Magistracy
Speciality Code:
7M05402
Speciality Name:
Mathematics
Faculty:
Mechanics and Mathematics
Qualification:
 Scientific and pedagogical direction  Master of Natural Sciences
 Model of graduating student
 Mandatory disciplines
 Elective disciplines
 Professional
ON1. Apply innovative educational technologies, methods in teaching mathematical disciplines; develop assessment tools, guidelines;
ON2. Give applied interpretations and on the basis of deep system knowledge in the subject area to analyze the degree of complexity of spectral problems;
ON3. Develop kinematic manipulator circuits, critically evaluating the dynamics of robotic systems;
ON4. Competently use linguistic and cultural linguistic knowledge for communication in a multilingual and multicultural society of the Republic of Kazakhstan and in the international arena;
ON5. Develop software packages for solving problems in the natural sciences, using modern programming languages and computer modeling;
ON6. Transform models using linear and nonlinear operators in various functional and topological spaces;
ON7. Conduct research on the sustainability of the operation of electric power systems;
ON8. Construct an application research process using mathematical and statistical methods;
ON9. Create search algorithms for various queries in databases using numbering theory;
ON10. Plan and carry out experiments, evaluating the accuracy and reliability of the simulation results;
ON11. Create constructive methods for solving boundary value problems of integral and differential equations;
ON12. To conduct laboratory and numerical experiments, to assess the accuracy and reliability of the simulation results in own scientific research.
ON2. Give applied interpretations and on the basis of deep system knowledge in the subject area to analyze the degree of complexity of spectral problems;
ON3. Develop kinematic manipulator circuits, critically evaluating the dynamics of robotic systems;
ON4. Competently use linguistic and cultural linguistic knowledge for communication in a multilingual and multicultural society of the Republic of Kazakhstan and in the international arena;
ON5. Develop software packages for solving problems in the natural sciences, using modern programming languages and computer modeling;
ON6. Transform models using linear and nonlinear operators in various functional and topological spaces;
ON7. Conduct research on the sustainability of the operation of electric power systems;
ON8. Construct an application research process using mathematical and statistical methods;
ON9. Create search algorithms for various queries in databases using numbering theory;
ON10. Plan and carry out experiments, evaluating the accuracy and reliability of the simulation results;
ON11. Create constructive methods for solving boundary value problems of integral and differential equations;
ON12. To conduct laboratory and numerical experiments, to assess the accuracy and reliability of the simulation results in own scientific research.

Foreign Language (professional)
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim оf discipline: development of professional communication competence language proficiency and language of professional communication for using foreign language in day to day situations and professional spheres. During the study of course, masters should be competent in: 1. speak quickly and spontaneously without much difficulty on every day or professional topics; 2. to practice discussion on general and professional topics; 3. to demonstrate presentations on various topics and make reports including professional; 4. to explain lectures and reports on a familiar topic and to understand the content of the text; 5. to translate authentic texts on the specialty from a foreign language to the native language using a dictionary and reference books; 6. write essays and reports on general and professional topics, arguing the point of view and write detailed reports on a wide range of issues of interest. During the study of the discipline masters will learn following aspects: The "Foreign Language (Professional)" course presupposes the development of communicative competences and skills necessary for the sphere of everyday professional communication, reading and translating of authentic literature in the specialty, the formation of a terminological dictionary, writing reports and speaking at a conference, etc. The course is oriented towards the development of masters' perception of a foreign language as a source of information and a means of communication with the purpose to expand and deepen systemic knowledge in the specialty and as a means of improving their professional qualifications. In studying the discipline, the following aspects will be considered: reading, listening, translating productive: speaking and writing skills, writing reports, speaking at a conference.

History and philosophy of science
 Number of credits: 3
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim оf discipline: studying regularities and trends in the development of special activities for the production of scientific knowledge, taken in their historical dynamics and examined in a historically changing sociocultural context. During the study of course, students should be competent in: 1. to know the nature, a structure, the principles of the organization and functioning of science, genesis and history of science from a position of formation of its models, images and styles of thinking; 2. to be able to formulate and solve the problems arising during research activity and demanding profound professional knowledge; 3. to be able to choose necessary methods of a research, to modify existing and to develop new methods proceeding from problems of a concrete research; 4. to analyze and comprehend realities of the modern theory and practice on the basis of history and philosophy of science, methodology of naturalscience, sociohumanistic and technical knowledge; 5. to apply methodological and methodical knowledge in carrying out scientific research, pedagogical and educational work. During the study of the discipline masters will learn following aspects: the subject of history and philosophy of science, the worldview foundations of science, the functions of science, the emergence and formation of science. Science in the Ancient World, Middle Ages and in the Renaissance, New European Science  the classical stage of the development of science, Basic concepts and directions of the nonclassical and postnonclassical stage of the development of science, Structure and levels of scientific knowledge, Science as a profession. Ideals and norms of science, Philosophical foundations of science and the scientific picture of the world, Scientific traditions and scientific revolutions, History and philosophy of natural and technical sciences, History and philosophy of social and human sciences, Philosophical problems of development of modern global civilization.

Mathematical analysis on metric spaces
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to explain the students of the basics of modern mathematical analysis on metric spaces, stochastic analysis and martingale theory, and also some of their applications. Basic concepts and the most important fundamental results of the general theory of stochastic processes; Fundamentals of the theory of martingales and semimartingales; Definitions of a stochastic differential equation and its solution; Conditions for the existence and uniqueness of solutions of stochastic differential equations; Definitions of the diffusion process; The direct and inverse Kolmogorov equations; Connection of diffusion processes with the Cauchy problem for partial differential equations of parabolic type; Connection of diffusion processes with solutions of stochastic differential equations.

Methods of Тeaching Higher Education Mathematics
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: To study the methods of proof, methods of solving problems; methods of teaching mathematics; organizational forms of teaching mathematics in high schools; aware of the contents of the mathematics in high schools; arming the future teacher with specific knowledge in teaching high school mathematics, widening the pedagogical outlook of the student, but correctly mastering general provisions on the forms and methods of organizing the high school's mathematical activity, familiarization with the peculiarities of teaching mathematics in high schools. The subject of the methodology of teaching mathematics: the subject, content, goals, tasks of the methodology of teaching mathematics; the content of the methodology of teaching mathematics: the state and prospects, the trend of development of methodical training for the future teacher of mathematics; the appointment of methodological science; connection of methodological science with other sciences; system of methodical training (concept, structure, content). The main components of the content of the school course of mathematics. Goals of teaching mathematics: educational goals; educational goals; developing goals. Principles of teaching: the concept of the principle of teaching; system of principles of training; the implementation of the principles of education. Methods of teaching mathematics: the concept of teaching method; scientific and traditional methods of teaching; innovative methods of teaching mathematics. Means and forms of teaching mathematics: classification; didactic functions; forms of organization of training. Mathematical concepts, suggestions and methods for their study: axioms, theorems; axiomatic method; evidence; the method of introducing mathematical concepts in a concreteinductive way; the technique of introducing mathematical concepts in an abstract deductive method. Psychological and pedagogical foundations in teaching mathematics: psychological foundations in teaching mathematics; pedagogical bases in teaching mathematics; formation of cognitive interest in mathematics; education in the learning process. Methodology for teaching mathematics through tasks: the role of problems in teaching mathematics; Classification of tasks by content and functions; general methods of teaching problem solving. Specificity of teaching discipline: the organization of the activities of students, planning and conducting training sessions. Application of innovative technologies in mathematics lessons.

Pedagogy of higher education
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim оf discipline: mastering the fundamentals of the professional and pedagogical culture of a higher school teacher, familiarizing future teachers with general issues, methodological and theoretical foundations of higher school pedagogy, modern analysis technologies, planning and organizing training and education, communicative technologies of subjectsubject interaction between a teacher and a student in the educational process of the university. During the study of course, masters should be competent in: 1. master the basics of professional and pedagogical culture of a higher school teacher; 2. analyze the system of higher professional education in Kazakhstan; 3. determine the content of higher education; 4. to apply traditional and innovative methods and forms of organization of education, new educational technologies in higher education; 5. to evaluate the communication technologies of the subjectsubject interaction of the teacher and student in the educational process of the university. During the study of the discipline masters will learn following aspects: Pedagogical science and its place in the system of human sciences. The modern paradigm of higher education. The system of higher professional education in Kazakhstan. Methodology of pedagogical science. Professional and communicative competence of the teacher of higher education. Theory of education in higher education (didactics). Content of higher education. Organization of the learning process based on the credit system of education in higher education. Traditional and innovative methods and forms of organization of training. New educational technologies in higher education. Organization of independent work of students in the conditions of credit technology. Technology of compilation of teaching materials. Theory of scientific activity of higher education. Scientific and research work of students. Higher school as a social institution of education and personality formation specialist. Curator in the system of higher education. Management in education.

Psychology of management
 Number of credits: 3
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim оf discipline: is the formation of master students in the development of modern scientific information using the fundamentals of the psychology of science and practice necessary to improve the effectiveness of professional activities. During the study of course, masters should be competent in: 1. to interpret basics of psychological science and practice necessary to improve the effectiveness of professional activities; 2. to explain the psychological patterns and phenomena arising in professional and scientificpedagogical activities; 3. to analyze life and professional situations from the point of view of psychology, to see the relationship between the behavior and results of individual and collective activities, resources development opportunities; 4. to integrate psychological information into general knowledge and formulate judgments; 5. to use psychological knowledge to make effective decisions, implement successful communication strategies in personal life and professional activities; effectively use psychological knowledge in order to develop own personality and team. During the study of the discipline masters will learn following aspects: This course will familiarize master degree students with scientific knowledge corresponding to the current level of development of psychological science and practice. The subject, branches, methods of psychology, patterns of development and functioning of mental processes, properties and states of personality, communication, basics of effective interaction and conflict overcoming, selfdevelopment and selfpresentation will be examined, taking into account the professional and scientificpedagogical activity of master degree students.

Master’s dissertation preparation and defense (MDPaD)
 Number of credits: 12
 Type of control: Master Dissertation
 Description: The main purpose of "The implementation of a Master Thesis": the formation of master students in preparation for the defense of the thesis for the Master in specialty (by industry). During the study of course, master student's should be competent in: 1. demonstrate the progress of solving problems arising in the course of research activities and requiring indepth professional knowledge; 2. to argue for carrying out theoretical or experimental research within the framework of the tasks, including a mathematical (simulation) experiment; 3. can choose the necessary research methods, modify existing methods and develop new methods, based on the tasks of the specific study; 4. to use foreign languages for independent work on normative sources and scientific literature; 5. formulate the goals and objectives of the dissertation research, determine the scientific novelty and practical significance of the results of research activities; to develop a structurally methodological scheme for performing research.

Multidimensional Complex Analysis
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: To study multidimensional complex analysis, the theory of holomorphic functions of several variables and holomorphic mappings of complex manifolds. Multidimensional complex analysis, the theory of holomorphic functions of several variables and holomorphic mappings of complex manifolds. Holomorphic mappings, which in the multidimensional case are just as basic concept as functions. Concepts of algebra and topology, the results of the geometric theory of functions of several complex variables.

Effective computability
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: acquaint students with modern problems and research trends in computability theory, attracting students to modern research in mathematical logic by formulating and discussing open topical issues. The concept of algorithmically solvable and insoluble problems; The theorem on parametrization; Kleene's theorem on a fixed point; Rice's theorem; The undecidability of the problem of recognizing the properties of functions by the leading Their programs; Computable isomorphism of any two universal computable functions; Basic classes of computably enumerable sets; Properties of Post and Kleene numbering.

General algebra
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The main goal of the course is to introduce the graduate students in all fields of mathematics into basic algebraic structures: groups, rings, modules. To introduce to general notions of algebraic structure, substructure, homomorphism, and isomorphism. To give typical examples of that kind of structures and study algebraic constructions that allow for a given structure to create a new one of the same type. To be able to recognize basic algebraic structures and algebraic constructions in any field of mathematics and apply knowledge on them into particular domains.

NikolskyBesov Spaces and Their Applications To Boundary Value Problems for Generalized Analytic Functions
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: W l The theory of generalized analytic functions was constructed by Academician I.N. Vekua in the framework of Sobolev spaces p , p> 2. Generalized analytic functions, possessing the basic classical properties of analytic functions of a complex variable, have found numerous real objects of application. The theory of these functions, having deep connections with numerous sections of analysis, geometry and mechanics, has been organically intertwined with the problems of differential geometry and continuum mechanics.

Singularly Perturbed Integrodifferential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: to study the basic problems of the theory of initial and boundary problems for singularly perturbed integrodifferential equations and methods for solving such problems. During the study of course, students should be competent in:  Have a modern theoretical understanding of the role and place of singularly perturbed problems;  Freely possess asymptotic methods for solving singularly perturbed integrodifferential equations and clarify the scope of these methods;  the skills of mathematical modeling of applied problems described by singularly perturbed integrodifferential equations and the interpretation of the results obtained;  To conduct intensive research work and publicly present their own new scientific results;  Work in a team and to defend the correctness of the choice of solving the problem;  To critically evaluate yourself activities, the activities of the team, and to be capable of selfeducation and selfdevelopment. Initial and boundary value problems with an initial jump for singularly perturbed integrodifferential equations. Initial and boundary functions of singularly perturbed homogeneous differential equations. Analytical formula and asymptotic estimate of the solution of initial and boundary value problems for singularly perturbed linear integrodifferential equations. Estimation of the difference between solutions of singularly perturbed and unperturbed problems. Uniform asymptotic expansion of solutions of initial and boundary value problems.

Computability in the Hierarchies
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of this course is to examine modern, unscientific scientific achievements in the field of computable numbering and to use them. GoncharovSorbi approach. Syntactic and algorithmic criteria for the computability of numberings in the classical case of families of computably enumerable sets. The general approach of GoncharovSorbi and its application to the introduction of the concept of arithmetic numbering. The syntactic criterion for the computability of numberings in the arithmetic hierarchy and the criterion in terms of uniform enumeration with respect to oracles. Rogers semilattices concept. Khutoretsky theorems and their generalizations. The classical Khutoretsky theorem on the power of semilattices of computable numberings. The Khutoretsky theorem on the impossibility of decomposition into a main ideal and a main filter. GoncharovSorbi theorem on minimal pairs of arithmetic numberings. The concept of the BadayevLemp theorem on the decomposition of Rogers semilattices for families of differences of computably enumerable sets. Computability in the Ershov hierarchy. A criterion for computability of the numbering of the family of sets of the Ershov hierarchy. Effective discreteness and power of Rogers semilattices. Badaev's theorem on the existence of finite nondiscrete families with the trivial Rogers semilattice. Families without minimal computable numberings.

Computable Functions
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: To introduce students to main approaches of the computable function notion, show their equivalence and acquaint them with decidable and undecidable problems in mathematics. Informal description of an algorithm. Problems that has no algorithmic solution. Gödel’s approach. Primitive recursive functions. Closure properties of the class of primitive recursive functions. Pairing functions. The Gödel function. Partially computable and computable functions. Kleene’s theorem on normal form. Numbering of Kleene. smn theorem. Recursion theorem. Turing machines. Functions that are computable on Turing machines. Different flavors of Turing machines. A universal Turing machine. Equivalence of the approaches of Gödel and Turing. The Church thesis. Decidable and undecidable problems. Diagonalization. The halting problem. Reduction. Rice’s theorem. Undecidable problems about contextfree languages.

Additional Сhapters of Differential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: indepth study of some sections of the asymptotic theory of differential equations, the formation of knowledge necessary for the effective use of asymptotic methods for constructing and analyzing solutions of ordinary differential equations and partial differential equations with a small parameter for higher derivatives and the ability to apply these methods in the study of fundamental and applied problems. During the study of course, students should be competent in:  Use fundamental knowledge in the field of mathematical analysis, complex and functional analysis, differential equations in future professional activities;  Freely possess asymptotic methods for solving singularly perturbed ordinary and partial differential equations and clarify the scope of these methods;  Use the methods of mathematical modeling in solving theoretical and applied problems;  Conduct intensive research work and publicly present their own new scientific results;  Work in a team and to defend the correctness of the choice of solving the problem;  To critically evaluate yourself activities, the activities of the team, and to be capable of selfeducation and selfdevelopment. Dependence of solutions on parameters; the method of small parameter for solving differential equations; finding periodic solutions of linear differential equations; asymptotic integration; ordinary differential equations with a small parameter at the derivative; limit transition theorem; asymptotic behavior of solutions of differential equations with respect to a small parameter; singularly perturbed partial differential equations; a singularly perturbed first boundary problem for systems of linear hyperbolic equations.

Ideals and Diversity
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The goal of the discipline is to teach students the basics of linear algebra, in which one of the most important ideas of mathematics  the idea of linearity  realized in such concepts as linear operations, linear dependence and independence, rank, linear space, linear and bilinear transformations, And also initial acquaintance of students with the basic algebraic structures such as group, ring, field, which have applications in the most diverse branches of modern science and technology. Abstract Affine variety and polynomial ideal. The algorithm for dividing polynomials in several variables in a ring. Hilbert's theorem on the basis. Bases of Grebner. Theorems on exclusion and continuation.

Iterative methods for solving nonlinear equations and their applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: To study of iterative methods for solving nonlinear equations and systems of nonlinear equations, numerical solution of boundary value problems for differential equations. This course explores the most important class of methods for solving nonlinear systems  iterative methods. The construction of a general theory of such methods is associated with the consistent application of functionaltheoretical ideas and, above all, with the use of the principle of contraction mappings. Note that iterative methods are widely used in the numerical solution of boundary value problems for differential equations. The course is aimed at studying both the classical methods of Newton and secants, as well as generalized linear methods, in particular, the methods of consistent upper relaxation. Much attention is paid to the convergence of iterative methods. Of particular interest here is the study of semilocal and global convergence, that is, convergence in cases where the initial approximation is not assumed to be close enough to the desired solution or is generally chosen arbitrarily. Iterative methods with damping factors will be applied to finding a solution to a nonlinear twopoint boundary value problem for ordinary differential equations.

The Qualitative Theory of Differential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: "Qualitative theory of differential equations" is a mathematical science, which is the Foundation of mathematical and natural science education. Therefore, the aim of the discipline Qualitative theory of differential equations is to study the basic concepts of qualitative theories of differential equations. In the course of studying the course to form students ' abilities: – Explain the properties of solutions to a system of differential equations using qualitative theory of differential equations (singular points, classification of integral curves and trajectories, classification of singular points, etc.) on the plane and in space; – Calculate typical problems (finding special points, study of types of special points, study of special points of the system of differential equations on the plane, finding the direction of points and trajectory) using the methods of qualitative theories of differential equations; – Organize the solution of applied problems using geometric and mechanical meanings of singular points; – Classify singular points on the plane by methods of qualitative theories of differential equations; – Describe the study of special points of a linear Autonomous system of differential equations on the plane by methods of qualitative theories of differential equations. – To design the research process of the applied problem using methods of qualitative theories of differential equations; – To work in a team, to defend the correctness of the choice of the solution of the problem Autonomous systems of differential equations. Properties of solutions. Autonomous systems of differential equations on the plane. Linear autonomous systems of differential equations on a plane. Special points. Types of singular points and phase portraits. Nondegenerate singular points of a nonlinear system of differential equations and phase portraits. Followup function. Cycle. Limiting cycle. Theory of indices. Limit points of trajectories of an autonomous system of differential equations. The notion of Poisson stability. The concept of Lagrange stability

Constructive Theory of Problems of Ordinary Differential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The aim of the course is to study complex constructive methods for solving boundary value problems of optimal control, formulate optimality conditions in the form, prove their convergence, and obtain estimates of the rate of convergence. In the course, we consider complex constructive methods for solving boundary value problems, i.e. where there are besides the objective functional and boundary conditions, phase constraints and integral constraints on the phase coordinates of the system, as well as constraints on the control values. The main task is to determine such boundary conditions from given sets and controls from a given functional space that satisfy the constraints on controls that ensure the achievement of the main control objective when performing phase and integral constraints.

Boundary value problems for differential equations in partial derivatives
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Boundary value problems for parabolic and elliptic equations in Holder and Sobolev spaces. First and second boundary problems for parabolic equations in the Hölder space. Existence, uniqueness, solution estimates. The method of constructing a regularizer for proving the existence of a solution, Schauder's method for deriving estimates of a solution. The Dirichlet problem for elliptic equations in the Sobolev space. Existence, estimates of the problem solution. Fredholm's theorems. As a result of the training, students must possess the technique of obtaining a priori estimates of the Holder and Sobolev spaces, and also the solvability of boundary value problems of the parabolic type by modern methods (the method of constructing a regularizer for proving the existence of a solution, the Schauder method, and the Fredholm property of differential operators).

Boundary Value Problems for Ordinary Differential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: We study boundary value problems for ordinary differential equations of arbitrary order with a small parameter with the highest derivative. Estimates of solutions and solutions of perturbed problems are given. We will obtain asymptotic expansions of solutions with a work of degree of accuracy with respect to a small parameter. As a result of the training, the undergraduates will be introduced to the methods of investigating boundaryvalue problems for ordinary differential equations with a small parameter with the highest derivative. Be able to detect the influence of a small parameter on the asymptotic behavior of solutions, to determine the order of growth of solutions at the point of the initial jump. To have skills in solving boundary value problems for ordinary differential equations

Mathematical foundations of optimal control
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: Masters should have fundamental knowledge in differential calculus in Banach space, differential controls in Banach space will fly into the basics of convex analysis in Banach space, practical skills for calculating the functional gradient, be able to apply minimization methods to solve applied problems in Banach space. In the process of the study course to form students' abilities: – Explain the general problem statement of optimal control problem with restrictions in the context of the appropriate theories; – Compute the typical tasks using the main determinations; – To order a solution of the applied problems by differentiating of nonlinear operators, differentiating of nonlinear functionals; – Use existence and uniqueness solution of the differential equations in Banach space, theorem about global minimum; – Describe optimality conditions, Weierstrass theorem in Banach space; – Construct a process of the study of applied problems by methods of minimization of functionals in Banach space: – To work in a team, to defend the correctness of the choice of the solution of the problem Contents of the discipline: To solve actual problems of the natural sciences, new mathematical methods are needed to solve complex scientific and technical problems. One of the characteristic features of the modern era is the increasing attention to the problems of g of management. In this discipline, the theoretical foundations of optimal control are studied, including: bases of differential calculus in a Banach space, convex analysis, methods of minimization in a Banach space.

The Method of Compactness and Monotony for Nonlinear Problems of Mathematical Physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is devoted to the study of nonlinear problems of mathematical physics from the point of view of modern functional analysis. Therefore, the concept of nonlinearity prevails throughout the course. Here we consider the following modern methods for studying initialboundary value problems for equations of mathematical physics: a priori estimates method, variational methods, monotonicity and compactness methods, the Pokhozhaeva identity, and a regularization method. In the course of studying the course to form students' abilities:  Explain the basic concepts of generalized solutions of initialboundary value problems;  Identify weak and strong generalized solutions for nonlinear problems of mathematical physics, using modern methods of theories of functional analysis;  To prove the solvability of applied problems using the method of a priori estimates, variational methods, methods of monotonicity and compactness, and the method of regularization;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the unique solvability of nonlinear problems of mathematical physics;  Design the process of studying an applied problem using the methods of monotonicity and compactness;  Work in a team, reasonably defend the correct choice of the problem. As a result of training, undergraduates should know the theory of generalized functions and its application in boundary value problems for nonlinear equations. Namely Model nonlinear equations. Derivation of some model nonlinear elliptic, parabolic and hyperbolic equations that have a physical meaning. Weak derivative H1 and H01spaces. The trace functions from the spaces H1 and W21,1. Nemytskyi operator. Statement of the Dirichlet problem for a semilinear elliptic equation. The method of upper and lower solutions. The method of compactness in combination with the methods of monotony and Galerkin. Leray – Schauder method. Classic solutions. The degree of the Leray – Schauder map. The existence of a solution to the heat equation with a nonlinear source. Weak maximum principle for weak solutions of the Laplace and Poisson equations. Weak maximum principle for weak solutions of the heat equation.

Methods for solving extremal problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: undergraduates must have fundamental knowledge of the theory of extremal problems in Banach space for partial differential equations, possess the basics of convex analysis, practically have the skills to calculate the gradient of a parabolic equation that is functional on a set of solutions, a hyperbolic equation, be able to apply elastic flexible strings to solve applied problems of heat conduction; know the methods of minimization of functionals in a Banach space. During the course study, form undergraduates' abilities: On the basis of deep system knowledge in the subject area to solve modern problems and problems in mathematics. Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts. Quickly find, analyze and correctly process scientific and technical, naturalscience and general scientific information contextually, leading it to the problemproblem form. Apply modern methods for independent research and interpret their results; Apply skills to write reviews, reports and research articles. Independently formulate the problem and select the appropriate mathematical model. Contents of the discipline: To solve theoretical problems of mathematical physics, natural sciences, new mathematical methods of optimal control of the processes described by partial differential equations are needed. In this discipline, we study the basic methods of minimizing functionals in the form of multiple integrals on a set of partial differential solutions; methods of minimizing functionals: gradient method, gradient projection method, conditional gradient method, NewtonKantorovich method, integral functionals method.

Optimal control of systems with partial derivatives
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Contents of the discipline: Undergraduates should have fundamental knowledge in the theory of optimal control of systems described by partial differential equations, master the basics of the theory of control of systems described by partial differential equations of elliptic, parabolic, hyperbolic types, theorems on the existence of optimal control and practical skills in solving applied problems. During the course study, form undergraduates' abilities: On the basis of deep system knowledge in the subject area to solve modern problems and problems in mathematics. Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts. Quickly find, analyze and correctly process scientific and technical, naturalscience and general scientific information contextually, leading it to the problemproblem form. Apply modern methods for independent research and interpret their results; Apply skills to write reviews, reports and research articles. Plan and carry out experiments, evaluating the accuracy and reliability of the simulation results. Analyze the results and draw reasonable conclusions. Contents of the discipline: To solve actual problems of mathematical physics, in particular, the problem of magnetic hydrodynamics, the theory of elasticity of gas dynamics, we need new mathematical methods for optimal control of progress by the partial differential equations described. In this discipline, we study methods for solving an optimal control problem for equations of elliptic, parabolic and hyperbolic types separately, necessary firstorder optimality conditions.

Approximations for Functions of Several Variables
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to present an interpolation formula, to give an estimate of the best best approximation by partial best approximations for a function of several variables Interpolation is the approximation of a function of a curve passing through all N points. The main disadvantage of interpolation algorithms is that when changing the value of a function at one point, it is necessary to completely recalculate the interpolation formulas. Approximation is the approximation of a curve that does not necessarily pass through all points. The basic approximation methods have the property 'local control': changing the value of a function at one point entails recomputing only 13 formulas. In the course, an estimate is given of the best best approximation by particular best approximations for a function of several variables.

AppIied Statistic
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of studying the discipline is the development of basic probabilistic knowledge of random processes in finance, as well as the formation of practical skills in the application of stochastic methods and models and economic interpretation of results. During the course, students should have the following abilities:  explain the key concepts of stochastic financial mathematics (basic and derivative financial instruments; stock and option pricing models; BlakeScholes model; portfolio of financial instruments; Markowitz model; diversification and optimization of the portfolio, etc.) in the context of the corresponding theory;  solve typical tasks (forecasting the price of a financial instrument; estimating the profitability and risk of a financial transaction; hedging; diversification and optimization of a portfolio, etc.) using the methods of stochastic financial mathematics;  to optimize the solution of applied problems using the tools of stochastic financial mathematics;  to classify the basic concepts of stochastic financial mathematics (financial instruments and their properties; portfolios of financial instruments, etc.);  describe the study of stochastic processes in finance using stochastic financial mathematics;  to design the process of studying an applied problem using the methods of stochastic financial mathematics;  work in a team, reasonably defend the correctness of the choice of solving the problem. The content of the discipline is aimed at studying such concepts and definitions as financial instruments; binomial models of price evolution; BlakeScholes model; Markowitz theory; types and properties of options; stochastic models of price dynamics.

Application of Approximate Calculations To the Problems About Eigenvalues
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: A survey of papers devoted to approximate calculation of eigenvalues and eigenfunctions of differential operators of SturmLiouville type by methods of the theory of regularized traces is proposed. The method of A.A. Dorodnitsyn and its development in the form of a theory of regularized traces of differential operators is decribed. In the course, we give known classical methods of asymptotic computation to problems on eigenvalues generated by differential operators on finite domains with a punctured single point. In the course regularized traces of differential operators, completeness of the system of root functions are considered.

Direct and Inverse Problems for Nonclassical Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to familiarize undergraduates with new results on the theory of direct and inverse problems for nonclassical equations and with modern methods for solving them. In the course of studying the course to form students' abilities: Have a high theoretical training in direct and inverse problems for nonclassical equations, in order to further perform independent scientific work;  Be able to apply and have practical skills in solving applied problems; Be able to independently carry out research and write articles; To analyze and formulate typical direct and inverse problems of natural science;  To carry out teaching of special disciplines in universities and colleges. Any differential equation is a mathematical model of a real physical, chemical or biological process. The achievements of modern scientific research show that most of these processes model by nonclassical equations of mathematical physics. In this course consider the direct and inverse problems for nonclassical equations of mathematical physics which describe these real processes.

Reducibility and completeness
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to form the ability to build different numbering for different families of sets and functions. mastering categorytheoretic approaches in numbering theory, learning how to work with subobjects and main subobjects in the category of numbered sets. Categorytheoretic concepts. The numbering of the set and its subsets. The category of numbered sets and its properties. Subobjects of a numbered set. Complete and fully numbered sets. Positively numbered sets. Numbered sets with approximation. Structural theorems on fully numbered sets. Creativity and muniversality for computable numberings.

Singularly perturbed differential equation with piecewise constant argument
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: to study the basic problems of the theory of initial and boundary problems for singularly perturbed differential equations with piecewiseconstant argument and methods for solving such problems. During the study of course, students should be competent in:  Have a modern theoretical understanding of the role and place of singularly perturbed problems;  Freely possess asymptotic methods for solving singularly perturbed differential equations with piecewiseconstant argument and clarify the scope of these methods;  the skills of mathematical modeling of applied problems described by singularly perturbed differential equations with piecewiseconstant argument and the interpretation of the results obtained;  To conduct intensive research work and publicly present their own new scientific results;  Work in a team and to defend the correctness of the choice of solving the problem;  To critically evaluate yourself activities, the activities of the team, and to be capable of selfeducation and selfdevelopment. Initial and boundary value problems for singularly perturbed integrodifferential equations with piecewiseconstant argument. Initial and boundary functions of singularly perturbed homogeneous differential equations with piecewiseconstant argument. Analytical formula and asymptotic estimate of the solution of initial and boundary value problems for singularly perturbed linear integrodifferential equations with piecewiseconstant argument. Estimation of the difference between solutions of singularly perturbed and unperturbed problems. Uniform asymptotic expansion of solutions of initial and boundary value problems.

Statistics of Random Processes
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to familiarize undergraduates of the specialty “Mathematics” with basic concepts and results of statistics and random processes, with considerable attention being paid to the theory of optimal nonlinear filtering. The objectives of the course are: Successful mastering of the main results of this discipline by undergraduates so that they can subsequently use them effectively in the course of their future scientific and educational activities; Acquisition of practical skills in educational and scientific literature in various sections of the course under study; Learning Outcomes: Gets enough information on the theory of optimal nonlinear filtering for both discrete and continuous time cases; Familiarize with the tasks of consistent assessment. Preliminary information: the theory of random processes; martingale theory; mathematical statistics. Introduction to the theory of optimal nonlinear filtering: the case of discrete time; case of continuous time. Questions of application to problems of sequential estimation, to linear filtering (Kalman filter  Bucy), interpolation and extrapolation of some components of random processes by others.

Stochastic Differential Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of teaching this discipline is to thoroughly familiarize undergraduates with basic concepts,results, and some of the most important, both theoretical and practical, applications of the modern theory of stochastic differential equations. The objectives of the course are: Successful mastering of the main results of this discipline by undergraduates so that they can subsequently use them effectively in the course of their future scientific and educational activities; Acquisition of practical skills in educational and scientific literature in various sections of the course under study; Learning Outcomes: The undergraduate who successfully mastered the program of this course: Receives skills in the application of methods of the theory of stochastic differential equations and is able to apply these methods to solve typical standard problems; Receives a clear understanding of there lationship of this discipline with other disciplines of the selected educational program; Will be able to freely navigate in the main directions of further development of topics of this discipline; Stochastic integrals from nonrandom and random functions on process with orthogonal increments; Ito Integral; Stochastic differential; Ito's Formula: onedimensional and multidimensional cases; Definitions of a stochastic differential equation and its solution; Conditions for the existence and uniqueness of solutions of stochastic differential equations; Connection of diffusion processes with solutions of stochastic differential equation.

Stochastic analysis and equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of teaching this discipline is to thoroughly familiarize undergraduates with basic concepts, results, and some of the most important, both theoretical and practical, applications of the modern theory of stochastic analysis. The objectives of the course are: Successful mastering of the main results of this discipline by undergraduates so that they can subsequently use them effectively in the course of their future scientific and educational activities; Acquisition of practical skills in educational and scientific literature in various sections of the course under study; Learning Outcomes: The undergraduate who successfully mastered the program of this course: Receives skills in the application of methods of the theory of stochastic analysis and stochastic calculus and is able to apply these methods to solve typical standard problems; Receives a clear understanding of the relationship of this discipline with other disciplines of the selected educational program; Will be able to freely navigate in the main directions of further development of topics of this discipline; Random function. Basic concepts of the general theory of stochastic processes; Convergence; Continuity; Derivatives; Integrals; Conditional expectation with respect to partitions and sigma algebras Fundamentals of the theory of martingales Diffusion process; The direct and inverse Kolmogorov equations; Connection of diffusion processes with the Cauchy problem for partial differential equations of parabolic type. Stochastic differential; Ito's Formula: onedimensional and multidimensional cases; Definitions of a stochastic differential equation and its solution; Conditions for the existence and uniqueness of solutions of stochastic differential equations; Connection of diffusion processes with solutions of stochastic differential equation.

Sums of Independent Random Variables
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to familiarize undergraduates of the specialty “mathematics” with the problem of researching the sums of independent random variables, to put it differently, familiarizing them with limit in different senses theorems for such sums and their numerous theoretical and practical applications. The objectives of the course are: Successful mastering of the main results of this discipline by undergraduates so that they can subsequently use them effectively in the course of their future scientific and educational activities; Acquisition of practical skills in educational and scientific literature in various sections of the course under study; Learning Outcomes: Receives sufficient information about the history of the development and development of limit theorems of probability theory, can distinguish between the conditions for their fulfillment and can apply their results to solve practical problems; knows different types of convergence of sequences and series of random variables and about the relations between them; has an idea of the main modern directions of development of the theory of summation of independent random variables. Limit theorems in the Bernoulli scheme. Different types of convergence of a sequence of random variables and their relationship. The method of characteristic functions of proof of limit theorems. Central limit theorems (CLT) for identically and differently distributed sequences of random variables. Laws "zero or one". Weak and strengthened laws of large numbers.

Numbertheoretic Methods in Approximate Analysis and Their Applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is the exposition of a function of many variables, functional classes, multiple integrals, quadrature formulas and recovery operators and their errors. In the middle of the 20th century, in connection with the needs of the military industrial complex, as well as other tasks of national economic importance, it became necessary to develop optimal and computerimplemented methods for approximate calculation of integrals of large multiplicity and recovery of functions of many variables. In Kazakhstan significant results were obtained in this direction (Prof. N. Temirgaliyev and his school). The course outlines the results of one of the founders of numbertheoretic methods in the approximate analysis of NM. Korobov, as well as listeners are offered new unsolved problems.

Theoretical and computational problems of mathematical physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of mastering the discipline "Theoretical and Computational Problems of Mathematical Physics" is to prepare undergraduates to solve boundary problems of mathematical physics and develop effective computational algorithms for numerical solution. In the course of studying the course to form the ability of undergraduates Create mathematical methods for solving boundary mathematical physics.  Develop effective mathematical methods for solving applied problems in various fields of science;  Develop efficient computational algorithms for the numerical solution of boundary value problems of mathematical physics. Have fundamental knowledge in modern sections of mathematical modeling and numerical solution.  Perform scientific work on current problems of differential equations to control theory. The course content is aimed at applying modern analytical and computational methods to solving boundary problems of mathematical physics and partial differential equations. The course covers the following topics: Basic problems of mathematical physics, basic methods for solving boundary problems of mathematical physics. Modern computational methods and their applications.

The Theory of Identification of the Boundary Conditions and Its Applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is the presentation of research on a new scientific direction  the theory of identification of boundary conditions for spectral eigenvalue problems. The course provides a systematic presentation of research on a new scientific direction  the theory of identification of boundary conditions of spectral problems in eigenvalues. As applications of the theory, methods are developed for diagnosing fastenings of mechanical systems based on their own frequencies of their oscillations, as well as methods for creating fastenings that provide the necessary (safe) frequency range for oscillations of a fixed mechanical system.

The Theory of Finite Fields
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The goal of the discipline is to teach students the basics of linear algebra, in which one of the most important ideas of mathematics  the idea of linearity  realized in such concepts as linear operations, linear dependence and independence, rank, linear space, linear and bilinear transformations, And also initial acquaintance of students with the basic algebraic structures such as group, ring, field, which have applications in the most diverse branches of modern science and technology. The course is an introduction to finite fields, which is one of the main Subjects in algebra, computer science and cryptography. The course will cover the main Topics on linear algebra (the first half of the semester) and analytical geometry (the second half). During the course, we formulate the basic concepts and results that have become Classic these days and try to describe the current trends and achievements.

Theory of boundary value problems of optimal control
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to acquaint undergraduates with new results on methods for solving boundary value problems of optimal control for the processes described by ordinary differential equations, that differring from the known methods based on the Lagrange principle. Masters must have a high theoretical background and be able to apply for solving applied problems on computers: – Elicitation of controlability theory role in optimal control; – Existence criteria test methods statement for boundary value optimal control problems with different kinds of constraints; – Solution construction methods statement for boundary value optimal control problems with linear and quadratic performance criteria; – Constructive optimal control theory application in solving applied problems ways description; – To work in a team, to defend the correctness of the choice of the solution of the problem. Content of the discipline: The principle of immersion. The existence of a solution to a boundary value problem. Construction of minimizing sequences and the definition of the lower bound of the functional. The construction of an optimal solution by narrowing the range of admissible controls.

Theory of martingales
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of teaching the discipline is to familiarize undergraduates of the specialty "Mathematics" with the basics of one of the most modern areas of the theory of random processes  the theory of martingales. The goal of the course is not only to communicate a known stock of information (definitions, theorems, their proofs, relationships between them, methods for solving traditional problems), but also to teach undergraduates to the skills of their application in various branches of science and practice, Learning outcomes: Knows the definitions and properties of the main objects of study of the theory of martingales, the wording of the most important statements, methods of their proofs, possible areas of application; Gets ideas about the connections between martingales and semimartingales (submartingales, supermartingales); Will be able to freely navigate in the main directions of further development of topics of this discipline; Acquires practical skills in solving standard problems of the theory of martingales. Conditional expectations: with respect to the partition; relative to sigmaalgebra; one random value relative to another random variable; Martingale definition; Moment of stopping; Application of martingales to random walks; Preservation of martingale property when replacing time with a random moment; Wald's identity; Fundamental inequalities; Martingales and semimartingales (discrete and continuous time); Convergence theorems. Wiener process as a square integrable martingale.

Theory of Statistical Estimation
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of teaching this discipline is to thoroughly familiarize undergraduates with basic concepts, results, and some of the most important, both theoretical and practical, applications of the modern theory of theory of statistical estimation The objectives of the course are: Successful mastering of the main results of this discipline by undergraduates so that they can subsequently use them effectively in the course of their future scientific and educational activities; Acquisition of practical skills in educational and scientific literature in various sections of the course under study; Learning Outcomes: The undergraduate who successfully mastered the program of this course: Receives skills in the application of methods of the theory of theory of statistical estimation and is able to apply these methods to solve typical standard problems; Receives a clear understanding of the relationship of this discipline with other disciplines of the selected educational program; Will be able to freely navigate in the main directions of further development of topics of this discipline; General introduction. Sufficient statistics. Unbiased estimation (parametric and nonparametric cases). Efficiency estimates for quadratic loss function. Maximum likelihood estimation. Asymptotic normality of the estimate. Trust evaluation. Tolerant assessment.

The Theory of the NavierStokes Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: The purpose of the discipline "Theory of NavierStokes Equations" is the study by undergraduates of generalized solutions of boundary and initial boundaryvalue problems for the NavierStokes equations. The problems of solvability and stability of solutions, boundary value problems for the NavierStokes equation are investigated. The method of a priori estimates, the LeraySchauder method, the FaedoGalerkin method will be considered. Special attention is paid to the applied side of the studied problems. In the course of studying the course to form students' abilities:  Explain the key concepts of the theories of the NavierStokes equation;  Calculate tasks (Obtaining a priori estimates in Hölder and Sobolev spaces. Solvability of initialboundary problems) using modern methods of functional analysis;  To prove solvability of boundary value problems of the Stokes equation, initial boundary value problems of the NavierStokes equation and various applied problems using the method of a priori estimates, the LeraySchauder method, the FaedoGalerkin method .;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the solution of the linear and nonlinear problem of the NavierStokes equation by methods of the theory of generalized functions;  Design the process of studying an applied problem using the methods of the theory of theories of the NavierStokes equation;  Work in a team, reasonably defend the correct choice of the problem. As a result of training, students should know the theory of generalized functions and its application in problems of the theory of partial differential equations. Namely, the space of basic functions and generalized functions. The completeness of the space of generalized functions. Replacing variables in generalized functions. Differentiation of generalized functions. Properties of generalized derivatives. Transformations of the generalized function. Direct product and convolution of generalized functions and their properties. Convolution algebra of generalized functions. Generalized functions of slow growth. The structure of generalized functions with point carrier. Fourier transform of generalized functions and their properties. Laplace transform of generalized functions (operational calculus) and their properties. Application of the theory of generalized functions to the construction of a fundamental solution and to the solution of the Cauchy problem for the wave equation and for the heat equation. In addition, in the course of study, students should and could build examples from quantum physics and other continuum mechanics problems. For this purpose, various examples of mathematical physics and the application of the theory of generalized functions when presenting solutions to the basic problems of mathematical physics will be considered in the course of study.

Theory of stability of dynamic systems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: To acquaint undergraduates with new research on the theory of stability of solutions of equations with differential inclusions of dynamic systems. In the course of studying the course to form the ability of undergraduates Get knowledge on the study of sustainability of regulated systems.  Create mathematical methods for studying the stability of solutions of dynamic systems.  Apply knowledge to the study of the stability of solutions of differential equations of other fields.  Perform scientific work on current problems of differential equations. The course content is aimed to exploring new methods for studying the absolute stability of the equilibrium state of nonlinear controlled systems. The methods for studying the global asymptotic stability of phase systems with a countable equilibrium position are consider.

The theory of stability regulated systems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: Masters must have fundamental knowledge of the theory of controllability of solving equations with differential inclusions, when the righthand side contains nonlinear functions from a given set, must know unsolved problems of the theory of dynamical systems, own the theories of stability of regulated systems in the basic, in simple critical and critical cases. In the process of the study course to form students' abilities: – Explain the general problem statement in the context of the appropriate theories; – Compute the typical tasks using the main determinations; – To order a solution of the applied problems by equilibrium state, nonuniqueness of a solution. Investigate of absolute stability of regulated systems in the main case. Nonsingular transformations. – Use solution's properties, improper integrals. – Describe absolute stability, investigation of absolute stability of regulated systems in a critical case. Nonsingular transformations. – Construct a process of the study of applied problems by solution's properties, improper integrals, absolute stability, investigation of absolute stability of regulated systems in a critical case. Nonsingular transformations. – To work in a team, to defend the correctness of the choice of the solution of the problem. Contents of the discipline: In this discipline, the general theory of absolute stability is studied on the equilibrium state of onedimensional and multidimensional regulated systems with limited resources for three cases: basic, simple critical, critical. For these cases, the conditions of absolute stability in the parameter space of systems were obtained separately.

Evolution Equations of the Second Order
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to study methods for solving boundary value problems for evolution equations using functional analysis. The theory of partial differential equations is not part of the functional analysis. In spite of the fact that some classes of equations can be interpreted in terms of abstract operators acting in Banach spaces, the insistence in taking a superficially abstract point of view and the consequent ignoring of subtle theorems, computations, and the derivation of a priori estimates is ultimately a great loss in the study of the required problems. During the study of course, students should be competent in: – The basic laws of the development of science and technology; – Know the Basic concepts and methods of the theory of equations of mathematical physics; – Know he main types of special functions; Evolution equations that characterize processes occurring in a continuous medium, and, as a rule, containing time derivatives. Equations that can be interpreted as the writing of the differential law of development (evolution) in time of some process.

Elements of theory of numberings
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to form the ability to build different numbering for different families of sets and functions. mastering categorytheoretic approaches in numbering theory, learning how to work with subobjects and main subobjects in the category of numbered sets. Categorytheoretic concepts. The numbering of the set and its subsets. The category of numbered sets and its properties. Subobjects of a numbered set. Complete and fully numbered sets. Positively numbered sets. Numbered sets with approximation. Structural theorems on fully numbered sets. Creativity and muniversality for computable numberings.

Second Order Elliptic Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Secondorder elliptic equations are one of the most beautiful and soughtafter sections of mathematics. A classic example of such equations is the Laplace equation, which describes the stationary temperature distribution. The course is dedicated to the general elliptic equation. Will be presented: The classical principle of maximum; Estimates by S.N. Bernstein; Harnack's inequality; Liouville theorem; The space of Sobolev, Helder; Concepts of weak solution; Fredholm theorem; Schauder method. Thus, the goal of the course "Elliptic equations of the second order" is to form core competencies for students (general scientific, instrumental, general professional, specialized) on the basis of indepth study of methods for studying boundary value problems for elliptic equations. The purpose of the course: In the course of studying the course to form students' abilities:  Explain the basic concepts of the formulation of boundary value problems for an elliptic equation;  Calculate problems (Findings of the fundamental solution. Maximum principle. Method of potentials) using modern methods for solving the theory of partial differential equations (integral equations, embedding theorems and Schauder Method);  To prove the solvability of applied problems using the theory of partial differential equations;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the existence and uniqueness of a boundary value problem for an elliptictype equation by methods of the theory of generalized functions, the theory of functional spaces, integral equations, embedding theorems and the theory of partial differential equations;  Design the process of studying an applied problem using the methods of the theory of partial differential equations;  Work in a team, reasonably defend the correct choice of the problem. As a result of training, undergraduates should know the theory of generalized functions and its application in the problems of the theory of the NavierStokes equation. Namely, General information about linear functionals and linear bounded operators in Hilbert spaces. Compact sets. Completely continuous operators. Linear equations in Hilbert space. Selfadjoint completely continuous operators. About unlimited operators. Generalized derivatives and averaging. Definition of Sobolev spaces and their basic properties. Embedding theorems for Sobolev spaces. Equations of elliptic type. Setting boundary value problems. Generalized solutions from . The first (energy) inequality. Investigation of the solvability of the Dirichlet problem in the space (three Fredholm theorems). Expansion theorems in eigenfunctions of symmetric operators. The second and third boundary problems. The second main inequality for elliptic operators. Solvability of the Dirichlet problem in space . Approximate methods for solving boundary value problems.

Inverse Problems in Hydrodynamics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The objectives of mastering the discipline "Inverse problems in hydrodynamics" are  to acquaint undergraduates with the theory and new results on methods for solving inverse problems hydrodynamics; study and skill of the basic settings of inverse problems; study of the features of their solution, some algorithms for their solution. In the course of studying the course to form students' abilities:  Know and explain the basic concepts and methods for solving the inverse problem of hydrodynamics;  Estimate and research the current state and achievement of the area of the inverse problem of hydrodynamics; Apply theoretical knowledge to solve applied problems of natural science. To be able to independently set the formulation of tasks and select effective methods for solving them; To analyze the results and compare with other modern results of the researcher Basic concepts of the theory of inverse problems. Formulations of direct and inverse problems of hydrodynamics. Classification of inverse problems. Known results on the direct problem of hydrodynamics. The main methods for solving inverse problems. Inverse problems for Stokes equation. Inverse problems for linearized and nonlinear NavierStokes equations. Inverse problems of heat convection, magnetic hydrodynamics. Inverse problems for nonNewtonian fluids.

Dissertation Writing
 Type of control: Защита НИР
 Description:

Scientific Internship
 Type of control: Защита НИР
 Description: The main purpose of "Scientific Internship": is the formation in the students of the ability to independently conduct research and development in the professional sphere using modern research methods and information and communication technologies on the basis of a foreign university. During the study of course, student should be competent in:  to substantiate the fundamentals of the methodology for performing scientific research, planning and organizing a scientific experiment, processing scientific data;  to argue methods of solving research and practical problems, including in interdisciplinary areas;  can analyze alternative solutions to research and practical problems and assess the potential benefits of implementing these options;  apply theoretical knowledge on methods of collecting, storing, processing and transmitting information using modern computer technologies;  choose the methods of presentation and methods of information transfer for different contingents of listeners.

Research Seminar III
 Type of control: Защита НИР
 Description:

Research practice
 Type of control: Защита НИР
 Description: Рurpose of practice: to develop the ability of theoretical generalization and practical research of data for creation of an original research portfolio of scientific work based on methods of planning, organization of research activities, application of scientific methods of studying objects and analytical system of their forecasting. During the study of practice, doctoral students should be competent in:  critically evaluating the scientific literature on the subject of scientific research in the field of fundamental and applied marketing;  formulating actual scientific problems, directions, hypotheses of research, including interdisciplinary ones, within the framework of own scientificexperimental and managerial activities;  plan and organize research activities: collection, processing and analysis of the main technical and economic, financial, marketing, statistical and empirical data, using a modern analytical system to develop strategic and forecast solutions for the development of the research object;  evaluate the validity of methods used in scientific research on marketing issues;  carry out scientific experiments to study the market potential, competitive advantages, consumer behavior, innovative development and correctly analyze their results;  creating research products both independently and as a part of research teams on the basis of observance of scientific ethics, correlating own scientific interests with public needs;  broadcasting their own results of research in the field of marketing to a wide range of specialists in the relevant field, also in an audience that does not have the appropriate training.

Teaching Internship
 Type of control: Защита практики
 Description: Aim оf discipline: formation of the ability to carry out educational activities in universities, to design the educational process and conduct certain types of training sessions using innovative educational technologies. During the study of course, masters should be competent in: 1. develop the taught discipline in an amount sufficient for the analytical assessment, selection and implementation of the module of the academic discipline, taking into account the level of preparedness of students, their needs, as well as the requirements of the State Educational Establishment of the Republic of Kazakhstan; 2. to develop the specifics of the organization and conduct of various types of classes in higher educational institutions (lecture, seminar, laboratory and practical); 3. apply the basic means of assessing students' academic achievements; 4. analyze the educational and methodological literature and use it to build your own presentation of program material; 5. prepare plans for seminars, practical classes, laboratory work in accordance with established methodological and methodological approaches. During the study of the discipline masters will learn following aspects: Acquaintance with the goals, objectives and content of teaching practice; drawing up of the schedule of consultations, kinds of the reporting and terms of their granting. Harmonization of the individual plan of the scientific and pedagogical practice of the undergraduate. Implementation of educationalmethodical tasks coordinated with the head of practice. Visiting and analyzing the training sessions conducted by the teachers of the department. Acquaintance with the organization of scientific, methodical and educational work (plans, normative documents regulating the pedagogical process) at the faculty / at the university and the department. Development (at least 10 lessons) and conducting classes with students. Carrying out activities on the designated activities (scientific and methodological seminars, conferences, scientific circles, educational events). Preparation of an article of a scientific and methodical nature. Drawing up a report on scientific and pedagogical practice.

Publication in the Proceedings of International Conferences
 Type of control: Защита НИР
 Description: The main purpose of "Publication in the Proceedings of International Conferences": is the formation of master candidates in the possibility of presenting the results of research work to the scientific community, receiving feedback, and exchanging experience in the field of professional activity. During the study of course, master student's should be competent in: 1. demonstrate current trends in scientific research; 2. to argue the annotated results of research in scientific journals, materials of international conferences and symposia; 3. they can apply new, scientifically grounded, theoretical or experimental results that allow solving a theoretical and applied problem; 4. analyze scientific results, the data of their colleagues and opponents in the sphere of the chosen professional activity; 5. generate ideas for the use of proposed developments in scientific research of the professional field of activity.

Research Seminar I
 Type of control: Защита НИР
 Description: Upon successful completion of this module, undergraduates should be able to: 1.To carry out an analytical review of the scientific literature on the subject of undergraduate scientific research; 2. To substantiate and formulate the relevance, theoretical and practical significance of the chosen topic of scientific research; 3. To apply modern methods of data analysis and processing, information technology and software products related to the research topic; 4. To acquire the skills of independent research work, as well as work as part of research teams; 5. To critically evaluate the results of their own scientific research, identify scientific priorities and formulate relevant scientific problems and problems on the topic of their research; 6. To present the results of the research in various forms of oral and written activities (presentation, abstract, analytical review, critical review, report, report, presentation, scientific article of a review, research and analytical nature, etc.); 7. To realize the provision of a direct connection of the research work with the dissertation; 8. To publish articles on the results of the research in domestic and foreign scientific journals.

Research Seminar II
 Type of control: Защита НИР
 Description: Upon successful completion of this module, undergraduates should be able to: 1.To carry out an analytical review of the scientific literature on the subject of undergraduate scientific research; 2. To substantiate and formulate the relevance, theoretical and practical significance of the chosen topic of scientific research; 3. To apply modern methods of data analysis and processing, information technology and software products related to the research topic; 4. To acquire the skills of independent research work, as well as work as part of research teams; 5. To critically evaluate the results of their own scientific research, identify scientific priorities and formulate relevant scientific problems and problems on the topic of their research; 6. To present the results of the research in various forms of oral and written activities (presentation, abstract, analytical review, critical review, report, report, presentation, scientific article of a review, research and analytical nature, etc.); 7. To realize the provision of a direct connection of the research work with the dissertation; 8. To publish articles on the results of the research in domestic and foreign scientific journals.