All articles - Mathematics

Subjects of the inquiry: nonlinear evolution equations with self-consistent source.
Aim of the inquiry: To deduce the scattering data of spectral problem connected with nonlinear evolution equations with self-consistent source.
Method of the inquiry: used research methods include the methods of mathematical physics, theory differential equations, the theory of functions of complex variables, spectral theory of differential and difference operators.
I hc results achieved and their novelty: The main results of this work are new and consist of the following:
1) the law of changing on t of spectral data of Stunn-Liouville operator with potential which is the solution of general Korteweg - de Vries equation with source in the class of ‘rapidly decreasing’ functions is deduced;
2) the evolutions of scattering data of Sturm-Liouville operator with potential which is the solution of general Korteweg - de Vries equation in the class of “steplike” functions is defined;
3) the integration of general Korteweg - de Vries equation with source from “steplike” initial data is studied;
4) the inverse scattering method is used to solve various nonlinear evolution equations with self -consistent source in the case the simple eigenvalues of corresponding spectral problems;
5) it is shown that the inverse scattering method may be used for integration of sin-Gordon equation with self-consistent source in the case the multiple eigenvalues of Dirac’s operator;
6) the solution of Toda lattice with self-consistent source is expressed in terms of the inverse scattering method for the discrete Sturm-Liouville operator.
Practical value: the work has a theoretical character.
Degree of embed and economic cffcctivity: a special course will be read for post-graduate students on the basis of the received results.
Sphere of usage: the obtained results may be used in mathematical physics for integration of nonlinear evolution equations.

Topicality and relevance of the subject of the dissertation. Researchs related to the theory of nonlinear problems are one of the topical directions in the modern mathematics. Source of productions of such problems are mathematical models used in applied mathematics, biology, economics, hydrodynamics, elasticity and plasticity theory, theoretical and mathematical physics. When solving nonlinear problems, an important factor is the phenomenon of bifurcation and branching in problems, which leads to the emergence of new solutions in cases of transfer of controlling parameters of equations by means of the critical values. Among these new solutions, there are stable solutions, as well as solutions that are either immediately go out, or do not occur in a practical situation. Study of new solutions of nonlinear problems emerging at points of branching is the direction, which is called the “theory of stability and bifurcations”. The most striking examples of bifurcation (critical) phenomena are divergence (static bifurcation) and flutter (dynamic oscillatory buckling of plates and shells, in particular aircraft wings) in a stream of gas or liquid (hydro elasticity). This problem of a flutter has become particularly important in supersonic aerodynamics. In the middle of the last century, to study problems of aerodynamics only variation and grid methods were applied. And only in the twenty-first century, methods of the bifurcation theory have been used in this area.
Stability of produced both static and dynamic solutions is studied by the methods of the perturbation theory. More precisely, the spectrum of the Frechet derivative of the nonlinear equation (system of equations) is studied on the branched solution. Assuming that the eigenvalues of linearization , i.e. values of the Frechet derivative on the trivial solution are known, they lock for the Frechet spectrum on the branched solution that allows the use of perturbation theory from the spectral theory of linear operators.
That is why the stream of research related to solving nonlinear problems of perturbation theory rises (from the middle of the last century) with exponential speed, and any new deep result in the perturbation theory is relevant both for the perturbation theory, and for its applications to solving nonlinear problems.
Closely relation of the bifurcation processes to problems of describing the perturbations of the discrete spectrum of linear operators is one of the main causes for the need of researches connected with the subjects of the dissertation. Researchs of situations pertaining to the perturbation of multiple eigenvalues are associated with certain difficulties, which, unfortunately, can not always be overcome. For example, in the perturbation problem of Fredholm eigenvalues, it is found that the number of the eigenvalues branching of these points of the perturbed operator will be as much as a root number of the operator, but it is necessary in this case to require the completeness of the generalized Jordan set (GJS). In the case of an incomplete GJS, degeneracy of branching equation is arisen. In this situation, additional calculations on a specially-built algorithm of replenishment of GJS are needed. In addition, coefficients of the branching equation are determinants of the п-th order, that’s why the process of their finding requires a huge amount of computing.
Such studies could not be carried out in the perturbation problem for Noether points of the discrete spectrum. This is due to the fact that the branching equation of an eigenvalue for these operators can not be built because of inequality of dimensions of zero and defect subspaces.
This situation leads to necessity in the construction of special operators, for which considered multiple eigenvalues would have been simple or multiple but with a complete GJS. The constructing process of such operators is said to be regularization of linear operators.
The regularization procedure of linear operators allows to transform the Noether points of operators into the Fredholm ones, and it gives the possibility to construct the branching equation, which allows determining all the eigenvalues and corresponding eigenvalues of the perturbed operator. In addition, multiple eigenvalues are reduced to simple ones, that allows capturing the condition of degeneration for branching equations.
The mentioned methods of reducing the great volume of calculations explain the necessity and need of attraction of researches related to the subject of the dissertation.

Objects of research: homogeneous and multilayered systems
Purpose of work: reception of effective approximately-analytical decisions for the quantitative and qualitative analysis at an estimation mass-transfer in multilayered systems.
Methods of research: widely used methods of mathematical physics, the theory of function complex variable, asymptotic methods are applied, integrated transformations Laplace, a sine and cosine transformations Fourier, method of fission and as a method of final elements.
The results obtained and their novelty: all of the main results of this work are new and consist of the following:
1. In the work the problem decision of a mass-transfer consisting of the equations in private derivatives of parabolic type of subordinates to certain initial and boundary conditions is received. The stream was from the outside carried out at a non-stationary mode from rectangular area with the account and without compressibility of environment in the bottom layers.
2. It is received problem decisions mass-transfer by a method of final elements when the equation contained the first derivative on a spatial variable.
3. Is received analytical decisions of a two-dimensional problem mass-transfer with method of the fission.
4. At use of method of final elements, the way of definition of an internal point for an element containing functions of forms of the second order at which for the fixed moment of time the approached decision has coincided with the exact is specified.
Practical value: the work has theoretical character.
Degree of embed and economic effectiveness: the received results can be used at reading of special courses for post-graduate students of faculties of a natural profile.
Field of application: it is considered in all problems of the mathematical physics, leading by the equation in private derivatives of parabolic type (thermal, gas diffusion, oil and gas extraction, and so forth).

Subject of inquiry: Weak periodic Gibbs measures of the Ising model and weak periodic ground states of Ising model with competing interactions.
Aim of the inquiry: We study weak periodic Gibbs measures of the Ising model and weak periodic ground states of Ising model with competing interactions.
Methods of the inquiry: Methods of Markov random fields and recurrent equations of this theory. Also methods of measure theory and contractive maps, Pirogov-Sinay theory.
The results achieved and their novelty: The main results of work arc the following:
о For Ising model on a Cayley tree under some conditions it is proved that there arc five weak periodic Gibbs measures corresponding to arbitrary normal subgroup of index two.
о In case of normal subgroups of index four under some conditions on parameters of Ising model it is shown that there arc seven weak periodic Gibbs measures.
о An uncountably many new non periodic (and non weak periodic) Gibbs measures arc constructed.
о For Ising model with competing interactions sufficient and necessary conditions arc obtained under which there arc four weak periodic ground states.
о For arbitrary normal subgroup of index r sufficient and necessary conditions arc given under which a configuration is ground state of the Ising model with competing interactions on Cayley tree of order к > 1.
Practical value: the results of the dissertation work have theoretical character. They can be applied in problem of statistical physics.
Sphere of usage: results of the work can be used in measure theory, theory of phase transitions, theory of probability, theoretical and mathematical physcs.

Subject of the inquiry: scparatcly-analytical functions, holomorphic functions, pluriharmonic functions, separately-harmonic functions, subharmonic function.
Aim of the inquiry: to determinate of the domain of holomorphicity of the scparatcly-analytic functions from the piece of the boundary;
to study analytically continuability of functions defined on a pencil of boundary complex line;
to study continuation of the pluriharmonic functions in a fixed direction;
Z” 1 < P< °O
to describe structure of singular sets of subharmonic functions from p ,
class.
Methods of inquiry: methods of theory of functions of several complex variables, complex theory of potential and theory of analytical spaces.
Achieved results and their novelty:
- determined a domains of holomorphy of scparately-analytic and separately-harmonic functions defined on a piece of boundary;
studied analytic continuation of holomorphic and pluriharmonic functions in a fixed direction;
described the structure of singular sets of subharmonic functions from
Lmp,\<p<<oo class through Cq.m - capacity. All proved theorems are new 
Practical value: dissertation has a theoretical character.
Applications and economical efficiency: presented methods and results can be used for the further developing of the functions theory. They also can be useful in the applications of the complex analysis.
Area of application: the theory of functions of complex variable and its application.

Actuality and demand of the theme of dissertation. Algebraic instruments are very useful in the study of elementary particles in quantum mechanics, the properties of solids and crystals, in the analysis of model problems of the economics, in the problems of population biology, etc. Since associative algebras defined by specific identity, have been considered when identifying properties of closeness with respect to the usual multiplication of square matrices, further development of algebras leads to the theory of alternative, Lie and Jordan algebras, which are very closely related to each others and have many connection with different areas of mathematics. Since Leibniz algebras are generalizations of Lie algebras, many results of the theory of Lie algebras have been extended to Leibniz algebras. One of the priority directions of research related to this subject is to prove analogues of theorems of the Lie algebras theory in the Leibniz algebras case and to investigate of the inherent properties of Leibniz algebras which are not valid for Lie algebras.
From the classical theory of Lie algebras it is known that an arbitrary finitedimensional Lie algebra over a field of characteristic zero is decomposed into the semi-direct sum of maximal solvable ideal and its semi-simple subalgebra. On the other hand finite dimensional Leibniz algebras are also decomposed into the semidirect sum of the maximum solvable ideal and a semi-simple Lie algebra. The investigation of solvable algebras with some special types of nilradicals comes from different problems in physics. Therefore, similarly to Lie algebras, the investigation of solvable Leibniz algebras with given nilradical is one of the actual problems.
Recall that the class of nilpotent Lie algebras is the special subclass of solvable algebras. Since the description of all nilpotent Lie algebras seems too complicated, their study should be carried out with additional restrictions. In particular, in the investigation of nilpotent algebras one of the main restrictions is to restriction to the index of nilpotency. It should be noted that the maximal nilindex for Lie algebras coincides with the dimension of the algebra, and such type of algebras are called filiform algebras. Though, filiform Leibniz algebras have a relatively simple restriction in the class of nilpotent algebras, they have a sufficiently complicated structure, which is convenient to investigate with an additional condition of gradation. The effectiveness of the maximal gradation specify that it most accurately provides information about the structure constants of the algebra in the multiplication table.
The notions of degeneration, contraction and deformation of the algebra appeared from physics. Namely, the notion of contraction of Lie algebras in physical terms means: two physical models are related by a limiting process, under the action of the associated invariants groups. Deformations characterize the local behavior in a small neighborhood in the variety of given type objects. Thus, the study of the deformations of these algebras is a special case of the study of the local geometric properties of varieties. According to algebraic geometry an algebraic variety is a union of irreducible components. The closures of orbits of  rigid algebras give the irreducible components of the variety. That is why the finding of rigid algebras is a crucial problem from the geometrical point of view. The main reason for the demand of the theme of the dissertation is a close relationship of Leibniz algebras and their cohomological properties with the problems of Jordan, Lie algebras and their other generalizations.
Motivation of studying Lie superalgebras as a generalization of Lie algebras came from supersymmetry in mathematical physics. The theory of Lie superalgebras has established itself as a universal subject in modern algebra. Leibniz superalgebras are generalizations of Leibniz algebras and, on the other hand, they naturally generalize Lie superalgebras. Thus, the investigation of Leibniz superalgebras should take place to some parallel studies of these varieties. Similarly to Leibniz algebras, the study of finite-dimensional nilpotent Leibniz superalgebras with the maximal index of nilpotency and Leibniz superalgebras with nilindex equal to the dimension of the superalgebras, is an actual problem.
The aim of the research is the development of the structure theory of finitedimensional complex Leibniz algebras and their derivations, further development of the theory of degeneration and deformation of non-associative algebras and the description of nilpotent Leibniz superalgebras.
Scientific novelty consists of the following:
a characterization of the nilpotency of finite-dimensional Leibniz algebras in terms of Leibniz derivations is obtained;
non-characteristically nilpotent filiform Leibniz algebras and n - dimensional filiform Leibniz algebras of length n-1 are classified;
it is shown that the classical result of the decomposition of a semi-simple Lie algebra into a direct sum of simple ideals is not true for Leibniz algebras;
description of four dimensional complex Leibniz algebras is obtained, and five-dimensional complex solvable algebra Leibniz with three-dimensional nilradical are classified;
classification of solvable algebra Leibniz, whose nilradical is the direct sum of the null-filiform ideals is obtained;
classification of algebras of level one and a description of algebras of level two in the varieties of finite-dimensional complex associative, Jordan and Lie algebras are obtained;
the second cohomology groups of null-filiform Leibniz algebras are described, and a description of infinitesimal deformations of naturally graded filiform Leibniz algebras is obtained;
classification of null-filiform and filiform complex Leibniz superalgebras with nilindex n+m is obtained;
Leibniz superalgebras with the nilindex n+m are described and it is proved that, Leibniz superalgebras except null-filiform and filiform Leibniz superalgebras and Leibniz superalgebras with characteristic sequence (n | m-1, 1), have nilindex strictly less than n+m.
CONCLUSION
1. Properties of certain semi-simple Leibniz algebras are obtained and it is shown that the classical result on decomposition of a semi-simple Lie algebra into a direct sum of simple ideals is not true for Leibniz algebras.
2. A characterization of the nilpotency of finite-dimensional Leibniz algebras is obtained and it is proved that Leibniz algebra is nilpotent if and only if it admits invertible Leibniz-derivation.
3. Classifications of non-characteristically nilpotent filiform Leibniz algebras and n - dimensional filiform Leibniz algebras of length n-1 are obtained.
4. A description of four complex Leibniz algebras up to isomorphism is obtained and five-dimensional complex solvable Leibniz algebras with three-dimensional nilradical are classified.
5. A classification of solvable Leibniz algebras, whose nilradical is the direct sum of the null-filiform ideals is obtained.
6. Certain results on degeneration of solvable Leibniz algebras are obtained, and it is proved that if the algebra degenerates to another one, then the dimension of nilradical of the second algebra is less than the dimension of the nilradical of the first one.
7. Algebras of lowest level are investigated, and classified the algebras of the level one. A description of algebras of the level two in the varieties of finitedimensional complex associative, Jordan and Lie algebras is obtained.
8. Infinitesimal deformations of Leibniz algebras are investigated and the description of second cohomology groups of null-filiform Leibniz algebras is obtained. It is proved that closure of the union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras.
9. A description of infinitesimal deformations of naturally graded filiform Leibniz algebras is obtained;
10. A classification of complex Leibniz superalgebras with nilindex n+m is obtained and it is proved that Leibniz superalgebras except null-filifom and filiform Leibniz superalgebras and Leibniz superalgebras with characteristic sequence (n | m-1, 1), have nilindex less than n+m
The results of the dissertation have theoretical character. The main results and methods presented in the work can be used in investigations of other algebras and superalgebras, in the theory of categories, in the study of algebras with various types of gradation, in calculation of cohomology and homology groups and in investigation of various processes in theoretical physics.

Importance and relevance of the dissertation topic. Classical potential theory is based on Laplace operator and class of subharmonic functions. Built in the 80s of last century, the pluripotcntial thcoryis related with nonlinear Monge -Ampcre operator and plurisubharmonic functions. The pluripotential theory is intensively developing and has a numerous applications in the geometry of manifolds, in Einstein’s theory of relativity, in particular, to prove the existence of Einstein metrics in the theory of PDE. Naturally, there is a need to study the original extension of the class of phirisubharmonic functions and construction of potential theory for such extensions is actual direction of the complex analysis.
The class of plurisubharmonic functions is a subclass of subharmonic functions. It is naturally to study original extensions of the class of plurisubharmonic functions and construction of potential theory for such extensions.
To construct a theory which covers both classical potential theory and pluripotcntial theory it was expected using operators in hessians which generalize both Laplace operator and nonlinear Monge-Ampcrc operator. However, it was not known until recently a class of functions which expected potential theory will based on. In studying Dirichlet problem for equation in hessians was introduced a notion of class of m-subharmonic functions which was suitable class for constructing potential theory, it plays same role in the solution of equations in hessians as the class of plurisubharmonic functions for Monge-Ampcrc equation. Therefore, it is important deep investigation of the class of m-subharmonic functions, also the class of weakly m-subharmonic functions, in particular, establishment of potential-capacity properties of these classes.
The relevance of the scientific direction of the dissertation is also characterized by the fact that, in the dissertation justified the potential theory, which based on the operators in hessians, developed a method of solution of Dirichlet problem in the class of m-subharmonic and weakly m-subharmonic functions, proved m-subharmonicity of supremum of m-subharmonic functions and (m-1) subharmonicity of restriction of m-subharmonic functions on the hypcrplanc. Definition of weakly m-subharmonic functions which subharmonic in complex planes, proof of their potential-capacity properties, quasi-continuity of m-subharmonic functions, comparison principle, continuity of operators in hessian for a standard approximations and proof of other fundamental theorems arc important results of the dissertation.
The estimates of main characteristic functions of Nevanlinna’s theory, a simple description of m-convex hull in Riemann geometry, an application of the theory of m-subharmonic functions in establishing criteria of pluriharmonicity (analogue of Lelong’s theorem) and in series of applications of theories of m-subharmonic and weakly m-subharmonic functions in multidimensional complex analysis indicates importance and relevance of the dissertation topic.
Aim of research is constructing potential theory on m-subharmonic functions, proving potential properties of weakly m-subharmonic functions and demonstrating of applications of constructed theory to problems of multidimensional complex and harmonic analysis.
Scientific novelty of the research. The dissertation work is a new scientific direction. In it:
m-subharmonicity of supremum in the class of m-subhanninic functions and (m-l)-subharmonicity of restriction on complex hypcrplanes were proved;
A complete construction of potential theory based on hessian operator which includes well-known classical and complex potential theory was given;
Important potential-capacity properties of subharmonic on complex planes weakly m-subharmonic functions were introduced and studied;
The methods of solution of Dirichlet problem in the class of m-subharmonic and weakly m-subharmonnic functions was developed;
Quasicontinuity and comparing principle for m-subharmonic functions were proved;
Continuity of operators in hessians and other fundamental theorems of potential theory in the class of m-subharmonic functions were proved.
CONCLUSION
The main obtained results of the investigation arc following:
1. The m - subhannonicity of supreme in the class of m - subharmonic functions and m - subhannonicity of restriction to the complex hypcrplane were proved;
2. The notion of condenser capacity in the class of m - subharmonic functions was introduced and a series of important properties of capacity were proved;
3. Quasicontinuity and comparing principle for m - subhannonic functions were proved;
4. Convergence of currents for a standard approximations and fundamental theorems of the potential theory in the class of m - subharmonic functions were proved;
5. The class of weakly m - subharmonic functions was defined and a series of potential properties of this class were proved;
6. The method of application of the class of m - subharmonic functions in multidimensional complex analysis and potential theory was developed. In particular, in Nevanlinna’s theory - to estimate characteristic functions, in convex geometry — to describe m-convex hulls, in theory of pluriharmonic functions — to set pluriharmonicity of functions (analogue of Lelon’s theorem).
In general, the obtained results allow us to speak about achieving the goals of research of dissertation work. Constructed potential theory in the class of m -subharmonic functions is a new research direction which has an important application in Nevanlinna’s theory, in complex projective space, in theory of nonlinear elliptic equations and etc.

Subjects of inquiry: Algebra of measurable operators, non commutative Arens algebras, local derivations.
Aim of the inquire: Description of local derivations on algebras of measurable operators.
Methods of the inquire: In the work general methods of functional analysis, of theory operator algebras are used.
The results achieved and their novelty: a description of local derivations on the non commutative Arens algebras associated with von Neumann algebra and faithful normal semi-finite trace is obtained; it is proved that every tT-continuous linear operator Д on the algebra 5(Л/,т) satisfying the identity A(p) = A(p)/? + /?A(p) is a derivation, where AY be a von Neumann algebra with a faithful normal semi-finite trace r; it is proved that every linear operator D: 4(X) —> B(X) satisfying the identity Z)(x") = У^хА 'D(x)x"~*, хбЛ(Т) *=i
is a spatial derivation, where n > 3 - some fix number; necessary and sufficient conditions for the existence of local derivations which are not derivations on algebras S(M) and S(M,r) affiliated with a commutative von Neumann algebra are obtained; a description of local derivations of the algebras LS(M), S(M) and S(M,r) concerning type I von Neumann algebras without abelian direct summands is obtained.
Practical value: The results of the dissertation have a theoretical character.
Degree of embed and economic effectivity: The results, presented in the work can be used in special courses on functional analysis and theory of operator algebras for masters and post-graduate students.
Field of application: Functional analysis, theory of operator algebras, mathematical physics and its applications.

Subject of the inquiry: learning activity of leavers of the 9Ih forms of the secondary schools and students of academic lyceums in the classes of mathematics.
Aim of the inquiry: revealing gifted pupils in the lessons of mathematics and creation scientifically-pedagogical bases of teaching process.
Methods of the inquiry: to study and using the methods of the inquiry directed on revealing gifted pupils in the research functioning and literature on the theme; analyses of the textbooks, school programs of the state educational standard in mathematics of secondary and specialized schools; analyses and observation of lessons of mathematics; questioning and conversation with teachers, pupils and students; leading pedagogical experiments and mathematical-statistical analyses of the results, and their generalization.
The results achieved and their novelty: there have been classified the essence of the notion of being mathematical gifted at pupils in continuous education; there have been also worked out the tests oriented on revealing mathematical gifted pupils, and worked out the method of development of mathematical gifted pupils, and have been led the experiment as well.
Practical value: the results of the inquiry represent to diagnose leavers of secondary schools and direct them on the following stage of continuous education in academic lyceums, teaching gifted pupils in academic lyceums.
Degree of embed and economical effectivity: results of the inquiry were published as a manuals of the author, articles in the journals «Pedagogik ta’lim (pedagogical education)» and «Kasb-hunar ta’limi (professional education)», as well as in the form thesis report in the Republic scientific-practical conferences.
Sphere of usage: gained results can be used in diagnosing leavers of secondary schools and direct them onto the academic lyceums, also in training teachers of mathematics of pedagogical institutes and as well as in the Republican centre of the diagnostics.

Subject of inquiry: Gibbs measures for q-eomponenl model and Potts model with competing interactions on a Cayley tree.
Aim of the inquiry: We study Gibbs measures and periodic ground states of the Potts and q-componcnt models with competing interactions on a Cayley tree.
Methods of the inquiry: Methods of contours on a Cayley tree, methods of Pirogov-Sinay theory, measure theory and contractive maps.
The results achieved and their novelty: The obtained results arc new. They consist of the following:
• For q-componcnt models on a Cayley tree contours and ground states arc constructed.
• For q-componcnt models, at sufficiently low temperatures, by a contour method on a Cayley tree existence of at least q different Gibbs measures is proved.
• For a Potts model with competing interactions on a Cayley tree the set of periodic ground states is constructed.
• It is shown that the Pcicrls’s condition is satisfied for the Hamiltonian of the Potts model.
• At sufficiently low temperatures, for the Potts model with competing interactions and three spins existence of at least three Gibbs measures is proved.
• On parameters of a model with the interaction radius two a sufficient conditions arc found under which the periodic configurations arc the ground states of this model.
Practical value: the results of the dissertation work have theoretical character. They can be applied in problems of statistical physics.
Sphere of usage: results of the work can be used in measure theory, theory of phase transitions, theory of probability, theoretical and mathematical physcs.

Objects of the investigation: the modified Korteweg-dc Vries equation.
Aim of the investigation: integration of the modified Korteweg-dc Vries equation with the self-consistent source in the class of rapidly decreasing functions.
Method of the investigation: in this work the methods of mathematical physics, differential equations, functional analysis, the theory of complex variable functions and the spectral theory of differential operators arc used.
The results achieved and their novelty: all of the main results of this work arc new and consist of the following:
1. The law of spectral data of spectral characteristics of Dirak operator with potential changing on t, which is the solution to the modified Korteweg-dc Vries equation in the class of rapidly decreasing functions is deduced.
2. The evolutions of scattering data of Dirak operator with simple eigenvalue potential of the solution to the modified Korteweg-dc Vries equation with the self consistent sources rapidly decreasing functions in case of moving eigenvalues arc defined.
3. The evolutions of scattering data of no self-joined Dirak operator with multiple eigenvalues potential of the solution to the modified Korteweg-dc Vries equation with the self consistent different sources arc defined.
Practical value: the work has theoretical character.
Degree of embed and economic affectivity: on the basis of the received results a special course will be read for the students of masters’ department and postgraduate study.
Sphere of usage: the obtained results may be used in mathematical physics for integration of the equations nonlinear evolution.

Subject of research: earned one's living (PR) on the rolling base.
Purpose of work: Define and value a dynamic inexactness of motion path and develop mathematical models and algorithms of optimum management, allowing enlarge a speed and PR positional accuracy on the rolling base.
Methods of research: methods of mathematical modeling of technological processes, theories of probability, mathematical statistics, algebras, theories and theory of optimum management.
The results obtained and their novelty: Determined mistake existing motion models PR on the rolling base, newly built equation of its motion, on the base which designed mathematical models, algorithms and software programs, allowing enlarge a speed and PR positional accuracy.
Practical value: Software programs of optimum governing the explored robots, due to increasing a speed and positional accuracy, can be used in all branches of public facilities, which provided with by the system that promotes minimization of general time of production and spare an energy facility.
Degree of embed and economical effectivity: Developed on the base mathematical optimum management models PR on the rolling base algorithms and software programs are introduced in the Join-stock company "Technologist". On the example of introduction in the process of assembly of units proved that arc vastly enlarged speed and positional accuracy operated by PR; their possible use mechanical processing in the process of. Annual cost-performance of introduction on one PR forms 535 thousand (on prices 2009).
Field of application: Developing mathematical models and algorithms can be used under optimum governing the different branches of public facilities, which provided with by systems.

Subjects of research: orc deposit, exploited methods of UL on the conditions of using horizon mining system.
Purpose of work: developing computer model of UL in heterogeneous environments in the realization of horizon mining systems for the analysis and decision making support in the control of technological processes of UL.
Methods of research: methods of control theory, mathematical modeling, finite-difference methods and computational experiment.
The results obtained and their novelty: mathematical model of controlling UL processes was developed on the conditions of horizon mining systems; dynamics of pressure change and the values of reagent concentration in the various values of outcome parameters, influencing on the behavior of technological process of UL on the conditions of horizon mining systems, was studied; computer model for carrying out computational experiments and visualizing the results in two dimensional and three dimensional graphics was developed; software package of UL process in the conditions of horizon mining systems for supporting making technological decisions in the control of mine workings was developed.
Practical value: developed computational algorithms and computer model can be applied for the analysis, prognosis of the parameters of leaching process and decision making for controlling its parameters on the purpose of optimal extraction of minerals from real deposit mines on the conditions of horizon mining systems of mine workings.
Degree of embed and economical effectivity: obtained results were applied in the mine North Bukinai NMSK, the act of applications was taken. Developed software was registered by Government Patent Committee of Uzbekistan.
Field of application: mineral deposit mines exploited by the method of UL.

Subjects of research: orc deposit, exploited methods of UL on the conditions of using horizon mining system.
Purpose of work: developing computer model of UL in heterogeneous environments in the realization of horizon mining systems for the analysis and decision making support in the control of technological processes of UL.
Methods of research: methods of control theory, mathematical modeling, finite-difference methods and computational experiment.
The results obtained and their novelty: mathematical model of controlling UL processes was developed on the conditions of horizon mining systems; dynamics of pressure change and the values of reagent concentration in the various values of outcome parameters, influencing on the behavior of technological process of UL on the conditions of horizon mining systems, was studied; computer model for carrying out computational experiments and visualizing the results in two dimensional and three dimensional graphics was developed; software package of UL process in the conditions of horizon mining systems for supporting making technological decisions in the control of mine workings was developed.
Practical value: developed computational algorithms and computer model can be applied for the analysis, prognosis of the parameters of leaching process and decision making for controlling its parameters on the purpose of optimal extraction of minerals from real deposit mines on the conditions of horizon mining systems of mine workings.
Degree of embed and economical effectivity: obtained results were applied in the mine North Bukinai NMSK, the act of applications was taken. Developed software was registered by Government Patent Committee of Uzbekistan.
Field of application: mineral deposit mines exploited by the method of UL.

Subject of research: homogeneous processes with independent increments and the generalised renewal processes.
Aim of the research: obtaine the complete asymptotic expansions for the distribution of the number of a rectilinear strip by trajectories of homogeneous process with independent increments and the generalised renewal process.
Methods of research: in the dissertation a used the analytical factorization method.
The results achivved and their novelty: all main outcomes of the thesis are new and consist of the following:
- complete asymptotic expansions in t —> oo for the distribution of the number of crossings of a rectilinear strip till the moment t by a trajectory of homogeneous process with independent increments have been obtained. Thus it is supposed that strip borders grow together with t and are imposed on condition process, basically, Kramer’s type;
- the first members of asymptotic decomposition are written out in an explicit form and the algorithm of calculation of the subsequent members is specified;
- the results specified above are transferred in case of the generalised process of restoration.
Practical value: the thesis has theoretical character.
Field of application: the received results can be used at the decision of various problems of mathematical statistics, the theory of mass service, the theory of storage of stocks and others.

Subjects of the research: creation of object-oriented software programs for forecasting and optimum control of underground leaching.
Purpose of work: creation of direction’s models, methods and program means for analyses and decision making in the control of ore mines technological process UL.
Methods of research: numerically-drawn near and approximate-analytical methods, methods of idle time and stream running, methods of variable directions, monotonous schemes, characteristic target function, computer methods of decision making in control.
The results obtained and their novelty:
• the two measured mathematical models direction and algorithms calculation for the decision making in the control of technological process UL, were accepted;
• dynamics of concentration changes corresponding to different values of parameters, which could change technological processes, influenced to the current of technological process of UL;
• program processes for the realization of calculating experiment and calculation of parameters decision making at controlling of UL process control and visualizing calculation results were accepted.
Practical value: Correctness of worked out mathematical models and algorithms was approved on the bases of information from real mines and the reference was got about giving approbation from OSC “Andijonneft”.
Degree of embed and economic effectivity: On the bases of historical information from the real deposit mines 3 blocks of 5-ore uranium getting direction which belongs to Navoi, mountain-metallurgical factory was corroborated authenticity and availability of getting research information. The results of the thesis accepted for the use in the process of gas production control “GissarNefteGaz”.
Field of application: Software is possible to use for calculation of the concentration of the useful component decision making in control on process of underground leaching.

Subjects of research: local and non-local boundary-value problems for parabolic-hyperbolic equations with three lines of type changing.
Purpose of work: formulation of local and non-local boundary-value problems for parabolic-hyperbolic equation with three lines of type changing and investigation for the existence and uniqueness of solution of formulated problems.
Methods of research: methods of integral equations and energy integrals are used.
The results obtained and their novelty: local and non-local boundary problems for parabolic-hyperbolic equations with three lines of type changing are formulated and the existence, the uniqueness of solution for formulated problems is proved.
Practical value: the results of the dissertation work have a theoretical character.
Degree of embed and economic effectiveness: on the base of achieved results, the special course for the master- students can be taught and may be used in the subsequent theoretical development of this field.
Field of application: results of the dissertation work can de used at future development of the theory of partial differential equations and also at studying mathematical questions of problems of physics, mechanics and biology.

Subjects of research: the formation of stress-strain state of thin-walled structures.
Purpose of work: development of mathematical model and algorithms for the solution the static problem of plates bending with physical and geometrical nonlinearity, which would allow automating process of the problem solution, and giving a chance to spend multiple experimental researches.
Methods of research: In this paper for solving the problem was used the following approximate methods: the variation Ritz’s method, the method of clastic solutions by A. Ilyushin, the method of consecutive approximations.
The result obtained and their novelty: the mathematical model solutions for physically and geometrically nonlinear bending problem of plates based on which was made the algorithm of calculating the stress-strain state of plates using the Ritz’s method, the elastic solution by Ilyushin and the method of successive approximations were found. The novelty of the proposed work is as follows: was derived a mathematical model for solving the nonlinear problem on the basis of the Ritz’s method, an algorithm for solving the problem, and a set of software tools to automate the process of solving the problem have been created.
Practical value: the developed mathematical model, algorithm and software can be recommended for research and design institutes.
Degree of embed and economic affectivity: the results of research can be used in various industries that use different type of the metal plates for design. The significant economic can be achieved by reducing the time and complexity of design-development using the algorithm and software tools have been created.
Field of applications: mechanical engineering, shipbuilding, aircraft, power engineering, construction.

Subjects of research: direct and inverse problems for even-order equations and mixed-type equations of even order.
Purpose of work: formulation and investigation of direct and inverse problems for even-order equations and mixed-type equations of even order.
Methods of research: the method of a priori estimates, Fourier method, the theory of linear operators and the methods of functional analysis arc used.
The results obtained and their novelty:
- various new direct and inverse problems for even-order equations and mixed-type equations of even order arc formulated;
- under certain conditions to given functions and in some problems depending on size of domain, the uniqueness and the existence of regular solution for those problems arc proved, the unique strong solvability of direct problems arc proved;
- operator equations, which arc equivalent to considered problems arc studied and conclusion on a spectrum of the problems arc obtained;
- a priory estimates for some problems, from which uniqueness and continuous dependence of regular solution from the right hand of the equation and existence of inverse operators arc obtained.
Practical value: the results of the dissertation work have got theoretical character.
Degree of embed and economic effectiveness: on the base of achieved results, the special course for the master-students can be teachcd and these results may be used in the subsequent theoretical development of this trend.
Field of application: results of the dissertation can be used in the studying of boundary value problems for mixed type equations, in the further development of the theory of partial differential equations and at solving problems of mathematical physics which arc reducing to such equations.

Subject of the inquiry: <t-smooth weakly additive, order- preserving, normed functionals and their space and functor.
Aim of the inquiry: to investagc topological and categorical properties of the spaces of cr -smooth order- preserving functionals.
Methods of inquiry: methods of general topology, covariant functors theory and functional analysis have been used.
The results achieved and their noveltyj. results obtained in the thesis arc new and consist of the following: It is proven that the construction On is a covariant functor, acting in the category of Tychonoff spaces and their continuous maps. A criteria is given for cr -smoothness of weakly additive, order- preserving functionals. It is shown that if f : X —> Y is a z -embedding between Tychonoff spaces, then the map Oa (f) : On (X) —> On (%) is an embedding. It is shown that the functor Oa forms a monada. It is proven that the space O^iX) is Hewitt complete for every Tychonoff space X.
A description is given for the Hewitt completions of Tychonoff spaces in terms of the spaces of a -smooth order-preserving functionals. We give a condition for coincidence of the spaces of r-smooth weakly additive and cr-smooth weakly additive functionals. It is shown that weight of the space O^X) of a -smooth weakly additive functionals is between the weight of Hewitt completions and z -weight of the given Tychonoff space X.
The practical value: the results of the thesis have a theoretical character.
Degree of application and economic effectivity: Results and methods introduced in the work can be used in special courses on general topology, functional analysis and theory of covariant functors.
Sphere of usage: the results of the thesis may be used in general topology, covariant functors theory and functional analysis.

Subject of research: gas supply systems which operate in normal working mode and under uncontrollable random disturbance influence.
Purpose of work: development of methods and algorithms of efficient management and decision-making analysis for technical and economical gas supply objects factors.
Methods of research: have been used: mathematical programming, probability theory, information systems construction and computing experiment methods.
The results obtained and their novelty: for the first time have been developed methodology and algorithms of interpretation of graphic predetermined functions. Also have been stated and summarized principles of definition and estimation of models of structured identification of gas supply objects accuracies. Devised and researched the information-logical model of management and operational decision-making automation on regulation of work of gas produce, transport and supply objects. Has been developed informational-logical model of automation of technological and economical computation of factors of objects of gas production and supply. Has been shaped mathematical model of gas distribution network as determined serving system, based on scheduling theory conception.
Practical value: have been developed algorithms and software tools that make possible efficiently to manage and regulate gas supply system, which may operate as in normal working mode and as under uncontrollable random disturbance influence.
Degree of embed and economic effectivity: developed software tools were accepted and adopted by state direction of “Samarkand Gas” (which now is SG “Samarkand Gas Supply”) and became component of different scientific research activities of “SCST RUz №-14.3”. Economical effect of cunent work’s adoption is about 9220000 sums per annum. Obtained three certificates of the State Patent Office of the Republic of Uzbekistan: №DGU 01936, №DGU 02011, №DGU2181.
Field of application: objects of production, transport and gas supply, and also in planned-economical departments of industrial factories.

Objects of research: the formation of stress-strain state of thin-walled structures
Methods of research: In this paper for solving the problem was used the following approximate methods: the variation Ritz’s method, the method of clastic solutions by A. Ilyushin, the method of consecutive approximations.
The findings and their novelty: the mathematical model solutions for physically and geometrically nonlinear bending problem of plates based on which was made the algorithm of calculating the stress-strain state of plates using the Ritz’s method, the elastic solution by Ilyushin and the method of successive approximations were found.
The novelty of the proposed work is as follows: was derived a mathematical model for solving the nonlinear problem on the basis of the Ritz’s method, an algorithm for solving the problem, and a set of software tools to automate the process of solving the problem have been created.
Practical significance: the developed mathematical model, algorithm and software can be recommended for research and design institutes: the method of mathematical model constructing also as well as algorithm development and software can be used in designing organizations and specialized departments of the Universities of the Republic of Uzbekistan.
The degree of implementation and economic efficiency: the results of research can be used in various industries that use different type of the metal plates for design. The significant economic can be achieved by reducing the time and complexity of design-development using the algorithm and software tools have been created.
Range of applications: mechanical engineering, shipbuilding, aircraft, power engineering, construction

Subject of research: multi-phase filtration of fluids in porous media.
Purpose of work: The development of computational algorithms and their basis the creation of mathematical software an automated system of calculating oil and gas fields.
Methods of research: methods of computational mathematics, mathematical modeling, development and testing of software systems.
The results achieved and their novelty: investigated mathematical models and developed computational algorithms for solving the problems of filtration of multi-phase fluids in porous media; developed software an automated system for solving the problems of filtration of multi -phase fluids in porous media; conducted computational experiments to calculate main indicators of oil and gas fields.
Practical value: proposed mathematical and special software that allows us to development and quickly carry out serial calculations to predict the main indicators of oil and gas fields.
Degree of embed and economic affectivity: The developed software is protected by certificate of evidence of the Agency for Intellectual Property of the Republic of Uzbekistan, № DGU 02001. The software tool implemented in the "Mubarak neftgaz" and showed its cost-effectiveness.
Field of application: The developed method of calculation and software tools allow us to investigate the development of the oil and gas fields, theoretical positions and research findings can be used to delivery special courses for undergraduate and master students on specialty "Mathematical and software of computers, computer systems and networks."