Mathematics

Это великое событие в истории человечества произошло примерно десять тысяч лет тому назад, когда ледяной покров в Европе и Азии начал таять и уступать место лесам и пустыням. Постепенно прекращались кочевые странствия в поисках пищи. Рыболовы и охотники больше вытеснялись первобытными земледельцами. Такие земледельцы, оставаясь на одном месте, пока почва сохраняла плодородие, строили жилища, рассчитанные на более долгие сроки. Стали возникать деревни для защиты от непогоды от врагов-хищников. Немало таких неолитических поселений раскопано. По их остаткам видно, как постепенно развивались такие простейшие ремесла, как гончарное, ткацкое и плотничье. Существовали житницы, так что население могло, производя излишки, запасать продукты на зиму и на случай неурожая.

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

The aim of the research 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.

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 .

Понятие компакта Дугунджи, введенное А.Пелчинским [1], оказалось весьма плодотворным и привело к созданию важных новых методов в общей топологии. Отвечая на вопрос Пелчинского, Р. Хэйдон показал [2], что всякий компакт Дугунджи диадичен [3] те. непрерывный образ обобщенного канторова дисконтиниума DT. С другой стороны компакты Дугунджи - это в точности компакты класса AE(G) . Теория АЕ(фТ) компактов была распространена А.Н. Дранишниковым [4] па абсолютные экстензоры в размерности и. Так же в этой работе определены некомпактные аналогии пространства Дугунджи и пространства Милютина. Изучены их топологические свойства и геометрические свойство с применением некоторых ковариантных функторов. Терминология и обозначение, нс разъясняемые ниже, такие же, как в книгах [1,3,5].

Наша главная задача обеспечить формирования высокой общей и профессиональной культуры учителя, его готовности к педагогическому творчеству. Об этом свидетельствует практика преподавания в высшей школе, освещенная в научно методической литературе не хватает разноуровневое, мобильности, гибкости, непрерывности, преемственности, вариативности.До сих нор остается неразрешенной проблема установления оптимального соотношения учебных форм работы, по-прежнему просматривается диспропорция между лекционными, семинарскими, лабораторнопрактическими занятиями и практикой в школе. Вербализм является доминирующим принципом всей подготовки. По-прежнему не преодолен разрыв между теоретической и практической подготовкой студентов. Многие курсы, которые ведутся по специальным дисциплинам, читаются в отрыве от школьной практики. Вопросы по математике: аксиоматический метод; математические доказательства; элементы, множества, отношения, отображения, числа; комбинаторика; конечные и бесконечные множества; основные идеи математического анализа; математика случайного; элементы теории вероятностей; роль математики в гуманитарных науках.

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.

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.

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

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

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

The aim of investigation in the perfection of the methodical system of the development of the technical style of thinking at the pupils of academic lyccums of the technical directions by means of profile differentiation of the mathematic training.
The object of investigation is the process of differential mathematic training in academic lyccums of technical direction.
The methods of investigation. This is critical analysis of the native and foreign pedagogical experiments, working out teaching and methodic materials for teachers- experimenters and the model- practical controf of their efficiency, and also mathematic- static treatment of receiving results.
The receiving results and their novelty arc contained to the carrying out of the criteria of selection of the content of the mathematic education, of the variant of content and methods of mathematic training in the academic lyceums taking into consideration methodic peculiarities, connected with technical directions of the education.
The practical significance is contained in the possibility of using the formulated criteria of selection of the content of mathematic education for the preparation of the training programmers and training- methodical appliances by mathematic.
The degree of introduction and economic significance. Not only in the technical but also in the natural- scientific directions can be guidebook in the working out of training - methodic complex by educational profiles.
The field of application: academic lyccums of the technical direction of Ministry of Higher and Secondary-Specialized Education of the Republic of Uzbekistan.

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.

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

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

XXI asr - texnologiyalar asri hisoblanadi. Shunday ekan o‘quv jarayonida turli zamonaviy axborot vositalaridan o‘rinli foydalanish, kompyuterli ta’lim jarayonida darslarni o‘quvchi-talaba va kompyuter orasidagi munosabatlarga ko‘ra tashkil etish, boshqarish, nazorat qilish bugungi kunda dolzarb masalalardandir.Tabiiy fanlar hamda texnika fanlarida uchraydigan ko‘pgina masalalar differensial tenglamalarga keltiriladi,ya’ni ularning analitik yechimini topish nihoyatda murakkab masala,shu sababli taqribiy yechish usullaridan foydalanish ko‘proq samara beradi.Bunday muammolarni hal qilish uchun esa matematik paketlar mavjud bo‘lib,ushbu maqolada differensial tenglamalarni Maple dasturida yechish haqida gap boradi.Ya’ni, birinchi tartibli chiziqli oddiy differensial tenglamani Maple dasturida analitik yechimini topish dasturi tuzilib natija olingan.

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.

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 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.

Tiklovchi yadro usulini qo‘llab sfera sirti bo‘yicha kubatur formula qurishni va algebraik aniqlik darajasi uncha kata bo‘lmagan kubatur formulalar qurish va ulaming tugunlari sfera ichiga chizilgan muntazam ko‘pyoqlik (simpleks, giperoktaedr) ning uchlari, koeffitsentlari o'zaro teng bo‘lib chiqdi, ya’ni invariant kubator formula singari bo'ldi.

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