value: results of the dissertation have theoretical character.
Degree of embed and economic efficiency: the results of the dissertation can be used in modeling nonlinear problems of mathematical physics and further in developing on the theory of nonlinear parabolic equations.
Field of application: the results of the dissertation can be used in modeling nonlinear processes heat conductivity, filtration, diffusion and on the base of achieved results, the special courses for students can be tcachcd.

Subjects of research: low-frequency digital signals, and architecture of digital signal processors.
Purpose of work: To develop high-speed signal processing techniques to be submitted in the form of an algebraic polynomial on the basis of the signal spectrum, and their software implementation on modern signal processors.
Methods of research: a theory of functional analysis, spectral analysis in the Fourier bases, methods of calculating polynomials and elementary functions, theory of numbers and matrices.
The results obtained and their novelty: A method for transforming the signal in the area of polynomial representation, and finding the polynomial coefficients using the spectral approach, the algorithms and software for polynomial signal processing using digital signal processors; the qualitative characteristics of the developed algorithms was examined, A polynomial approach for calculating the biosignal parameters and to solve the problems of compression and audio signals smoothing.
Practical value: The algorithm for calculating the coefficients of algebraic polynomials, a set of digital signal processing was established applications, developed applications are protected by the Patent Office evidence of the Republic of Uzbekistan.
Degree of embed and economic effectivity: the basic theoretical and practical results of the thesis inculcated in the Institute of Physiology and Biophysics, Academy of Sciences of Uzbekistan and Institute of Microelectronics, as well as embedded in the learning process at the Department of «Computer Systems» Tashkent University of Information Technology. The total economic effect is 10 million sum in a year.
Field of application: the thesis methods, algorithms and software designed in dissertation work can be used in medicine, biology, geophysics, ecology, seismology, speech processing and audio signals.

Subjects of research: Fiber-optical systems of transfer of the information, optical units and elements of fiber-optical communication lines (FOCL).
Purpose of work: Research and development of methods of regeneration of optical signals for restoration of their spectral characteristics and strengthenings of intensity by means of AOTF.
Methods of research: Methods of improvement of spectral characteristics FOST, linearization their through passage characteristics with use of filters AORF, testing of separate devices and units FOCL, experimental definition of sizes of distortions of spectral characteristics, statistical processing of experimental results.
The results obtained and their novelty: Basic bases of use akusto optical effects for research of spectral characteristics of fiber-optical systems of transfer of the information are established. The technique of optimization of spectral characteristics FOST on the basis of use acoustic optical filters is created. The functional structure of the measuring stand and a complex is developed for researches of parameters FOCL and imitation of the phenomena occuring in real high-speed fiber-optical systems of transfer of the information. The model of the optical radiation approached on the spectral structure to radiation in FOST is created.
Practical value: On application of the received results practical recommendations are developed for indemnification of distortions of a contour of through passage spectral characteristics FOST. Developed the stand and a measuring complex are recommended for researches of spectral characteristics of elements and units FOCL and selection of optimum operating modes.
Degree of embed and economic effectivity: Results of work are given for introduction in enterprises of the UzACI, in the telecommunication companies and in educational process of the TUIT.
Field of application: In FOST for improvement of spectral characteristics and for testing elements and units FOCL.

Objects of study: Degenerating equations of high odd order with multiple characteristics.
Purpose of the work: Studying the existence and uniqueness of boundary problems for degenerating equations of high odd order with multiple characteristics, finding private values, constructing automodcl solution for degenerating equation of high odd order.
Method of study: It was applied Fourier method, method of similarity and other methods in solving the equations with private derivative.
Obtained results and their novelty: the existence and uniqueness of stated boundary problem for degenerating equations of high odd order with multiple characteristics was studied, private values was found, and automodcl solution for studied equation was constructed.
All of the results of dissertation arc new.
Practical importance: the dissertation has theoretical importance.
Fields of application: The results of the dissertation can be applied to studying degenerating differential equations with private derivative, and to the problems of physics and mechanics.

Subjects of research: matrixes of proximity, images, directed and undirected graphs
Purpose of work: the development and study of minimax model of layered clusterization based on conception of monotonous proximity function
Methods of research: the methods of discrete mathematics, linguistic data analysis, classification, cluster analysis, image processing. Software was realized on C++, MATLAB, C#.
The results obtained and their novelty:
- development of core clusterization parametric model. Based on conception of monotonous proximity function;
- the method of image segmentation based on core clusterization, and its’ efficiency in comparing with method of normalized cut and k-mcans;
- the procedure of protein sequences multiple alignment based on core clusterization was evaluated in comparing with known software such as CLUSTAL and DIALIGN.
Practical value: developed software can be used in different applications in area of data and image processing, particularly in bioinformatics problems solution.
Degree of embed and economic effectivity: the results of dissertation were used as software package for image processing and teaching needs in information system department of Murom Institute of Vladimir state University University and as software for computing diagnostics of cholecystitis different forms in Republican Scientific Center of Emergency Medicine, Uzbekistan Ministry of Healthcare. Efficiency is social impact on research, treatment of patients, and education.
Field of application: Data Mining and decision-making systems in biology and healthcare.

In this work                                                                  

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0  ?=1

= 1, 2, ⋯ ?; ? ≤ ?

The approximate solution of the system of integral equations under the conditions ??(?) ∈ ?¹(?1), ??? ∈  was built using an evenly distributed grid and the error of the solution was estimated.

In this work, combining the method of optimal coefficients with the iteration method, the following

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= ?(?)

1          1

+ ? ∫ ⋯ ∫ ?(?, ?)?(?)??

0          0

An approximate solution of the Fredholm integral

equation of the 2nd type was found and the residual was evaluated. Let the free term and kernel in this equation satisfy the following condition:         ?(?) ∈ ?^?(? )

?       1

?(?, ?) ∈ ?^? (?₂)    .

2?

Subject of the inquiry: methods and algorithms of dynamic filtering and estimation conditions of dynamic object control.
Aim of the inquiry: developing algorithm of firm estimation conditions of dynamic objects control on the basis of the concept of adaptive filtering and their practical application in the solution of the problems of automation by a certain manufacturing process.
Method of inquiry: methods of system analysis, identification, dynamic filtering, adaptive control and solution of problems set forth incorrectly.
The results achieved and their novelty: regular iterative algorithms of the adaptive evaluation of the elements of matrix factor of the Kalman filter; the algorithms of firm adaptive estimation of the vector of the condition of apriori uncertainty covariations matrixes of the noise of the object and hindrances of the measurements; the regularized algorithms of adaptive estimation of the conditions of auto- and mutual correlated noise of the object and hindrances of the measurements; the adaptive regulation system by technological process of granulations-drying of steamed pulps in production of granulated ammofos. Novelty of the work is in the development of algorithm of firm estimation of conditions of dynamic objects control on the basis of the concept of adaptive filtering and computing schemes and their practical realization.
Practical value: the practical value of the results of the research is the development of mathematical and algorithmic solution of the problems of adaptive filtering and syntheses of regulation system of a wide range class of technological objects. The designed algorithms of firm estimation of conditions of operated object can be widely used in building of functional structures and automations of the designing adaptive regulation system of technological process with ceaseless production.
Degree of embed and economic effectivity: the results of the research arc accepted for introduction in design works on system development of adaptive management of technological process of granulations-drying ammofos pulps on Almalik plant "AMMOFOS". Expected annual economic benefit makes 4 million and 680 000 sum.
Sphere of usage: results of research and development can be used at the enterprises of chemical and processing industries with continuous manufacturing.

Subject of the inquiry: semigroup operators in Banach - Kantorovich spaces and Markov processes in Banach - Kantorovich spaces E[LP].
Aim of the inquiry: The aim of the thesis is generalization of the theory of semigroup operators for Banach - Kantorovich spaces.
Methods of inquiry: In the work methods of measurable Banach bundles, of functional analysis, of the theory of Banach - Kantorovich spaces, of Markov processes are used.
The results obtained and their novelty: All obtained results of the thesis arc new and consist of the following:
- representation of the Lo -bounded semigroup of Lo -bounded LQ -linear operators in Banach - Kantorovich spaces in the form of a measurable bundle of semigroups of bounded operators;
- representation of strongly continuous semigroups of operators in Banach -Kantorovich spaces with of strongly continuous semigroups of operators of bundles is discribcd;
- representation of infinitczimial operators of the semigroup of Lo -bounded Lo -linear operators with the help of measurable bundle of semigroups of operator is given;
- description of the semigroup operators appeared in the result of Markov processes in Banach - Kantorovich spaces E[Lp] and the variants of the static and individual ergodic theorem for all.
Practical value: The work has a theoretical character.
Degree of embed and economic effectivity: The results and methods introduced in the work can be used in special courses on functional analysis, of the theory of Banach - Kantorovich spaces and the of ergodic theory.
Field of application: The theory of Banach - Kantorovich spaces, the ergodic theory.

Subject of the inquiry: Non-commutative /Л-spaces of measurable operators associated with a Maharam trace, Orlicz-Kantorovich space.
Aim of the inquire: Construction theory non-commutative integration for Maharam trace with the values in a complex Dedekind complete Riesz spaces. Description non-commutative Lp -spaces associated with a Maharam trace. Construction theory of Orlicz-Kantorovich lattices.
Method of inquire: Methods of functional analysis, theory of operator algebras and theory of measurable Banach bundles are used.
The results achieved and their novelty: A complete description of Maharam trace on von Neumann algebra with the values in a complex Dedekind complete Riesz spaces is given; non-commutative integration for Maharam trace is constructed; new class Banach-Kantorovich space - non-commutative Lp-spaces associated with a Maharam trace - is defined and their dual spaces is described; new class Orlicz-Kantorovich lattices associated with a disjointly decomposable Z,0-valued measure is constructed; explained condition, in which they re flexed; some version of ergodic theorems for positive contractions in Orlicz-Kantorovich lattices is established; the class a complete Boolean algebras with disjointly decomposable L°-valued measure, representable as a measurable bundle of continuous (respectively, atomic) Boolean algebras is divided.
Practical value: The work has a theoretical character.
Degree of embed and economic effectivity: Results and methods introduced in the thesis can be used in reading special courses on functional analysis and theory of operator algebras.
Sphere of usage: Theory of operator algebras, the vector-valued measure theory, the ergodic theory, and theory of Banach-Kantorovich space.

The topicality and significance of the subject of dissertation. In the countries of the world there arc different levels of water availability. On average, every person on Earth is necessary to 24,646 m3 (24.65 million liters) of water per year.1 At the present time, with the increase of the world's population, grows the demand for drinking water. According to the statistics of the world population of the annual needs of drinking water consumption is 64 million m3. There is a trend of regular reduction of drinking water supplies. According to the results of research to the 2025-2030 year, 47% of the planet population countries there is a shortage of water.2 Providing the population of the world states with drinking water and the improvement of methods of the analysis of a condition of the hydrosphere of underground waters, increase of efficiency of carrying out hydrogeological experiences for operation of environmentally clean waters, to definition of information uncertainties connected with domination of information belonging to hydrogeological objects is paid separate attention.
In the Republic of Uzbekistan large-scale events for the effective organization of measures for formation and operation of water intaking underground waters arc held. In this area, including the development and improvement of the development of mechanisms for the rational use of water resources taking into account the characteristics of each region, an analysis of the needs and requirements of the population of drinking and household water, the creation of technology and desalination methods anomalies highly mineralized groundwater, the composition and volume of reserves one- and double-layer groundwater.
The world's attention is paid to the development of methods and algorithms of fuzzy-deterministic simulation of the formation and operation of underground water-bearing structures on the basis of seasonally-rcgional features and nextgeneration computerized system. In this area, the implementation of targeted research arc priority tasks, including scientific research in the following areas: creation of a complex of the software means and mathematical apparatuses intended for a solution of a problem of a freshening of anomalies of highly mineralized underground waters in strongly salted conditions one - and a two-layer structure of water-bearing layers; development of the indistinct determined mathematical models of dynamic supervision of water resources in the course of formation, operation and restoration of GWI in one - and two-layer water-bearing layers; algorithm elaboration and methods of the indistinct determined modeling of seasonal and territorial processes of formation, operation and restoration of GWI; determination of regularities of research of hydrogeological, technological and ecological bases of functional and structural formation one - or two-layer GWI; development of structure of the computerized monitoring system of GWI based on information integration of decision-making processes and the indistinct determined modeling of GWI on the basis of wireless sensor networks;
This dissertation research to a certain extent is the implementation of the tasks provided for in the law of the Republic of Uzbekistan "On water and water use" (ZRU-837-XII of May 6, 1993), in the Republic of Uzbekistan President Decree № PP-1989 "On measures for further development of the National information and communication system of the Republic of Uzbekistan "dated 27 June 2013, and in the Decree of the President of the Republic of Uzbekistan № PP-2264" on investment program of Uzbekistan for 2015 "dated 17 November 2014, as well as in the Resolution of the Cabinet of Ministers on 19 March 2013 №82 "The order of water use and consumption in the Republic of Uzbekistan".
The aim of the research is to create methods, algorithms and computerized system of fuzzy-deterministic modeling processes based on the seasonally-tcrritorial peculiarities of formation and functioning of intakes of groundwater and for making decisions on the rational use of groundwater resources.
Scientific novelty of dissertational research consists in the following:
complex algorithms and software of fuzzy-deterministic modeling of the formation of territorial and seasonal, seasonal maintenance and restoration of GWI were developed;
were designed information- identification and information technology models to establish information linkages between the GWI and its fuzzy deterministic models;
were offered the structure of a computerized system of information integration processes fuzzy-deterministic modeling and decision-making based on wireless sensor networks;
were developed fuzzy-deterministic models, algorithms and software to solve problems freshening anomalies highly mineralized groundwater for conditions of single and double layer structure of the aquifer;
the principles of parallelization algorithms and software for the not well-deterministic modeling gcofiltrational processes;
CONCLUSION
In the course of the research produced the following results: On the basis of the survey on his doctoral dissertation on the topic "The system of fuzzy-deterministic modeling of the formation and maintenance of the groundwater intake" presented the following conclusions:
1. Algorithms and program complex fuzzy-deterministic modeling of hydrogeological objects of groundwater intakes of natural and man-made character allows to forecast the dynamics of change and estimate groundwater resources with sufficient reliability.
2. Technology of formation of groundwater resources through the creation of GWI conditions for one and multi-layer structure of aquifers makes it possible to operational study of formation of elements of groundwater resources and them operational management.
3. Are proposed fuzzy deterministic models for the formalization of the processes of formation, maintenance and restoration of GWI in conditions of a heterogeneity of filtration area in plan and in section, fuzziness of initial and boundary conditions, uncertainty of operating sources of the surface water and water wells.
4. Arc proposed fuzzy-detcrministic mathematical models of filtration and salt transport in the subsurface hydrosphere to restore groundwater quality GWI, based on the hypothesis Myatiev - Girinsky, according to which the flow of salts in the section is taken as a two-dimensional at quasi- two-dimensional considering the flow of groundwater in interacting layers. Terms of the applicability of such a mathematical model for the restoration of groundwater quality GWI in conditions of two-layer structure of the aquifer.
5. The proposed algorithms and complex programs implementing technological schemes GWI allow reclosure adequately to take into account in the fuzzy-detcrministic mathematical models designs, styles, parameters and infiltration of water intake structures.
6. Arc considered the problems associated with the decomposition of the modeled process into a number of tasks running simultaneously on the basis of segmentation into physical processes, tasks, data strategy, visualization and computational processes. For example, studies of the process of GWI projects, arc examined aspects of parallelism of computing in the process of FDM GWI. The results of the parallel solution of the problem on the justification of groundwater sampling mode under various boundary conditions arc obtained for the four options. At the same time, for a consistent solution of this problem it took 83 msec, and solutions in parallelization mode received 2 msec.
7. The principles, algorithms and program codes for information modeling GWI, taking into account technological components - canals, rivers, infiltration basins and wells GWI, allowing to establish the relationship between the GWI and its FDM, as well as allowing to organize the computational experiments in order to ensure the possibility of varying the different parameters of environment and the boundary conditions environment during the numerical modeling.
8. It is proposed an information base for information integration processes of fuzzy deterministic modeling GWI and decision-making, allowing to organize the continuous measurements of the parameters of GWI (levels, salinity and temperature of groundwater), eliminate the human factor from the process of measurements, to provide rclevantness data by of transmission wireless sensor networks.
9. Arc conducted computational experiments on the use of gcoinformation technologies to determine geofiltrational parameters in terms of heterogeneity filtration on the basis of data on the distribution of groundwater levels and degree their mineralization, as well as the boundary conditions on the basis of their registration as a separate thematic layers of GIS GWI model.

The topicality and significance of the subject of dissertation. Recently science has seen a rapid development of theory related fields, i.e. the mutual influence of two or more physical fields, in particular, a typical example of this direction of research is magneto-elasticity. Electromagnetic sensors arc in high demand at the present time in the world, according to forecasts only for the automotive market the proceeds of their sales in 2012 amounted to 812,2 million US dollars, next 2013 year they increased by 9,5%, and in subsequent two years this indicator grew by 6-7%, and the end of 2016 it is expected that the amount of revenue will reach 1,1 billion U.S. dollars1.
In Uzbekistan, held large-scale activities on the use of magneto-thin bodies in the technical designs and identify the influence of electromagnetic fields on the on deformation state of thin electro-conductive bodies. In this area, it is important to develop methods for determining the effects of electromagnetic fields on the deformation state of thin electro-conductive bodies of complex configuration, development of methods and algorithms for solving systems of differential equation in partial derivatives with initial-boundary conditions defining the magnctoelastic thin plates and shells of complex structural shapes, aimed at study the principles of creating magnctocumulativc generators for plasma confinement devices in fusion devices, magneto-hydrodynamic accelerators of contactless magnetic poles moving systems, high-quality and long-term use of measuring equipment, operating in the area of influence of electromagnetic fields.
In world practice, focuses on process modeling effects of electromagnetic fields on the deformation state of thin clcctro-conductivc bodies, development of mathematical models and numerical-analytical methods for solving partial differential equations derived from the initial and boundary conditions governing magnctoelastisity thin plates and shells of complex structural forms, using the method of R -functions formation of systems and structures of solutions satisfying the boundary conditions for the magnctoelastic plates and shells of complex configuration that is of particular interest from the scientific community. In this area, the implementation of targeted research arc priority tasks, including scientific research in the following areas: development of numerical and analytical methods and algorithms for solving systems of differential equations in partial derivatives with initial-boundary conditions, describing the influence of electromagnetic fields on the thin electro-conductive bodies (plates, shells) of complex configuration; development of complex software tools using the method of R-functions, magnctoelasticity thin bodies of complex shape, calculation algorithms of the class of problems of magnctoelasticity thin plates and shells of complex shape; conducting computational experiments to determine the degree of influence of electromagnetic fields on thin plates and shells with complex structural form, the development of algorithms for solving problems of statics and dynamics magnctoelasticity thin bodies.
In Uzbekistan, the modeling of the effects of electromagnetic fields on the state of deformation of thin clcctro-conductivc bodies, theory of magnetic elasticity for the interaction of the deformation field and the electromagnetic field in a solid clastic body is aimed at the study of the principles of creating magneto-cumulative generators, devices for plasma confinement in thermonuclear facility, magneto-hydrodynamic accelerators, contactless magnetic bearing of the moving systems, measuring equipment, working in the field of action of electromagnetic fields. In various industries technical and economic reliability from the practical application of magneto-elastic sensors is characterized by the error of their component errors of 2-3%.
The thesis is directly serve the implementation of the tasks set out in the following provisions of the President of the Republic of Uzbekistan: PP-1730 of 21 March 2012 «On measures for further implementation and development of modern information and communication technologies», the PP-1442 of 15 December 2010 «On the priorities of industrial development of Uzbekistan in 2011-2015 years», and in the decree of the Cabinet of Ministers of the Republic of Uzbekistan №64 of 7 March 2012 «On additional measures on decreasing production expenses and reduction of production cost in industry» and also in other standard legal documents accepted to the sphere.
The use of thin electro-conductive bodies in the elements of constructions of devices and machines under the influence of electromagnetic fields in modem electronic, medical and other measuring systems, as well as in communication devices, radio engineering and computer science establishes topicality of the research problems of mutual influence of electromagnetic fields and electro-conductive thin bodies having a complex configuration by R-function method (RFM).
The purpose of research is to develop algorithms and software tools of mathematical modeling of electromagnetic fields influence on the deformation state of the thin electro-conductive bodies of complex configuration using the R-functions and numerical-analytical methods.
Scientific novelty of dissertational research consists in the following:
a mathematical model describing the processes of influence of electromagnetic fields on the deformation state of thin electro-conductive bodies is built on the basis of the generalized variational Hamilton-Ostrogradsky principle with the terms of the linear theory of elasticity and Lorentz electromagnetic forces, the mathematical model of magnetic elasticity of thin plates and shells is constructed;
qualitative numerical-analytical methods and algorithms for solving systems of differential equations with initial-boundary conditions describing the effect of electromagnetic fields on clcctro-conductivc thin bodies (plates and shells) with a complex structural shape of the joint application of the variational Bubnov-Galcrkin method and the structural R-functions method is developed;
solutions structures and systems to practical boundary conditions at rigid-clamped, hinged-simply supported edge magnetoelastic plates and shells with complex configuration (with cuts) is formed;
complex software for calculation of magneto-elasticity of thin plates and shells is developed on conducted algorithms for solving problem classes of thin plates and shells magneto-elasticity with complex structural form;
the convergence of the computational algorithm concerning the number of coordinate functions of the solutions structure is shown, the practical applicability of the method and the reliability of the obtained numerical calculation results of magnetic elasticity of thin bodies by comparing with the exact solutions is proved;
the algorithms for carrying out computational experiments to study the static and dynamic effects of electromagnetic fields on the deformation state of the thin perfectly conducting bodies with complex structural form is developed.
CONCLUSION
On the basis of studies on the doctoral thesis « Mathematical modeling of processes of the electromagnetic fields’ effects on deformational condition of thin conductive bodies by the method of R-function» presented the following conclusions:
1. The fundamental geometric and physical relationships of the linear elasticity theory and linear electrodynamics arc defined taking into account properties of the structure and mechanical characteristics of the material for electro-conductive thin bodies under the influence of electromagnetic forces;
2. new mathematical models arc developed and a two-dimensional mathematical model of magnetic elasticity of thin shells and plates is built on the basis of generalized principle of Hamilton-Ostrogradsky using the Kirchhoff-Lyav hypothesis for thin bodies taking into account the linear Cauchy relations and Hooke's law of elasticity and relations of the linear theory of electrodynamics, in particular, Maxwell's equations, the influence of the electromagnetic field is determined by the volume of Lorentz pondcromotive forces but the surface and contour forces arc defined by Maxwell's electromagnetic tensor.
3. analytical and numerical methods and algorithms for solving systems of differential equations with initial-boundary conditions describing the effect of electromagnetic fields on the deformation state of the conductive thin bodies (plates and shells) complex shape with a joint application of the variational method of Bubnov-Galcrkin method and the structural R-functions method arc developed and the resolving equations (discrete model) arc obtained.
4. solution structure (sequence of coordinate functions) to the basic boundary value problem of magneto-clastic plates and shells with complex configuration area (a circle with two and four circular cutouts, polygon, rectangle with rounded comers, etc.) by the method of R-functions is formed and normalized equations for complex fields of the thin bodies, using card operations of algebraic the R-functions theory is constructed;
5. Vector-matrix equations for discrete models of magnetic elasticity of the subtle bodies, formed by the corresponding block of the matrix of damping, etc. when modeling thin-walled structures defined by systems of linear algebraic and ordinary differential equations with initial conditions and numerical methods for solving these systems of equations based on the use of quadrature sums, methods of Newmark and Gaussian elimination is developed;
6. software in the form of a complex of programs for calculation of magnetic elasticity of thin bodies by the method of R-functions on the computer, consisting of ten core modules is developed on the base of modular analysis of algorithms for solving problem classes of magnetic elasticity of thin plates and shells with complex shape;
7. numcrically-analytical methods arc developed and the validity of the obtained numerical calculation results of magnetic elasticity for thin plates of areas a classic shape (square, circle) by comparing exact and approximate solutions by the R-functions method is substantiated, moreover the plates having rigidly-clamped and hingcd-simply supported boundary conditions arc considered. The convergence of the computational algorithm of calculating the magnetic elasticity of thin shells and plates with complex structural form with regard to the number of coordinate functions of the structure of the solutions built by R-functions method and on the number of nodes (points) when calculating double integrals is studied. As the basic polynomial is selected by power polynomial, and good convergence is observed when the degree of the polynomial 3-4 (which corresponds to 10-15 coordinate functions).
8. computational experiments on the solution of problems of statics of magnetic elasticity for thin plates of complex configuration (with two circle and four circular cutouts, complex polygon shape, a ring) arc described on the basis of the developed algorithmic and software Toolkit (software package). The effect of static electromagnetic field on the deformation state of the plate with rigidly-clamped and hinged boundary conditions at a given magnetic field with different values and directions of the magnetic field is shown;
9. The dynamic effect of the electromagnetic field on the deformation state of the plate with rigidly-clamped and hinged boundary conditions on the basis of the developed algorithmic and software complex and computational experiments on problems of dynamics of magnetic elasticity of thin bodies for areas with a complex configuration by the R-functions method is studied. Plates of constant thickness, made of a material with finite electrical conductivity in an external electromagnetic field arc considered. This problem is solved in two stages: the first is the problem of electrostatics and determine the values of the magnetic field, the second solves the problem of magnetic elasticity using the values of the magnetic field. The dynamic effect of the electromagnetic field on the deformation state of the thin bodies of complex structural forms is defined.
10. The results obtained in the form of algorithmic and software tools arc implemented in the solution of specific problems of magnetic elasticity of thin shells and plates with complex configuration in the framework of the contract and economic efficiency in the amount of 127.8 million soums is obtained as a result of the implementation.

The urgency and relevance of the theme of dissertation. The solution of a number of fundamental problems in the field of various applications on a global level requires the establishment of science refined mathematical models of physical processes under study, the development of new methods of research and implementation of the results into practice. Based on the needs of practice, increased attention to the theory of equations of high order, in particular, to the theory of partial differential equations of the third order. Among the third-order equations occupy a special place of the equation with multiple characteristics thanks to its specific characteristics. To study the small but finite amplitude waves in dispersive media as a model equation is often used Kortcweg - de Vries equation, which is a nonlinear equation of the third order with multiple characteristics, comprising a first derivative with respect to time. The developed theory for these equations was the impetus for starting research and for other classes of equations - third-order equations with multiple characteristics, containing the second derivatives with respect to time. Due to the complexity of the processes associated with the above equations and the lack of sufficiently developed analytical methods, the study of third-order equations with multiple characteristics, containing the second derivatives with respect to time, is one of the priority areas.
The scientists of our country obtained significant results in studies of third-order equations with multiple characteristics, comprising a first derivative with respect to time. For such high-ordcr equations constructed fundamental solutions, expressed in terms of special functions, studied their properties and behavior, also solved the boundary value problems. Using the fundamental solution built L.Cattabriga, we investigated boundary value problems for a third order equations with multiple characteristics, containing the second derivatives with respect to time. According to the equations of the mixed type and the composite high-order and mixed-composite type achieved certain results recognized in the world. The equations of the third order with multiple characteristics, containing the second derivatives with respect to time, require the construction of fundamental solutions through a special function, the study of their properties and behavior and decisions with their help, boundary value problems, and this requires a search for new approaches to solving this problem.
Research processes of nonlinear acoustics, the hydrodynamic theory of cosmic plasmas, nonlinear vibrations, fluid flow in the channel, surrounded by a porous medium, etc. associated with the study of a third-order equation with multiple characteristics, containing the second derivatives with respect to time, as well as problems for equations of mixed parabolic - hyperbolic type, which explains the need to study these equations.
Research in this thesis to some extent arc the challenges identified in the Republic of Uzbekistan Presidential Decree nubmer. PR-436 of 7 August 2006 "On measures to improve the coordination and management of the development of science and technology", the number PP-916 from July 15, 2008 "On additional measures to stimulate innovative projects and technologies" and other normative-legal acts of fundamental sciences.
The aim ofresearch workis to construct the fundamental and analytic solutions for the third order equations with multiple characteristics which have second order derivative in time and in three dimensional space to solve the boundary value problems for parabola hyperbolic equations.
Scientific novelty of the research workis as follows:
the theory of analytical and fundamental solutions for equations of third order with multiple characteristics containing the second derivative in time has been constructed with the help of special functions;
for the first time the algorithm of solution for the boundary value problems by the Fourier method is worked out for the third order equations with multiple characteristics containing the second derivative in time;
the potential theory for equations of third order with multiple characteristics containing the second derivative with respect to time is developed;
in the solution of the boundary value problems, stated for equations of third order with multiple characteristics containing the second derivative in time, the Green functions are constructed;
the Fourier algorithm is applied to the solution of the boundary value problems for equations of third order with multiple characteristics containing the second derivative in time;
the unique solvability of Tricomi and Gellerstedt's problems for mixed parabolic - hyperbolic equation in three-dimensional space is showen;
the necessary and sufficient conditions of existence of the direct and inverse Fourier integral transforms in solution of the boundary value problems in three -dimensional space arc justified.
CONCLUSION
The dissertation work is devoted to development of the theory of fundamental solutions and theory of potential, construction of the constructive theory of the Fourier method for third-order equations with multiple characteristics containing the second derivative with respect to time, as well as proof of the unique solvability of boundary value problems for mixed parabolic-hyperbolic equations in a three-dimensional space
The main results of the resarch arc the following.
1. The theory of analytical and fundamental solutions for equations of third order with multiple characteristics containing the second derivative in time are constructed with the help of special functions.
2. For the first time the algorithm of solution for the boundary value problems by the Fourier method is worked out for the third order equations with multiple characteristics containing the second derivative in time.
3. The potential theory for equations of the third order with multiple characteristics containing the second derivative with respect to time is fully justified.
4. The Green functions are constructed to solution of boundary value problems for equations of the third order with multiple characteristics containing the second derivative in time.
5. The Fourier algorithm is applied to solution of the boundary value problems, for degenerate equations of third order with multiple characteristics containing the second derivative with respect to time.
6. The uniquely solvability of Tricomi and Gellerstedt's problems for mixed parabolic - hyperbolic equation in three-dimensional space is proved.
7. The necessary and sufficient conditions for the existence of the direct and inverse Fourier transform for solution of the boundary value problems in three-dimensional space are found.

Actuality and demand of the theme of dissertation. Due to the rapid development of scientific and technological progress in the world, required the development of new methods for fundamental research and implementation of the results into practice. According to the demand of practice on the intersection of differential equations and differential topology French scientists established fundamental foundations of the theory of foliated manifolds. It is proved the stability of compact foliations and that limit sets of leaves arc invariant sets. Scientists of USA and Russia investigated qualitative theory of foliations, in which geometric and topological properties of foliated manifolds arc investigated. Moreover, an application of theory of foliations to practice is one of the important problems of geometry.
After the declaring independence of our country attention to the actual directions in the field of natural and exact sciences has been significantly increased, in particular, especial attention is given to applications of methods and results of this theory to the theory of optimal control and dynamical systems. In this field arc obtained sufficient condition of stability for controllable systems, proved non-negativity of sectional curvatures of leaves of foliations generated by Ricmannian submersions in the spaces of non-negative curvature, besides arc obtained valuable results on investigating geometry of vector fields.
Nowadays, researches in the world in the area of geometry of orbits of vector fields on manifold related to the theory of dynamical polysystems and optimal control and very important for them. In this field, the essential task is a wide application of targeted researches, in particular, applications of results obtained on foliations, generated by Ricmannian submersions, to determine the structure of phase spaces of dynamical polisystem: application of methods of the theory of foliations to the theory of dynamical systems, optimal control and to various problems in other fields; investigation of geometry of foliations generated by Ricmannian submersions; investigation of geometry of Ricmannian foliations on surfaces of non-negative sectional curvature. Scientific researches on above-mentioned directions justify the relevance of the theme of current dissertation.
Research in this thesis to some extent arc the challenges identified in the Republic of Uzbekistan Presidential Decree number PP-436 of 7 August 2006 «On measures to improve the coordination and management of the development of science and technology», as well as PP-2204 from July 8, 2014 «On measures to further optimize the structure of the Academy of sciences of the Republic of Uzbekistan and to strengthen the integration of academic science and higher education of the Republic» and other normative-legal acts of fundamental sciences.
The aims of the research arc the investigations of the geometry and topology of the foliated manifolds, the structure of the group of isometries of foliated manifolds and the geometry of foliated manifolds of constant sectional curvature, and also application of the taken results for the investigations of the attainability set and establishing of the continuous dependence on the initial point of the attainability set.
Scientific novelty of the research work is as follows:
It is proved that any group of homeomorphisms of a smooth manifold is a topological group in the compact-open topology;
It is proved that isometry group foliated manifold is a topological group in the compact-open topology;
it is showed that if a sequence of isometries foliated manifolds converge on at the point on each leaf, then this sequence can be extracted to a convergent subsequence of isometrics of foliated manifold in the compact-open topology;
it is proved that if the foliation generated by Ricmannian submersion, then leaves of foliation arc manifolds of constant Gaussian curvature;
it is showed that the limit of the geodesic lines of foliated manifold is a geodesic line on the limit leaf of foliation;
It is proved the existence of the foliation, for which there is a isometry of foliated manifold which is non-isomety of the manifold;
It is proved the compactness of the attainability set, and the continuity of the multi valued mapping «the point - the attainability set» for a system of vector fields of a special form;
it is showed compactness of the closure of the attainable set for the time does not exceed a fixed time, and the continuous dependence of the attainable set from time for a certain class of vector fields;
it is found conditions in order to attainable set (the set of controllability) coincided with the fixed dimension planes for linear systems.
CONCLUSION
The dissertation examines the group of isometries foliated Ricmannian manifolds. To solve the problems studied topological and geometrical properties of foliated manifolds, it is studied geometry Ricmannian submersions. It is introduced a new notion of foliated compact-open topology, which depends on the foliation. We studied a group of isometrics foliated manifolds in the compact-open topology and in the foliated compact-open topology.
The main results of investigation arc as follows.
1) it is established that the group of homeomorphisms of any manifold is a topological group in the compact-open topology;
2) it is proved that the isometry group of a foliated manifold is a topological group in the compact-open topology;
3) it is established that if the sequence of isometries of foliated manifold converge on a one point on each leaf then this sequence can be extracted to a convergent subsequence isometries of foliated manifold in the compact-open topology;
4) it is proved that the Ricmannian submersion produces foliated manifold of constant Gaussian curvature;
5) it is established that limit of geodesic lines of foliated manifold is a geodesic line on the limit leaf of foliation;
6) it is showed that the four-dimensional manifold can not be immersed into five-dimensional Euclidean space;
7) it is proved the existence of the foliation, for which there is a Isometry of foliated manifold which is non-isomety of the manifold;
8) it is proved that the set of reachable system of vector fields of a certain class is compact, and it is a continuous function of time;
9) it is found a sufficient condition for linear control systems under which each set of attainability (controllability set) is a plane of fixed dimension.
Author brings his deep appreciation to the scientific adviser Professor Narmanov Abdigappar Yakubovich for posing problems, valuable tips and helpful advices in the discussion and and support.

The urgency and relevance of the theme of dissertation. Today in the world practice of the the natural sciences the development of methods of efficiency of the reaction-diffusion processes control system, study of nonlinear mathematical models is considered one of the most urgent tasks. According to the Elsevier information base the scientific works of scientists around the world devoted to the study of nonlinear reaction-diffusion equation, as the Cauchy problem and boundary-value problems to this equation and their practical applications.
In the Republic of Uzbekistan conducted extensive works on the effective organization of events dedicated to the development of automated systems for the computer visualization of diffusion processes, mathematical modeling of diffusion processes described by nonlinear equations with double nonlinearity in a heterogeneous environment. At the same time, carried out a series of research projects dedicated to the improvement of research methods and visualization of non-linear process, the creation of automated production systems, which play an important role in the study of mathematical models of nonlinear processes.
Currently, in the world a number of fundamental problems require mathematical modeling of nonlinear processes, the improvement of the methods and visualization tools, and applying to the practice of obtaining important results of the reaction-diffusion problems with double nonlinearity. At present, the study of equations with double non-linearity and practical application, conducting targeted research on the following areas is considered one of the most important tasks: the development of imaging methods in the study of nonlinear models; creating software systems that help the study of nonlinear processes; creating technology of computational experiment, monitoring the evolution over time of the process, the establishment of a computerized system of determining the properties of which depend on the parameters of the dynamics of change. Research carried out on the above areas of research, indicate the relevance of the topic of this thesis.
Research of this thesis, to a certain extent, serve to implement the objectives of all legal acts on this activity, the decree of the President on March 21, 2012 № PP-1730 «On measures for further implementation and development of modem information and communication technologies», dated December 15, 2010 № PP-1442 «On the priorities of industrial development of Uzbekistan in 2011-2015», Resolution of the Cabinet of Ministers dated February 1, 2012 № 24 «On measures to create conditions for further development of computerization and information and communication field of technology», as well as other legal documents adopted in this area.
The aim of research work is consists of mathematical modeling of reaction-diffusion, heat conductivity, luquids and gas distribution, filtration processes described degenerate parabolic equations and systems of equations with double nonlinearity with the influence of the source and absorption.
Scientific novelty of the research work is as follows:
the properties of finite speed of distribution of the disturbance and spatial localization, proof of the asymptotic behavior of blow-up solutions of reactiondiffusion systems with double nonlinearity and with variable density have been studied;
global solvability of the Cauchy problem for a model reaction-diffusion equation with double non-linearity to the source or absorption have been proven;
the estimates of the solution, and the front and for the class of systems of equations with double nonlinearity parabolic, asymptotic expressions of generalized solutions with compact support for degenerate nonlinear self-similar equations and systems have been produced;
the solution of type Zcldovich-Barcnblatt to the mutual diffusion system with double nonlinearity with a source and convective transfer have been found, the properties of finite speed of distribution of disturbations and spatial localization arc shown and an algorithm for determining the value of the critical exponent have been developed;
on based of the properties studied nonlinear mathematical models, have been built an iterative process that converges quickly;
nonlinear problem for a system or a degenerate parabolic equation with double non-linearity, take into account external factors and properties of the medium (variable density, medium conductivity, convective transfer, etc.) has numerically solved.
CONCLUSION
On the topic of the doctoral thesis «Mathematical modeling of reactiondiffusion systems with double nonlinearity» presented the following conclusions:
1. In solving problems of nonlinear models of reaction-diffusion, filtering, heat conductivity, as in homogeneous and in heterogeneous environments, based on the theoretical study by the self-analysis and the comparison principle, the analysis of the use of computational algorithms and software complexes isolated original properties and defined the further development of research.
2. The proposed methods arc used to study the properties of FSPD and localization solutions of nonlinear reaction-diffusion model with double nonlinearity for variable density environments by constraction solutions of type Zeldovich-Barcnblatt.
3. It is shown that the property FSPD and localization arc shown in moving nonlinear medium whose velocity depends on time.
4. For non-linear reaction-diffusion model in absorbing media or source found occurrence of localized wave structure.
5. Established property FSPD and spatial localization of a mathematical model of reaction-diffusion systems with double nonlinearity and with variable density.
6. It has been shown that there is a blow up property for the solutions of a system of self-rcaction-diffusion equations with double nonlinearity.
7. Built asymptotic behavior of generalized solutions with compact support and vanishing at infinity of solutions of self-similar equations or systems with double nonlinearity.
8. It is proved the global solvability of such problems for reaction-diffusion systems with double nonlinearity with the source or absorption.
9. At the critical exponent for the preparation of reaction-diffusion systems with double nonlinearity with the source or absorption and convective transfer use a universal algorithm.
10. A solution of type Zeldovich-Barcnblatt for nonlinear systems with cross property FSPD and spatial localization solutions.
11. The developed programs allows you to carry out computer simulations to study on the basis of the qualitative properties of nonlinear mathematical models of reaction-diffusion systems.
12. The developed computational schemes, algorithms and programm for solving a system of parabolic equations with double nonlinearity provide high performance in the study of the theory and process of numerical solution of such problems.

The urgency and relevance of the theme of dissertation. There is a great interest in the study of nonlinear models of a variety of phenomena and processes occurring in mechanics, physics, technology, biophysics, biology, ecology, medicine and other fields which arc described by nonlinear differential equations widely in science. The basis of these models in particular constitute arc parabolic type partial differential equations. In research of properties of studies and numerical solutions of the Cauchy problems and boundary value problems, approximation methods were applied. Here, the main place get degenerate equations and systems of parabolic type, which arc simulate different nonlinear processes occurring in the natural sciences.
In the independence years of our Republic research and the practical application of nonlinear models of a variety of physical, biological, and chemical processing that arc relevant areas of applied mathematics. From this point, scientific works carrying out on a number of mathematical models, which express the heat conductivity processes, filtration, biological population that have a practical application in the fields of energetic, medicine, oil and gas.
Is currently widely spread in the world of mathematical models of processes described received degenerate quasilincar parabolic equations, it is because they arc derived from the fundamental conservation laws. Therefore, it is possible when two physical processes that in common do not have seemingly anything arc described by the same nonlinear diffusion equation, only with different numerical parameters. Currently, the implementation of scientific research and practical application of these equations is one of the important problems that arc carried out in the following areas: development of methods for the study of qualitative properties of nonlinear mathematical models; finding accurate estimates of solutions in different spaces; definition of nonlinear effects; development of efficient numerical schemes; creating a set of programs for the study of mathematical models of nonlinear processes and evolution dynamics of the process in time. Scientific studies, which are conducted in all of these areas, explain the relevance of the topic of this thesis.
This dissertation research in a certain extent is the implementation of the tasks provided in the Resolution of the President of the Republic of Uzbekistan PP-1730 «On measures for further implementation and development of modem information and communication technologies», dated March 21, 2012, PP-1442 «On the priorities of industrial development of Uzbekistan in 2011-2015» dated December 15, 2010 and the Cabinet of Ministers of the Republic of Uzbekistan №24 «On measures to create conditions for further development computerizing and information communication technologies in the field» of 1 February 2012, and also in other legal instruments adopted in this area.
The aim of research work arc the numerical and analytical investigation of qualitative properties of nonlinear mathematical models describing degenerate quasilincar parabolic equations of heat conduction processes (filtration, diffusion) in homogenous and in a medium with variable density to the source and the nonlinear boundary condition, development complex programs for the numerical investigation of nonlinear boundary value problems.
Scientific novelties of the dissertation research are as follows:
the conditions of global solvability and nosolvability of solutions for nonlinear heat conduction model in a inhomogeneous medium without power with nonlocal boundary condition arc determined; 
determined the effect of heterogeneity of the medium at the conditions of global solvability and nosolvability for the whole time of the solutions of nonlinear problems;
it is found the value of the type Fujita critical exponent for the model describing the Neumann problem in the case of slow and fast diffusion;
it was found the value of the critical exponent of the global existence of the solution for the model described by the second type of boundary value problem in the case of slow and fast diffusion;
the upper and lower bounds for the generalized solutions of the problem of slow-diffiision heat conduction in homogeneous and inhomogeneous medium arc constructed;
were obtained the principal terms of the asymptotic behavior of various selfsimilar solutions of double and triple nonlinear heat conduction problem by the method of standard equations;
computational schemes have been proposed for the study of qualitative properties of nonlinear mathematical models of thermal conductivity with variable density, developed algorithms, complex programs in Visual Studio 2012 (C #) and visualized solutions of nonlinear problems.
CONCLUSION
On the basis of studies on the doctoral thesis "Mathematical modeling of the heat conduction processes in a medium with double nonlinearity" arc presented the following conclusions:
1. For nonlinear mathematical model of heat propagation, non-Newtonian polytrophic filtration, diffusion, described by nonlinear parabolic equations with nonlocal boundary condition and with variable density studied conditions for global solvability and no solvability solutions in time is established.
2. The critical exponent type Fujita and a critical exponent of solvability for nonlocal problem of heat propagation in an inhomogeneous medium arc found.
3. The upper and lower bounds of global and unbonded generalized solutions for nonlinear mathematical models of thermal conductivity with variable density and nonlocal boundary condition.
4. Established properties of finite speed of propagation of disturbances and spatial localization of solutions for nonlinear mathematical model of polytrophic filtration with double non-linearity and with variable density in the case of slow diffusion.
5. The properties of the infinite speed of propagation of disturbances of the nonlinear mathematical model for the polytrophic filtration with double nonlinearity and with variable density in the case of fast diffusion.
6. We prove the asymptotic behavior of generalized solutions with compact support of the Cauchy problem for a degenerate heat equation in an inhomogeneous medium with the source and with variable density.
7. The condition of the global solvability and no solvability solutions in time and asymptotic representation of solutions of systems of nonlinear equations for the modeling of polytrophic filtration with a nonlocal boundary condition with variable density is proved.
8. Installed above the qualitative properties of solutions and estimates solution of nonlinear problems with nonlocal boundary conditions allowed to conduct numerical calculations, giving new nonlinear effects.
9. The computing schemes, algorithms and software systems in the environment of Visual Studio 2012 (C #) for the numerical simulation of nonlinear problems of filtration and visualization arc developed.

Actuality and demand of the theme of dissertation. Numerous scientific and applied researches conducted around the world show the following fact: throughout physics stable composite objects are usually formed by way of attractive forces, which allow the constituents to lower their energy by binding together. Repulsive forces separate particles in a free space. However, in recent years scientists have proved that in a structured environment such a periodic potential and in the absence of dissipation, stable composite objects can exist even for repulsive interactions. The Bose-Hubbard models, which have been used to describe the repulsive pairs, i.e. the Schrodinger operators on lattices is the theoretical basis for the experimental observations and applications. In this regard, the study of Schrodinger operators, associated to Hamiltonians of systems of particles moving on lattices, which appear in models of solid state physics and lattice field theory, is one of the priority areas of science.
In our country in the years of independence, a great attention has been paid to scientific areas having a practical importance; in particular, a great emphasis has been placed on study of Schrodinger operators associated to Hamiltonian of a system of particles moving on integer lattices. Significant results have been achieved in finding conditions for the existence of bound states and for their number, the energy of which is located outside the essential spectrum, and also to the threshold effects of the essential spectrum for Schrodinger operators, associated to systems of two and three particles on lattices.
Since the spectrum of the family of the Schrodinger operators appears quite sensitive to a change of the quasi-momentum of system, solving problems related to the spectrum of these operators, in particular, to prove the existence of bound states as well as to determine their number depending to the quasi-momentum of system, for three particle discrete Schrodinger operators is of highly importance. In this regard, the implementation of investigations in the following directions is one of the main problems: to investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two identical particles (bosons or fermions) with short-range pair potentials on lattices; to establish the threshold phenomenon of the essential spectrum for these operators; to obtain an asymptotic formula for the number of eigenvalues for the three-particle Schrodinger operator associated to a system of three identical particles on the three-dimensional lattice with a short-range pair interaction; to show the existence of eigenvalues of the three-particle Schrodinger operator associated to a system of three identical particles on lattices of dimensions one and two. Many research activities carried out in the aforementioned scientific areas all around the world exhibit a great interest and motivation to the topic of dissertation.
The research conducted in this thesis corresponds to the tasks specified in the Decree of the President of the Republic of Uzbekistan № PD-436 on August 7, 2006 “On measures to improve coordination and management of the development of science and technology”, No. PD-916 from July 15, 2008 “On additional measures to stimulate innovative projects and technologies” and other normative and legal acts relating to the fundamental sciences.
The aim of the research is studying the essential and discrete spectrum of two and thrcc-particle Schrodinger operators associated to a system of two or three identical particles (bosons or fermions) with short-range pair potentials on lattice.
The scientific novelty consists of the following:
the conditions for existence of the eigenvalues outside the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential in all dimensions of the lattice is found;
the finiteness of the number of eigenvalues lying outside of the essential spectrum of the Schrodinger operator associated to a system of two identical particles (fermions) with a short-range potential on lattice is proved;
the number and location of eigenvalues of Schrodinger operator associated to a system of two particles (fermions), interacting on neighboring sites of lattice for all values of the parameters of the operator is determined;
the asymptotic behavior of eigenvalues lying below of the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice is studied;
the finiteness of the number of eigenvalues lying below the essential spectrum of the Schrodinger operator associated to a system of three particles (bosons) with short-range pair potentials in the three-dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero is shown;
the existence of eigenvalues of the Schrodinger operator associated to a system of three particles with pair two-particle zero-range potential in the one and two-dimensional lattices is proved. Our result is the first one in the theory of the thrcc-particle Schrodinger operators.
Conclusion
The thesis is devoted to investigate the essential and discrete spectra of two and thrcc-particle Schrodinger operator corresponding to the system of two or three identical particles (bosons or fermions) interacting via short-range pair potentials on lattices.
Basic results of the research arc as follows.
1. We introduce the notion of resonance for the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential on one-dimensional and two-dimensional lattice.
2. We find the conditions for existence of the eigenvalues lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of two identical particles (fermions) interacting via short-range potential for all dimensions of the lattice.
3. We determine the number and location of the eigenvalues of the Schrodinger operator corresponding to a system of two particles (fermions), interacting on neighboring sites of latticcfor all parameters of the operator.
4. We obtain an asymptotic formula for the number of eigenvalues lying to the left of the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials in the thrccdimcnsional lattice.
5. We show the finiteness of the number of eigenvalues lyingbclow the essential spectrum of the Schrodinger operator corresponding to a system of three particles (bosons) with short-range pair potentials on thcthrcc dimensional lattice for nonzero values of the quasi-momentum in the neighborhood of zero.
6. We prove the existence of an eigenvalue lying outside of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with pair contact potentials on one dimensional and two dimensional lattice, which is the only result in the theory of discrete Schrodinger operators.
7. Weestablish the finiteness of the number of eigenvalues lying below the bottom of the essential spectrum of the Schrodinger operator corresponding to a system of three particles with zero-range pair potentials on oncand two-dimensional lattices.

The aim of research work is investigation of the state of the Bochner-Martinelli integral on the boundary in domains with the piecewise smooth boundary and in domains with the singular boundary, and application of the obtained results to the problems of the holomorphic continuation of functions.
Scientific novelty of the research work is as follows:
Theorems on the holomorphic continuation arc proved, as well as analogs of the Hartogs-Bochncr theorems on the holomorphic continuation of functions in bounded domains with piecewise smooth boundaries and with the boundary containing conical edges;
theorems on holomorphy arc obtained for functions represented by the Bochner-Martinelli integral in domains with piecewise smooth boundary that reinforce previously known Aizcnbcrg-Kytmanov theorems;
formulas of rearrangement and composition arc obtained for the singular Hcnkin-Ramircz integral operator in strictly pscudoconvex domains;

The aim of research work is the study of translation-invariant limiting Gibbs measures for Potts and SOS models; the study of periodic limiting Gibbs measures for Potts model; the study of weakly periodic limiting Gibbs measures for HC model.
Scientific novelty of the research work is as follows:
for two state HC model conditions of the uniqueness of weakly two-periodic Gibbs measure are found.
The localization of translation-invariant Gibbs measures for Potts and SOS models is obtained.
For the antiferromagnetic Potts model (J <0) with zero external field on a Cayley tree of order two it is proved that on some invariant sets all periodic Gibbs measures are translation-invariant.
It is shown that all Gj.2)-periodic Gibbs measures are translation-invariant for the ferromagnetic Potts model (J > 0) on a Cayley tree of order к > 1.
For three state Potts model with non-zero external field on a Cayley tree of order к = 2 the existence of Gj2’-periodic (non translation-invariant) Gibbs measures is proved.
For q-state (3<q<k + \) Potts model on a Cayley tree of order k>3 a lower bound for number of Gp-periodic Gibbs measures is found.

The aim of the research is to increase the effectiveness of methods for mathematical modeling of gcofiltration processes of regional hydrogeological systems.
Scientific novelty of the research:
the concept of mathematical modeling of hydrogeological processes of a regional nature was developed, based on the principles of the theory of gcofiltration and gcomigration in complex hydrogeological conditions; methods for integrating the mathematical modeling of hydrogeological processes with information and communication technologies were improved;
numerical methods of mathematical modeling of hydrogeological processes of the regional plan were developed on the basis of modem G1S technologies, which allow to unite diverse models of gcofiltration within a single information and technological system;
a flexible system of geoinformation and mathematical modeling of gcofiltration processes of regional hydrogeological objects was proposed, based on the use of the principles of formation and joint application of models of different scale and spatial coverage;
the software, technologies and hardware-tools of the automated metering, registration and transfer of hydrogeological information, used for gcoinformation and mathematical modeling, as well as for monitoring the underground hydrosphere was devolcpcd;
the principles of organizing a database of geoinformation-and-mathematieal models of regional hydrogeological objects combining factographic and cartographic data with the possibilities of their subsequent integration into a single automated complex were developed;
principles and criteria for constructing a gcoinformation system in the integration of the mathematical model of salt transfer by interconnected flows of surface and groundwater for large-scale objects with complex hydrogeological conditions were developed.

The aim of the research work. The aim of the study is to develop numerical models describing the processes of multicomponent systems competing biological population quasilinear parabolic equations and their systems in homogeneous and inhomogeneous medium by the method of nonlinear splitting.
The tasks of research:
investigate the properties of two classes of models - models of nonlinear population, and a system of competing populations;
modeling on a computer the processes of one and multi-component systems competing biological population based on die algorithm of nonlinear splitting:
construct lower and upper solutions of die Cauchy problem by the algorithm of nonlinear splitting for multi-component systems competing biological population equation depending on the values of environmental parameters and the dimension of the space;
develop asymptotic expressions for solving systems of parabolic equations describing nonlinear process multicomponent competing biological populations:
create an initial approximation for the application of iterative methods and to construct a numerical scheme in the study of nonlinear processes in multicomponent systems competing biological population:
create algorithms and software for solving the foregoing problems, to determine new effects associated with die nonlinearity, visually present the decision to conduct a computational experiment.
The object of the research work. The object of the study are the nonlinear processes of the biological population, described by nonlinear parabolic equations and their systems.
Scientific novelty of the research work. The scientific novelty of die study is as follows:
developed methods to produce self-similar and approximately self-similar solutions for nonlinear models of multicomponent systems competing biological population based on the algorithm of nonlinear splitting;
identified new properties of a nonlinear madiematical model of the process of multicomponent competing biological population described by a system of Kolmogorov-Fisher type parabolic equations;
developed asymptotic expressions of solutions of self-similar equations and estimates of solutions of the Cauchy problem for multicomponent competitive systems of equations of biological population depending on parameter values the environment and the dimension of the space;
developed methods for constructing lower and upper solutions is needed for computer calculations of multicomponent tasks competing tasks biological populations;
created appropriate initial approximation, which provides the calculations with die required accuracy depending on the numerical values of the parameters using iterative techniques for fast and accurate numerical solution of the nonlinear task of Kolmogorov-Fisher type biological population;
developed computational schemes, algorithms and a software for performing numerical simulation of nonlinear mathematical models.
The outline of the thesis. The volume of the thesis is 105 pages typewritten text, illustrated by X drawings and 1 tables.

The aim of the research is development Lai-Massey network, encryption algorithms based on this network and generation resistance S-boxes.
The objectives of the research are the Lai-Massey networks, Nyberg construction.
Scientific novelty of the research is as follows:
created Lai-Massey networks form IDEAX-Y, RFWKIDEAX-Y using the structure of encryption algorithm IDEA and Lai-Massey scheme;
created Lai-Massey networks form PESX-Y. RFWKPESX-Y using the structure of encryption algorithm PES and Lai-Massey scheme;
developed encryption algorithms form AES-IDEAX-Y, AES-RFWKIDEAX-Y, AES-PESX-Y, AES-RFWKPESX-Y as a result of applying the round function of the encryption algorithm AES as the round functions of Lai-Massey networks:
developed encryption algorithms form GO ST28147-89-1DE AX-Y, GOST28147-89-RFWKIDEAX-Y, GOST28I47-89-PESX-Y, GOST28147-89-RFWKPESX-Y as a result of applying the round function of encryption algorithms GOST 28147-89 as round functions of Lai-Massey networks;
on the basis of Nyberg construction developed resistance S-box size of 8x8, 4x4.
Implementation of the research results. On the base of Lai-Massey network, based on a single algorithm:
encryption algorithm AES-IDEA32-4 created using round transformations of encryption algorithm AES, implemented in the software «Himfayl» in SUE «UNICON.UZ» (certificate of the Ministry of Information Technologies and Communications of May 29, 2017 No. 33-8 / 3256). Availability capabilities choice key length and number of rounds in the encryption algorithm AES-IDEA32-4. and application of encryption algorithms in file protection arbitrary format led to an increase in encryption speed by 17%.
encryption algorithm GOST28147-89-IDEA16-2 created using round transformations of encryption algorithm AES, implemented in the software «Himfayl» in SUE «UNICON.UZ» (certificate of the Ministry of Information Technologies and Communications of May 29, 2017 No. 33-8 / 3256). Availability capabilities choice key length and number of rounds in the encryption algorithm GOST28147-89-IDEA16-2, and application of encryption algorithms in file protection arbitrary format led to an increase in encryption speed by 21%.
The results of the dissertation encryption algorithms AES-PESI6-1, AES-RFWKPES16-1, AES-RFWKPES32-I, AES-RFWKIDEA32-1 arc used in foreign scientific works (International Journal of Network Security, vol. 19, No.6, pp.899-903, Nov. 2017; Internationa) Journal of Network Security, vol. 19, No.6, pp.984-994, Nov. 2017; Internationa) Journal of Network Security, vol. 19, No.3, pp.413-420, May 2017; Displays, vol.49, pp.l 16-123, Sep. 2017). The application of scientific results allows for the construction of further characterization of H-vectorial functions, hiding information in binary images, when protecting large images, cryptographically model for efficient multiple keyword-based search over encrypted data by secure index.
Publication of the results. On the topic of the dissertation published only 50 scientific papers, in t.ch. 21 articles (13 in the republican and 8 in foreign journals) published in scientific publications, recommended by the Higher Attestation Commission of the Republic of Uzbekistan for the main scientific results of doctoral dissertations. 6 certificates on registration of software for computers have been received.
The outline of the thesis. The thesis consists of an introduction, five chapters, conclusion, a list of used literature and applications. The volume of the thesis is 198 pages.