All articles - Mathematics

The urgency and relevance of the dissertation topic. Many scientific and applied studies (conducted at the world level) are in many cases brought to the problems of the theory of phase transitions in physics, biology, statistical mechanics, and so on. The theory of phase transitions is closely related to the theory of Gibbs measures. The development of the theory of such measures important because of the complexity of the description of Gibbs measures for classical models and the insufficient formalization of verification of their existence.
The aim of the research work is study existence of weakly periodic Gibbs measures and ground states for the Ising and Potts models. Construct some class of non (weakly) periodic limiting Gibbs measures, ground states and calculate free energies for the Ising and Potts models.
Scientific novelty' of the research work is as follows:
In the case of a normal subgroup of index four, for some conditions, it is proved that for the Ising model there are at least seven weakly periodic Gibbs measures;
Notions of (£0)-translation-invariant and (£0)- periodic Gibbs measure are introduced and the existences of such measures are proved;
General formula, for free energies for the Ising and Potts models are given, free energies corresponding to some known boundary conditions are calculated;
For the antiferromagnetic Potts model on the Cayley tree the existence of 2q -2 weakly periodic Gibbs measures is proved for normal subgroups of index two;
Under some conditions for the ferromagnetic Potts model on the Cayley tree the existence of at least two weakly periodic Gibbs measures is proved;
For the Potts model with external field, under defined conditions, the existence of at least two weakly periodic Gibbs measures is proved;
For the Potts model dependence between translation-invariant Gibbs measures and boundary conditions is found. Boundary conditions corresponding to the translation-invariant Gibbs measures are constructed;
For the Ising model with competing interactions on the Cayley tree sufficient and necessary conditions (on the order к of lattice and on parameters of normal subgrups of index two and four), for existence four of weakly periodic ground states are found;
For arbitrary normal subgrup of finite indices sufficient and necessary conditions are given under which a configurations is ground state of the Ising model with competing interactions on Cayley tree;
For the Potts model with competing interantion sufficient and necessary conditions are obtained under which there are weakly periodic ground states.

The aim of the research work is mathematical modeling of applied hydrodynamic processes of compressible two-phase media on the basis of analytic functions "A" of complex variable in two-phase media.
Scientific novelty' of the research work is as follows:
Methods for generalizing the theory of an analytic function of a complex variable operator "A", algorithms and methods for solving some applied problems for two-phase media;
Cauchy’s and Montel’s theorems, Picard's large theorem are proved, the mechanisms for expanding Taylor and Laurent series for a classical generalization of analytic functions "A (z)" are developed;
Methods for generalizing the integral Poisson formula for a stationary system of poroelasticity;
An algorithm for obtaining the solution of the analogue of the Mindlin problem for the stationary system of the poroelasticity equation in a half-space and numerical study of the influence of various dynamic characteristics on the wave field is found;
Differential identities connecting velocities, pressures, and mass forces in the equations of two-velocity hydrodynamics with phase equilibrium from pressure are proved in a divergent form;
Methods for constructing a general solution for the stream function in the planar case of the Monge-Ampere system of equations are developed;
A numerical model to study the flow of incompressible viscous two-velocity fluids, taking into account the equilibrium of the phases with respect to pressure is developed.

The aim of research work is the establishing of asymptotic representations for the likelihood ratio statistics in incomplete data models, obtaining by several types censoring of competing risks model.
The object of the research w ork is likelihood ratio statistics in a competing risks model under several types of random censoring.
Scientific novelty' of the research w ork is consist on follows:
proved a result on the approximation of stochastic integrals of the likelihood ratio statistics of two-parametrical Wiener process in the competing risks model;
proved the property of local asymptotic normality of the likelihood ratio statistics in competing risks model under hybrid censoring on the right;
using methods of strong approximation for empirical processes in competing risks model under a random and informative censoring from both sides established asymptotic representations for the likelihood ratio statistics;
proved properties of local and uniform local asymptotic normality for the likelihood ratio statistics in competing risks model under random censoring by nonobserving intervals;
proved an of asymptotic minimax efficiency of the maximum likelihood and Bayesian type estimates;
found the limit distribution of generalized chi-square statistics and likelihood ratio under random censoring from both sides.
Implementation of the research results. The results obtained during the dissertation research are practiced in the following areas:
The results obtained in the dissertation on generalized chi-square statistics for incomplete observations and the asymptotic properties of this statistics were used in Center for Retraining and statistical research for determining the distributions of the investigated random variables, and also in the training process for retraining courses. (Certificate of State Committee of the Republic of Uzbekistan on Statistics, August 25, 2017, No. 01/1-01-19/2-1039). The results of the dissertation are used in the educational process of the Department of "Theory of Probability and Mathematical Statistics" of the Mathematical Faculty of the NUUz (Certificate of the Ministry of Higher and Secondary Special Education of the Republic of Uzbekistan, August 2017).
The structure and volume of the thesis. The thesis consists of an introduction, four chapters, conclusion and bibliography. The volume of the thesis is 120 pages.

The urgency and relevance of the dissertation topic. Many scientific and applied studies conducted at the world level, in many cases, reduce to the study of ill-posed boundary-value problems for partial differential equations. The basis of the theory of ill-posed problems laid in the middle of the last century and they are associated with problems of great practical importance. The main object of the theory of inverse and ill-posed problems is the model of applied research in the field of geophysical observation, gas dynamics, the propagation of acoustic waves, etc. Since the study of ill-posed problems for mixed-composite equations on conditional correctness and the construction of an approximate solution, it is insufficient to develop a study of ill-posed problems for such equations is an actual problem.
The aim of the research work is to establish conditionally correctness and finding approximate solutions on the set of correctness of ill-posed problems for high order partial differential equations of mixed, composite, mixed-composite types.
The tasks of research work:
- investigation of ill-posed boundary value problems for high orders partial differential equations of composite and mixed-composite types;
-studies of ill-posed boundary value problems for high orders partial differential equations of mixed type;
-finding regular solutions, determining estimates of the norms of the difference between exact and approximate solutions in the corresponding function spaces;
-finding formulas for calculating regularization parameters, implementing numerical solutions and graphical results.
The object of the research work is the high order partial differential equation of mixed, composite and mixed-composite types.
Scientific novelty of the research work is as follows:
- a priori estimates of the solution of ill-posed boundary-value problems for partial differential equations of mixed, composite and mixed-composite types of high orders are obtained;
- sets of correctness defined for ill-posed boundary-value problems for partial differential equations of mixed, composite and mixed-composite types of high orders, and uniqueness and conditional stability theorems are proved;
-approximate solutions are constructed, estimates of the norms of the difference between the exact and approximate solutions in the corresponding spaces are determined, formulas for calculating the regularization parameters are derived;
- implemented programs in a visual c # environment that outputs numerical and graphical results of an exact and approximate solution based on computational algorithms.

This article provides a detailed analysis of teaching mathematics in general education schools based on the methodology of integrative approaches. It examines the effectiveness of integrative methods in enabling students to master theoretical and practical knowledge, as well as forming and consolidating their sustainable understanding. The article highlights concepts such as integration, the method of integrated lessons, challenges of integrative classes, the idea of integrative education, and the integration course, emphasizing their importance in the educational process. Special attention is given to the potential of these approaches to enhance students’ interest in mathematics and develop their mathematical thinking. The study demonstrates the pedagogical efficiency of incorporating modern integrative methods into the learning process.

The aim of the research work is to get the regularization in the bounded and unbounded domain, criteria for the decidability solution of the Cauchy problem for the linear elliptic system of the first order.
Scientific novelty of the research work is as follows:
- the Cauchy integral formula is created, for generalized holomorphic and generalized potential vector, generalized system Cauchy-Riemann equations with quaternion parameter and integral formula Stratton-Chu for a homogeneous system of Maxwell equations in the unbounded domain with non-compact boundary are obtained;
- for the generalized Cauchy- Riemann equation systems in the bounded and unbounded domain and regularization of the solution of the Cauchy problem are solved and the criteria of decidability is found;
- for the generalized system of Moiseil-Theodoresco equations an analogue Carleman formula is obtained and criterion for the solvability of the Cauchy problem is proved;
- it solved the Cauchy problem for generalized Cauchy-Riemann equations in multidimensional bounded and unbounded domain. Found analogue Carleman formula with which built the regularization of the Cauchy problem and proved the solvability criterion;
- constructed Carleman formula and regularization of the Cauchy problem for the homogeneous system of Maxwell's equations. Found analogue of Fock-Kuni theorem for a homogeneous system of Maxwell's equations;
- it solved the regularization solution of Cauchy problem for generalized Cauchy-Riemann equations and homogeneous system of the time-harmonic Maxwell and Dirac equations with a complex quaternion parameter.

Actuality and demand of the theme of dissertation. Numerous scientific and applied research conducted on a global level show that everywhere in physics stable complex objects arc usually formed as a result of action of attractive forces that allow the component parts to reduce the energy in their binding. However, recent years scientists have proved that in the ordered medium stable complex objects can exist even in the case of repulsive interactions. Bose-Hubbard model is used to describe the repulsive pairs, i.e. Schrodinger operator on a lattice is the theoretical basis of experimental observations and theoretical basis for the application. Therefore, the development of research of Schrodinger operators corresponding Hamiltonians of the systems of particles on a lattice, which arc found in models of solid state physics and lattice field theory is one of the priorities.
In our country in the years of independence much attention has been paid to directions of applied importance, in particular, special attention was paid to the study of Schrodinger operators corresponding to the system of particles on an integer lattice. For the Schrodinger operators significant results were achieved in determining the conditions for the existence of bound states which is located outside of the essential spectrum and for their number.
Since the spectrum of the Schrodinger operators associated to the systems of two quantum particles on lattice is quite sensitive to changes in the quasi-momentum of the system, solving problems related to studies of the spectrum of the operator and to show the existence of bound states as well as determine their number plays an important role . In this regard, the implementation of targeted investigation in the following areas is one of the most important problems: investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two arbitrary particles with short-range pair potentials on lattice, to establish the threshold phenomenon below the bottom or above the top essential spectrum for the operator. Research carried out in the aforementioned areas confirms the actuality of the dissertation topic.
This dissertation, to some extent, serves the tasks specified in the Decrees of the President of the Republic of Uzbekistan № DP-436 dated August 7, 2006 "On Measures for Improving the Coordination and Management of the Development of Science and Technology" and № DP-916 dated July 15, 2008 "Encouraging the introduction of innovative projects and technologies in production ", № DP -2789 dated February 17, 2017 "On measures to further improve the organization, management and financing of research activities and activities of the Academy of Sciences " and № DP -4947 dated February 8, 2017 "On strategy actions for the further development of the Republic of Uzbekistan ", as well as in other normative-legal acts on this activity.
The aim of the research is to study location of essential and discrete spectra as well as the number of eigenvalues of the Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact potential on lattice.
The scientific novelty of the research is as follows:
it is proven the existence of eigenvalue located to the right of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (//>0) potential on d>3 dimensional lattice and it is proven regularity of corresponding eigenstate finding its exact form;
it is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state is integrable;
it is defined for the fixed value of quasi-momentum the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant it is separated the set of quasimomentum which the operator has eigenvalue or has not eigenvalue or has a virtual level;
it is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;
it is proven the existence of eigenvalue located to the left of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive potential on d>3 dimensional lattice and it is shown regularity of corresponding eigenstate finding its exact form
it is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is d = 3,4 and it is established that the virtual state is integrable;
it is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5.
Conclusion
The dissertation is devoted to study essential and discrete spectra of the two particle Schrodinger operator corresponding to system of two arbitrary particles on lattice interacting via a pair contact potential.
The main results of the research arc as follows:
1. It is proven the existence of eigenvalue above the top of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (/z > 0) potential on d > 3 dimensional lattice and it is shown the corresponding eigenstate in coordinate representation exponentially decreases at infinity;
2. It is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state approaches to zero at infinity;
3. It is defined for the fixed value of quasi-momentum the set of the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant the set of quasimomentum;
4. It is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;
5. It is proven the existence of eigenvalue below the bottom of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive (// > 0) potential on 4>3 dimensional lattice and it is shown that the corresponding eigenstate in coordinate representation exponentially decreases at infinity;
6. It is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is <7 = 3,4 and it is established that the virtual state approaches to zero at infinity;
7. It is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;

The aim of the research work is study location of essential spectrum and a number of eigenvalues out of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on lattice.
Scientific novelty of the research work is as follows:
It is found location of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on lattice, interacting via creation and annihilation operators and two particle Schrocdinger operator, interacting via contact potentials;
It is shown existence at least one eigenvalue out of essential spectrum of certain generalized Friedrichs model, interacting via two particle Schrodinger operator, corresponding to a system of no more than two particles on one and two dimensional lattice and interacting via contact potentials;
It is shown the existence of eigenvalue out of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on a lattice, represented by two particle Schrocdinger operator, interacting via contact potentials on lattice with dimension no less than three, or its absence depending on parameters of the operator;
It is proved the existence at least one eigenvalue lying below the essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on one dimensional lattice, interacting via creation and annihilation operators and two particle Schrodinger operator, interacting via contact potentials, and established that existence or absence second eigenvalue depends on parameters of the operator.

The urgency and relevance of the dissertation topic.
At the present time, In the word, an expand the field of application of algorithms of the method Monte Carlo for various problems of mathematical physics, especially nonlinear boundary value problems is one of the important tasks. Numerical solution of such nonlinear problems is usually connected with considerable difficulties. Design, development and use of statistical modeling techniques, along with the deterministic methods is an actual problem and allows to obtain numerical results in the solution of applied tasks corresponding to increasingly complex models of the theory of gas dynamics, financial mathematics, biology and other fields. Research conducted in the aforementioned areas, confirm the relevance of the topic of the thesis
The aim of the research work is to construct and substantiate probabilistic models for solving boundary problems for nonlinear equations and systems of equations of elliptic and parabolic types in partial derivatives of the second order.
The scientific novelty of the research work is as follows:
numerical methods based on the probabilistic model for solving boundary value problems for nonlinear equations of parabolic type with constant and variable coefficients in the form of an infinite power series arc developed;
numerical methods based on the probabilistic model for solving the first and second boundary value problems for nonlinear equations of elliptic type arc developed;
numerical methods based on the probabilistic model for solving boundary value problems for systems of equations of elliptic and parabolic types arc developed.

The aim of the research work. The aim of the research is to develop and improve mathematical models, numerical algorithms and software for filtration processes in oil-, gas- and water-bearing beds.
The scientific novelty of the research work is as follows:
the mathematical model of the process of gas filtration in porous media was improved by taking into account various boundary conditions and the computational algorithm for solving the corresponding problem was developed on the basis of the finite difference method;
the mathematical model of the filtration process in the case of piston displacement was improved by taking into account the factor of oil production from the liquid phase region and the computational algorithm for solving this problem was developed on the basis of method of rectifying the phase fronts;
the mathematical model of the process of joint fluid and gas filtration was improved on the basis of the model of interconnected phases and the computational algorithm for solving this problem was developed on the basis of the variable direction method;
the effective numerical algorithm for solving the problem of gas filtration in porous media by the method of physical splitting was developed;
the parallel computational algorithm was developed to solve the problem of gas filtration in porous media for an arbitrary filtering region.

The aim of research work is to elaborate methods, models and algorithm of monitoring the scientific potential of higher educational and research institutions, as well as the complex of program tools on the basis of MVC technologies.
The scientific novelty of the research work is as follows:
information IDEF models of functional processes in the segment of the scientific potential indices of higher educational and research institutions, and a relative model of database have been elaborated;
an algorithm of relational algebra calculus in identifying a database of scientific potential of higher educational and research institutions and an assessment algorithm of indices of scientific potential of scientific-pedagogical personnel and their publications have been elaborated;
a program providing a monitoring of scientific potential of higher educational and research institutions possible to prepare and develop generalized final data in online regime on particular profile, and preparation of intentional segment at personalization level has been elaborated;
integration modules providing the interconnection of monitoring data of scientific potential of higher educational and research institutions with information system of electronic government, setting the data format and their exchange have been elaborated.

The aim of the research work is to develop functional dependencies between mathematical methods, algorithms and software for classification of symptocomplexes for early diagnosis of breast tumor diseases.
Scientific novelty of the research work. The scientific novelty of the study is as follows:
The classifier has been created for assessing the state of oncological diseases of the breast on the basis of statistical methods of pattern recognition;
A modified algorithm for classifying objects has been developed on the base of decisive rule "Apollonius ball";
The method and the algorithm of a private search for choozing of the most informative symptoms have been developed for solving the problem of classification of objects;
The hybrid algorithm has been developed based on the joint application of Bayesian algorithms, K.NN, "Apollonia ball" by optimizing the clinical features of breast tumor diseases;
The requirements for the architecture of the program and computing facilities, software based on algorithms, methods for selecting informative symptocomplexes and classification has been developed.

The aim of the research work is the development and improvement of mathematical models, computational algorithms and computer models for studying the regulatory mechanisms of the interaction of skin epidermal cells.
The scientific novelty of the research work is as follows:
the biological models that describe the interrelationships of the dividing, differentiating, fulfilling specific functions of skin epidermal cells were improved;
the system of equations for the regulatory mechanisms of skin epidermal cells based on biological models was developed taking into account the spatiotemporal organization;
the mathematical model of interaction of regulatory mechanisms of skin epidermal cells was created on the basis of a system of functional-differential equations with stumbling arguments;
the computational methods for mathematical models of regulatory mechanisms of skin epidermis were developed, taking into account the time of back action;
the software for computing experiments intended for solving medical problems in the field of construction of models of the interconnected processes of regulatory mechanisms of skin epidermal cells was created.

The aim of the research work is description of periodic p -harmonic functions and continuation of /^-harmonic functins from low order Cayley tree to the high order Cayley tree.
Scientific novelty of the research work is as follows:
periodic p-harmonic functions corresponding to normal divisors of the group representation of a Cayley tree are described in the casees of finite and infinite indexes;
linear combination of p-harmonic functions is not p-harmonic function in general. But it is proved that linear comination of periodic p-harmonic functions is a p-harmonic function;
p-harmonic functions given on a special Kurata tree arc extended to Cayley tree and p-harmonic functions given on a low order Cayley tree arc extended to high order Cayley tree;
the mean value theorem for harmonic funcions on Cayley tree is proved.

The aim of the research work is to study the existence of eigenvalues and also to obtain a convergent expansion for the eigenvalue lying outside the essential spectrum of the Schrodinger operator corresponding to system of two identical particles (fermions or bosons) interacting via pairwise short-range potentials on one and two dimensional lattices.
Scientific novelty of the research work is as follows:
the existence of eigenvalue and explicit form of the corresponding eigenfunction of the two-particle Schrodinger operator associated to a system of two identical particles(bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices is established;
a convergent expansion for eigenvalue at the coupling constant threshold of the two-particle Schrodinger operator associated to a system of two identical particles (bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices;
it is shown that the left threshold of the essential spectrum may be a virtual level (resonance) or an eigenvalue for the two-particle Schrodinger operator associated to a system of two particles (fermions) interacting at neighboring sites on one-dimensional lattice and the existence or absence of eigenvalue lying to the left of the essential spectrum is proved;
a convergent expansion at the coupling constant threshold and quasi-momentum threshold for the eigenvalue of the two-particle Schrodinger operator associated to the two-particle system (fermions) with pair interactions at neighboring sites on a one-dimensional lattice is found.

Actuality and demand of the theme of dissertation. Many scientific and applied research conducted on a global level show that everywhere in physics stable complex objects arc usually formed as a result of action of attractive forces that allow the component parts to reduce the energy in their binding. However, recent years scientists have proved that in the ordered medium stable complex objects can exist even in the case of repulsive interactions. Bosc-Hubbard model is used to describe the repulsive pairs, i.e. Schrodinger operator on a lattice is the theoretical basis of experimental observations and theoretical basis for the application. Therefore, the development of research of Schrodinger operators corresponding Hamiltonians of the systems of particles on a lattice which is reduced tothc generalized Friedrichs models that arc found in models of solid state physics and lattice field theory is one of the priorities.
At the present time in the world one of the important problems of mathematical analysis is the problem of studying the spectrum and resonances of self-adjoint operators. These problems have a close connection with the study of the spectrum and resonances of the generalized Friedrichs model corresponding to a system of two particles on a lattice. In most cases the numerous problems of mathematical physics and mechanics, in particular, the investigation of the spectral properties of the Schrodinger operator associated to asystem of two particles reduce to study the spectrum of the generalized Friedrichs models which arc defined as self-adjoint bounded operator. In this connection, to describe the essential spectrum of the generalized Friedrichs model corresponding to a system of two particles, to study the existence and number of eigenvalues depending on parameters and depending on the dimension of the space are implementation of targeted scientific research.
In our country much attention has been paid to directions of applied importance, in particular, special attention was paid to the study of generalized Friedrichs model which generalizes the Schrodinger operators corresponding Hamiltonians of the systems of two particles. For the Schrodinger operators and generalized Friedrichs model a number of results were achieved in determining the conditions for the existence of bound states which is located outside of the essential spectrum and for their number. The priority area of activity and the main task is the conduct of research in the main areas of such sciences as mathematics, physics, applied mathematics, in accordance with world standards2. The development of quantum field theory and the spectral theory of linear operators, in particular, the study of the spectral properties of the generalized Friedrichs model play an important role in the execution of the resolution.
This dissertation, to some extent, serves the tasks specified in the Decrees of the President of the Republic of Uzbekistan № DP-436 dated August 7, 2006 "On Measures for Improving the Coordination and Management of the Development of Science and Technology" and №DP-916 dated July 15, 2008 "Encouraging the introduction of innovative projects and technologies in production ", №DP -2789 dated February 17, 2017 "On measures to further improve the organization, management and financing of research activities and activities of the Academy of Sciences " and №DP -4947 dated February 8, 2017 "On strategy actions for the further development of the Republic of Uzbekistan ", as well as in other normative-legal acts on this activity.
The aim of the research is to show the existence of eigenvalues and to obtain the convergent expansions for these eigenvalues of the generalized Friedrichs model with the perturbation of rank one.
The scientific novelty of the research is as follows:
the location of the essential spectrum of the generalized Friedrichs model with the perturbation of rank is defined;
the conditions for existence of eigenvalues lying bellow the essential spectrum of the generalized Friedrichs model with the perturbation of rank one in the one and two-dimensional cases arc found;
the properties of the corresponding eigenfunction arc studied;
a criterion, for being the bottom of the essential spectrum a virtual level or virtual state of the generalized Friedrichs model with the perturbation of rank in the two-dimensional cases is given;
obtained and the explicit forms of the corresponding eigenfunction and virtual state arc found respectively;
the expansions for eigenvalue at the neighborhood of coupling constant of the generalized Friedrichs model with the perturbation of rank one in the one and two-dimensional cases arc found;
an asymptotic formula for eigenvalue as interaction energy tends to infinity is obtained.
Conclution
The dissertation is devoted to study the spectral properties, in particular, an expansion for eigenvalue of the generalized Friedrichs model with the perturbation of rank in one and two-dimensional case.
The main results of the research arc as follows:
1. It is given the conditions for existence of eigenvalue of the generalized Friedrichs model with the perturbation of rank one.
2. It is proved the analiticity of eigenvalue.
3. The implicit form of the corresponding eigenfunction is found and its analiticity is proved.
3. It is found the condition for being the bottom of the essential spectrum a virtual level or virtual state. The implicit forms of the corresponding eigenfunction and virtual state arc found respectevely.
4. It is found the expansions for eigenvalue at the neighborhood of coupling constant and uning these expansions it is obtained the asymptotic formulas;
5. It is obtained an asymptotic formula for eigenvalue as interaction energy tends to infinity.
The obtained results can be used to determine the quality of experimental investigations in mathematical physics, solid state physics and quantum mechanics.

The aim of the research work is to develop the mathematical models, effective computational algorithms and software for the processes of physically nonlinear strain of rods under spatially variable loading taking into account the damageability of materials.
Scientific novelty of the research work is a follows:
on the basis of the refined theory of V.K..Kabulov and the variation principle, mathematical models are developed for solving physically nonlinear rod problems under the influence of complex external forces, taking into account the damageability of materials;
the multi-parameter mathematical models in the form of a system of nine nonlinear differential equations of the second order are developed with natural boundary conditions for studying the stress state of rods in the case of spatially repeated loading in current and dummy coordinate systems;
by the A.A.Ilyushin method of elastic solution the computational algorithms for solving physically nonlinear problems of rods with different approximations are developed based on the central difference scheme and the modification of A.A.Samarsky-I.V.Fryazinov (MSF) method of finite differences;
the effective computational algorithms providing fast approximation to a stable solution, a high degree of accuracy, directed to the numerical calculation of some physically nonlinear rod problems, described by mathematical models of multi-parameter differential equations are developed;
an automated system has been created on the computer that allows the formation and solution of physically nonlinear rod problems for various variable loads and the planes with geometric, static and mixed boundary conditions.

The aim of research work is to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar; to prove pluripolarity of graphs of quasianalytic functions in the sense of Gonchar; to show pluripolarity of graphs of algcbroid functions; to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Dcnjoy; to define a class of quasiharmonic functions and prove a theorem about thinness of graphs.
Scientific novelty of the research work. All the results obtained in the dissertation arc new and consist of the following:
- Uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar is proved;
- Pluripolarity of graphs of quasianalytic functions in the sense of Gonchar is proved;
- Pluripolarity of graphs of algcbroid functions is proved;
- Pluripolarity of graphs of quasianalytic functions of several variables in the sense of Dcnjoy is proved;
- Pluripolarity of graphs of functions from Gevrey class is proved;
- The class of quasiharmonic functions is defined and the theorem about thinnes of graphs of functions from this class is proved.

The aim of research work is to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar; to prove pluripolarity of graphs of quasianalytic functions in the sense of Gonchar; to show pluripolarity of graphs of algcbroid functions; to prove uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Dcnjoy; to define a class of quasiharmonic functions and prove a theorem about thinness of graphs.
Scientific novelty of the research work. All the results obtained in the dissertation arc new and consist of the following:
- Uniqueness theorem for the class of quasianalytic functions of several variables in the sense of Gonchar is proved;
- Pluripolarity of graphs of quasianalytic functions in the sense of Gonchar is proved;
- Pluripolarity of graphs of algcbroid functions is proved;
- Pluripolarity of graphs of quasianalytic functions of several variables in the sense of Dcnjoy is proved;
- Pluripolarity of graphs of functions from Gevrey class is proved;
- The class of quasiharmonic functions is defined and the theorem about thinnes of graphs of functions from this class is proved.

The urgency and relevance of the dissertation topic. Many scientific and applied studies (conducted at the world level), in many cases, arc reduced to the study of ill-posed boundary-value problems for partial differential equations. The basis of the theory of ill-posed problems laid in the middle of the last century and they arc associated with problems of great practical importance. The main object of applied investigations on conditional correctness and creation of solution of boundary-value problems of elliptical equations becomes very important in hydrodynamics, geophysics and electrodynamics. A study of ill-posed problems for linear elliptical system of the first order in space domains is applicably important.
The aim of the research work is to get the regularization in the bounded and unbounded domain, criteria for the decidability solution of the Cauchy problem for the linear elliptic system of the first order.
Scientific novelty of the research work is as follows:
- the Cauchy integral formula is created, for generalized holomorphic and generalized potential vector, generalized system Cauchy-Riemann equations with quaternion parameter and integral formula of Stratton-Chu for a homogeneous system of Maxwell equations in the unbounded domain with non-compact boundary arc obtained;
- for the generalized Cauchy- Riemann equation systems in the bounded and unbounded domain and regularization of the solution of the Cauchy problem arc solved and the criteria of decidability is found;
- for the generalized system of Moiseil-Thcodoresco equations an analogue of Carlcman formula is obtained and criterion for the solvability of the Cauchy problem is proved;
- the Cauchy problem for generalized Cauchy-Riemann equations in multidimensional bounded and unbounded domains arc solved an analogue of Carlcman formula is detained. Using this formula the regularization of the Cauchy problem is constructed and the solvability criterion is found;
- Carlcman formula and regularization of the Cauchy problem for the homogeneous system of Maxwell's equations are constructed. An analogue of Fock-Kuni theorem for a homogeneous system of Maxwell's equations is proved;
- the regularization solution of Cauchy problem for generalized Cauchy-Riemann equations and homogeneous system of the time-harmonic Maxwell and Dirac equations with a complex quaternion parameter arc obtained .

The aim of the research work is the study of cardinal invariants of space of complete linked systems with compact elements.
Scientific novelty of the research work is as follows:
proved that following equalities hold for any infinite r, - space:
Id (X ) = ld(expn X) = ld(expa X ) = Id (expe X )
proved that for spaces x and ncx the density, n - weight, weakly density, net weight and the Souslin number arc equal;
proved that for Hattory space in the real line and its supcrextension spred, hereditary it - weight, hereditary Shanin number, hereditarily Souslin number, hereditarily calibre, hereditarily prccalibrc, hereditarily extent arc not equal;
proved that the topology i(r2) is an admissible extension of topology я(г,) iff the topology r, is an admissible extension of topology r,;
proved that the topology n (r,) is an admissible extension of topology v (r,) iff the topology r, is an admissible extension of topology r,;
proved that for Hattory space in the real line the density, weakly density, Souslin number, n - weight, character, n - character, Shanin number, preshanin number, tesnota, Lindclof number, extent arc countable;
proved that the topology exp(r,) is an admissible extension of topology exp(r,) iff the topology r, is an admissible extension of topology r,.

In this article, five-stage types of connections for teaching students school geometry in synchronous and asynchronous connection with physics are developed, and at each stage, connections through facts, connections through knowledge, and generalized skills are analyzed. In this approach, such aspects as concretizing the concepts of geometry and physics, revealing the processes and phenomena of one science using the concepts of another science, establishing connections between conclusions based on general concepts, and the ability to form connections between the concepts of different sciences when expressing one’s opinion are considered.