Mathematics

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.

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 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.

In this paper, importance of preprocessing and techniques in this field such as data cleaning, dimensionality reduction, smoothing, normalization are illustrated. During the research we mentioned some details of techniques above. However, our research includes only theoretical aspect of data preprocessing. The data preprocessing phase while arduous and time-intensive stands as the cornerstone of data science, possessing paramount significance. Neglecting the meticulous cleansing and structuring of data has the potential to undermine the integrity and efficacy of subsequent modeling endeavors.

Mexanika, fizika, biologiya, iqtisod va boshqa fanlar masalalari nochiziqli tenglamalarga yoki ularning sistemalariga keltiriladi. Bunday tenglamalarni yechimlari regulyar va singulyar yechimlarga bo‘linadi. Regulyar yechim yaqinida oshkormas funksiya haqidagi teorema yoki uning analogi qo‘llaniladi, u boshqa barcha yaqin yechimlarning tavfsifini beradi. Singulyar yechim yaqinida oshkormas funksiya haqidagi teoremani qo‘llab bo‘lmaydi. Ushbu ishda Darajali gometriyaning asosiy konsepsiyasi unga kiruvchi monom darajalari ko‘rsatkichlari bo‘yicha tenglamalar yechimlari xossalarini o‘rganish hisoblanadi.

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

Ushbu ishda turli sohadagi masalalarni chiziqli tenglamalar sistemasi orqali yechib, soha vakillarining fikrlashlarini rivojlantirishdan iborat.

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,.

Ushbu maqolada brayl matn tasvir sifatini oshirish usullari haqida so‘z boradi. Haqiqiy hayotda brayl alifbosidagi hujjatlarning koʻp tasvirlari sifatsiz boʻlgani uchun, bu maqolada koʻrib chiqish lozim: interpolyatsiya, shovqinni filtrlash, morfologik operatsiyalar kabi turli xil dastlabki ishlov berish algoritmlari va global konturlar haqida ma’lumot berilgan.

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

Ushbu ishda burish soni p − irratsional bo`lgan yo‘nalishni saqlovchi T aylana gomeomorfizmi qaralgan. T gomeomorfizmning burish sonlari

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

Subject of 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.

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.

In this work                                                                  

?  ?

?_?(?) = ∫ ∑ ?_??(?, ?)?_?(?)?? + ?_?(?),         ?

0  ?=1

= 1, 2, ⋯ ?; ? ≤ ?

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

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

?(?)

= ?(?)

1          1

+ ? ∫ ⋯ ∫ ?(?, ?)?(?)??

0          0

An approximate solution of the Fredholm integral

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

?       1

?(?, ?) ∈ ?^? (?₂)    .

2?

Subject of the inquiry: 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.

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

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 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.