Master degree program
Mathematics

Mathematics

QUALIFICATION

  • Scientific and pedagogical direction - Master of Natural Sciences

MODEL OF GRADUATING STUDENT

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 non-linear 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.

Program passport

Speciality Name
Mathematics
Speciality Code
7M05402
Faculty
Mechanics and Mathematics

disciplines

Foreign Language (professional)
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of the discipline is to form practical skills in various types of speech activity in a foreign language. The training course builds the ability to perceive, understand and translate information in the modern global space, participate in scientific events to test their own research. The discipline is aimed at improving competencies in accordance with international standards of foreign language education.

General Algebra
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The main goal of this discipline is the formation of skills and abilities to solve applied problems of an algebraic structure, to use the basic laws of algebraic construction, which allow this structure to create a new object of the same type, to apply the methods of algebraic structures in the field of Mathematics.

History and Philosophy of Science
  • Number of credits - 3
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of the discipline is to form a holistic systemic understanding of philosophy as a special form of cognition of the world, its main sections, problems and methods of their study in the context of future professional activity. The training course forms the theoretical and methodological basis of research work.

Mathematical analysis on metric spaces
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To familiarize undergraduates with the basic concepts of the theory of metric spaces; describe methods of working in metric spaces; apply the above methods to replenish spaces; analyze the class of compact sets; identify properties of continuous functions on compact sets; develop the concept of functional spaces with metrics; present arguments for introducing generic functions.

Methods of Teaching Higher Education Mathematics
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Contents of the discipline: Subject, content, goals, objectives of the methodology of teaching mathematics; content of mathematics teaching methods: state and prospects, development trends in methodological training of a future mathematics teacher in higher education; purpose of methodological science; connection of methodological science with other sciences; a system of methodological training (concept, structure, content) aimed at preparing a future mathematics teacher in higher education.

Organization and Planning of Scientific Research (in English)
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of the discipline to form the ability to apply practical skills in the organization and planning of scientific research. The discipline studies: forms and methods of planning, organization and design of scientific articles and dissertations; forms of summarizing the results of scientific research in presentations, speeches, projects, articles.

Pedagogy of Higher Education
  • Number of credits - 5
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of the discipline is to form the ability to teach in universities and colleges based on knowledge of higher school didactics, theory of education and management of education, analysis and self-assessment of teaching activities. The training course is aimed at studying the trends in the development of education and the Bologna process. The course will help you master teaching and curatorial skills, various strategies and methods of teaching and education.

Psychology of management
  • Number of credits - 3
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of the discipline is to provide scientific training of highly qualified specialists based on the study of fundamental concepts of management psychology, creating prerequisites for a theoretical understanding and practical application of the most important aspects of the field of management in the process of professional formation. The course is aimed at studying the patterns of development and functioning of mental processes, the basics of effective interaction and conflict resolution, self-development and self-presentation.

Theoretical and computational problems of mathematical physics
  • Number of credits - 5
  • Type of control - [RK1+MT+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. 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.

Theory of stability of dynamic systems
  • Number of credits - 5
  • Type of control - [RK1+MT+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.

Data for 2021-2024 years

disciplines

Additional Chapters of Differential Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying methods for solving boundary value problems for equations of mathematical physics using functional analysis, with approaches to solving some boundary value problems for equations of mathematical physics in function spaces.

AppIied Statistic
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the application of modern applied statistics. Contents of the discipline: Variation series of samples; Ordinal statistics; Selected characteristics; Point estimation of unknown distribution parameters; Methods for finding estimates; Interval estimation; Statistical hypotheses. Statistical criterion.

Application of Approximate Calculations To the Problems About Eigenvalues
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Introduce the concepts of approximation theory; describe methods for approximate calculations of eigenvalues; apply the above methods to calculate the eigenvalues ​​of matrices; analyze the possibility of approximate calculation of eigenvalues ​​of boundary value problems for differential equations; identify the spectral properties of individual classes of operators.

Approximations for Functions of Several Variables
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to construct simpler and more optimal formulas for approximating complex objects, such as a function of many variables, integral operators or series. The content of the discipline is aimed at studying the main basic concepts, theorems and problems of approximation theories for functions of many variables.

Boundary value problems for differential equations in partial derivatives
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the following topics: Boundary value problems for equations of parabolic and elliptic types in Hölder and Sobolev spaces. The first and second boundary value problems for parabolic equations in Hölder space. Existence, uniqueness, estimates of the solution. A method for constructing a regularizer to prove the existence of a solution, the Schauder method for deriving estimates of the solution.

Boundary Value Problems for Ordinary Differential Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study boundary value problems for ordinary differential equations of arbitrary order with a small parameter at the highest derivative. Will be studied: - Estimation of the difference between solutions of singularly perturbed and unperturbed problems. -asymptotic expansions of solutions with any degree of accuracy in a small parameter; - influence of a small parameter on the asymptotic behavior of solutions; - the order of growth of solutions at the point of the initial jump.

Computability in the Hierarchies
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to calculate the complexity of various sets relative to arithmetic, hyperarithmetic, analytical and Ershov hierarchies. The course content is aimed at studying various properties of the above-mentioned hierarchies and methods for determining the complexity of sets and their closedness with respect to computable reducibility. Much attention will be paid to the arithmetic hierarchy and the Ershov hierarchy.

Computable Functions
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to determine the computability of various functions. The content of the discipline is aimed at studying the computability of functions, primitive and partially recursive functions, computability on a Turing machine, computability with respect to oracles, numbering of computable functions, as well as stopping problems, recursion theorems and Rice's theorem.

Constructive theory of boundary value problems for ordinary differential equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Contents of the discipline: Statement of the problem of boundary value problems for linear and nonlinear ODEs. Integral equations. Two-point boundary value problem. Boundary value problems with phase constraints. Boundary value problems with phase and integral constraints. Boundary value problem with a parameter for the OAU. Sturm-Liouville problems. Boundary value problems with a parameter in the presence of phase constraints. Periodic solutions of linear and nonlinear autonomous dynamic systems

Direct and Inverse Problems for Nonclassical Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - 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 are modeled with non-classical equations of mathematical physics. The content of the discipline is aimed at studying direct and inverse problems for non-classical equations of mathematical physics.

Effective computability
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - TThe goal of the discipline is to form motivation to gain knowledge of the theory of effective computability, to develop the necessary practical skills for modern research in mathematical logic by formulating and discussing open topical problems.

Elements of theory of numberings
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to construct different numberings for different families of sets and functions. The content of the discipline is aimed at studying the basic concepts of numbering theory, in particular such concepts as complete, precomplete, minimal, main numbering and various subobjects such as n-subobject wn-subobject and so on.

Evolution Equations of the Second Order
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study methods for solving boundary value problems for evolutionary equations using functional analysis. The theory of partial differential equations is not part of functional analysis. Despite the fact that some classes of equations can be interpreted in terms of abstract operators acting in Banach spaces, insistence on adopting a superficially abstract point of view and the consequent ignorance of subtle theorems, calculations and derivation of a priori estimates is ultimately a great loss in the study of the desired problems.

General algebra
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Provide students in all areas of mathematics with basic algebraic structures: groups, rings, modules. Introduce the general concepts of algebraic structure, substructure, homomorphism and isomorphism. Give typical examples of this type of structure and study the algebraic constructions that allow a given structure to create a new object of the same type. Basic algebraic structures and algebraic constructions in any area of ​​Mathematics and apply knowledge of them in specific areas.

Ideals and Diversity
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to transform the basic concepts of commutative algebra and geometry from abstract theoretical ones into concretely computable ones. Course content: Polynomials and affine space; monomial ideals and Dixon's lemma; Hilbert's basis theorem and Gröbner bases; Buchberger algorithm; improvements to the Buchberger algorithm; exclusion geometry; implicit representation.

Inverse Problems in Hydrodynamics
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Contents of the discipline: Statements of direct and inverse problems of hydrodynamics. Classification of inverse problems. Well-known results on the direct problem of hydrodynamics. Basic methods for solving inverse problems. Inverse problems for the Stokes equation. Inverse problems for linearized and nonlinear Navier-Stokes equations. Inverse problems of thermal convection, magnetohydrodynamics. Inverse problems for non-Newtonian fluids.

Iterative methods for solving nonlinear equations and their applications
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study iterative methods for solving nonlinear equations and systems of nonlinear equations, numerical solutions of boundary value problems for differential equations. The course is aimed at studying both classical Newton methods and secant methods, as well as generalized linear methods, in particular methods of sequential upper relaxation.

Mathematical foundations of optimal control
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to use fundamental knowledge of differential calculus in Banach space. Contents of the discipline: General formulation of the optimal control problem with constraints. Differentiation of nonlinear operators and nonlinear functionals. Existence and uniqueness of solutions to differential equations in a Banach space. Global minimum theorem. Optimality conditions. Weierstrass's theorem in a Banach space. Methods for minimizing functionals in Banach space.

Methods for solving extremal problems
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the following topics: Methods for searching for minima of a function on given sets for problems of linear programming, nonlinear programming and convex programming, minimum (maximum) problems arise in various areas of human activity, where the choice of the best possible action is necessary.

Multidimensional Complex Analysis
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Introduce the concepts of multidimensional complex analysis; describe methods of multivariate complex analysis; apply complex analysis methods to solve partial differential equations; analyze the possibilities of complex analysis for studying individual function classes; identify the properties of analytical functions of many complex variables.

Nikolsky-Besov Spaces and Their Applications To Boundary Value Problems for Generalized Analytic Functions
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to use the theory of generalized derivatives to define metrics by function spaces. Contents of the discipline: Measurements of Lebesgue sets, features of cumulative functions; properties of Lp spaces, completeness; averaging operation, kernels, properties; generalized derivative, Nikolsky-Besov scale, features of the norm, embedding theorems for various metrics; bounded embedding theorems in isotropic B-spaces for functions defined in the complex plane.

Number-theoretic Methods in Approximate Analysis and Their Applications
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to find the best approximations of complex objects using different polynomials. Contents of the discipline: Statement of the approximation problem. Best approximation. Weierstrass's theorem. Basic concepts related to approximation of the best approximation by an algebraic polynomial. Space Hn. Borel's theorem on the existence of the best polynomial. Chebyshev's theorem on the existence of a unique polynomial that best describes a given function in the space Hn.

Optimal control of systems with partial derivatives
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study the theory of optimal control of systems described by partial differential equations. Contents of the discipline: Gradients of functionals on the set of solutions of a parabolic equation, hyperbolic equation. Lipschitz conditions for gradients. Optimality conditions. Basic minimization methods. Algorithms for constructing minimizing sequences. Estimating the convergence of minimizing sequences.

Reducibility and completeness
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to construct various reducibilities such as m-reducibility, table reducibility and Tureng reducibility. The content of the discipline is aimed at studying the above mentioned types of reducibility and complete sets for these convergences. A complete set in the class of a certain family of sets that can be reduced to any other

Second Order Elliptic Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Second-order elliptic equations are one of the most beautiful and popular branches of mathematics. A classic example of such equations is the Laplace equation, which describes a stationary temperature distribution. The course is devoted to the general elliptic equation. Will be studied: The classical maximum principle; Estimates by S.N. Bernstein; Harnack's inequality; Liouville's theorem; Sobolev, Hölder space; Concepts of a weak solution; Fredholm's theorem; Schauder method.

Singularly perturbed differential equation with piecewise constant argument
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to use the theory of singular differential equations with a piecewise constant argument to study objects of natural science. The content of the discipline is aimed at studying the analytical formula for the solution, the unperturbed problem, the theorem on passage to the limit, the initial jump of the solution, the uniform asymptotic expansion of the solution

Singularly Perturbed Integro-differential Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the following topics: Cauchy problems of initial jumps and initial jumps, local and nonlocal boundary value problems for singularly perturbed integro-differential equations. A constructive formula and estimates for the solutions are given, as well as the difference between the solutions of singularly perturbed and unperturbed equations.

Statistics of Random Processes
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to use the theory of optimal nonlinear filtering for statistics of random processes. The content of the discipline is aimed at studying the theory of optimal nonlinear filtering for both the case of discrete and continuous time; familiarization with sequential assessment tasks

Stochastic analysis and equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The purpose of mastering the discipline is the formation of the following competencies: the ability to apply the methods of the theory of stochastic analysis and stochastic calculus to solve typical standard problems; the ability to analyze and freely navigate in the main directions of further development of the subjects of this discipline; the ability to investigate the relationship of diffusion processes with solutions of stochastic differential equations.

Stochastic Differential Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to use the modern theory of stochastic differential equations in the study of both theoretical and practical problems. Contents of the discipline: Stochastic integrals of non-random and random functions over a process with orthogonal increments; Ito integral; Stochastic differential; Formula Summary: one-dimensional and multidimensional cases.

Sums of Independent Random Variables
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Develop the ability to use the theory of sums of independent random variables in different senses to justify theorems and practical applications. Contents of the discipline: limit theorems of probability theory, conditions for their implementation, types of convergence of sequences and series of random variables, and connections between them; main modern directions in the development of the theory of summation of independent random variables.

The Method of Compactness and Monotony for Nonlinear Problems of Mathematical Physics
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying nonlinear problems of mathematical physics from the point of view of modern functional analysis. Therefore, the concept of nonlinearity dominates the entire course. Here we consider the following modern methods for studying initial boundary value problems for equations of mathematical physics: the method of a priori estimates, variational methods, monotonicity and compactness methods, Pokhozhaev's identity and the regularization method.

The Qualitative Theory of Differential Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the following topics: Autonomous systems of differential equations. Properties of solutions. Autonomous systems of differential equations on the plane. Linear autonomous systems of differential equations on the plane. Special points. Types of singular points and phase portraits. Non-degenerate singular points of a nonlinear system of differential equations and phase portraits. Follow function.

The Theory of Finite Fields
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to use elements of finite field theory in mathematics and technology. The content of the discipline is aimed at studying the theory of groups and fields, finite and infinite groups, centers and series of normal subgroups. Group action on sets, stopping problems and computable numberings with respect to reducibility

The Theory of Identification of the Boundary Conditions and Its Applications
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Introduce the concepts of the theory of inverse problems; describe methods for reconstructing an object using additional boundary conditions; analyze the problems of restoring the boundary conditions of the coefficients; identify the correctness properties of identification problems; develop the concept of conditional stability of inverse problems; present arguments for solving technical diagnostic problems.

The theory of stability regulated systems
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - Contents of the discipline: General statement of the problem. Balance position. Not the only solution. Study of the absolute stability of controlled systems in the main case. Not special transformations. Properties of solutions. Absolute stability. Study of the absolute stability of controlled systems in a simple critical case. Study of the absolute stability of controlled systems in a critical case.

The Theory of the Navier-Stokes Equations
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the solvability and stability of solutions and boundary value problems for the Navier-Stokes equation. The course is designed to study generalized solutions of boundary value and initial-boundary value problems for the Navier-Stokes equations. The course also covers concepts from functional analysis and control theory.

Theoretical and computational problems of mathematical physics
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study methods for solving boundary value problems of mathematical physics and to develop effective computational algorithms for numerical solution. The content of the course is aimed at applying modern analytical and computational methods to solving problems of mathematical physics and partial differential equations. The main problems of mathematical physics and methods for solving them. Modern computational methods and their applications.

Theory of boundary value problems of optimal control
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to use theoretical training and apply it to solve applied problems on computers. The content of the discipline is aimed at studying the following topics: methods for solving boundary value problems of optimal control for processes described by ordinary differential equations, which differ from well-known methods based on the Lagrange principle.

Theory of martingales
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying the theory of martingales. Contents of the discipline: Conditional mathematical expectations regarding partitions and sigma algebras; one random variable relative to another random variable; Definition of martingale; The moment of stopping; Application of martingales to random walks; Wald identity; Basic inequalities; semimartingales (discrete and continuous time); Convergence theorems. Wiener process as a quadratically integrable martingale.

Theory of stability of dynamic systems
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - To develop the ability to study new methods for studying the theory of stability of solutions of dynamic systems. The content of the course is aimed at studying new methods for studying the absolute stability of the equilibrium position of nonlinear controlled systems. Methods for studying the global asymptotic stability of phase systems with countable equilibrium positions are considered.

Theory of Statistical Estimation
  • Type of control - [RK1+MT+RK2+Exam] (100)
  • Description - The content of the discipline is aimed at studying modern assessment theory. Contents of the discipline: Sufficient statistics. Unbiased estimation (parametric and nonparametric cases). Efficiency of estimates with a quadratic loss function. Maximum likelihood estimation. Asymptotic normality of the estimate. Confidential assessment. Tolerant assessment.

Data for 2021-2024 years

INTERNSHIPS

Pedagogical
  • Type of control - Защита практики
  • Description - Formation of practical, educational-methodical skills of conducting lectures, seminars, creatively apply scientific, theoretical knowledge, practical skills in teaching activities, conduct training sessions in the disciplines of the specialty; own modern professional techniques, methods of training, use in practice the latest theoretical, methodological advances, make educational, methodological documentation.

Research
  • Type of control - Защита практики
  • Description - The purpose of the practice: gaining experience in the study of an actual scientific problem, expand the professional knowledge gained in the learning process, and developing practical skills for conducting independent scientific work. The practice is aimed at developing the skills of research, analysis and application of economic knowledge.

Data for 2021-2024 years