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 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.
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.
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Psychology of management
- Number of credits: 3
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Formation of knowledge about the fundamental concepts of management psychology for the practical application of the most critical aspects of management in professional interaction. The following will be studied: basic principles of management psychology, personality in management interactions, management of personality behavior, modern ideas, psychology of managing group phenomena, motivation, and practical reflection.
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Foreign Language (professional)
- Number of credits: 5
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose is to acquire and improve competencies by international standards of foreign language education and to communicate in an intercultural, professional, and scientific environment. A master's student must integrate new information, understand the organization of languages, interact in society, and defend his point of view.
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History and Philosophy of Science
- Number of credits: 3
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose: Understanding of modern philosophy as a system of scientific knowledge, including worldview in rational-theoretical comprehension. The discipline includes aspects of the evolution and development of scientific thinking, historical moments, the contribution of scientists and scientific schools to the formation of science, and ethical and social aspects of scientific activity.
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Mathematical analysis on metric spaces
- Number of credits: 5
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose the discipline is to develop skills and abilities for solving non-standard, atypical applied problems of modern mathematical analysis on metric spaces and stochastic analysis, as well as the formation of readiness for independent professional activity by some of their applications.
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Methods of Teaching Higher Education Mathematics
- Number of credits: 5
- Type of control: [RK1+MT+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.
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Pedagogy of Higher Education
- Number of credits: 5
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose: To provide pedagogical theories and practical strategies for effective teaching in higher education, fostering critical thinking, and academic success. The course explores instructional methods, curriculum design, assessment techniques, and classroom management strategies preparing educators to create inclusive and stimulating learning environments.
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Constructive Theory of Problems of Ordinary Differential Equations
- Type of control: [RK1+MT+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.
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Additional Сhapters of Differential Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Aim: in-depth 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 self-education and self-development. 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.
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Computability in the Hierarchies
- Type of control: [RK1+MT+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. Goncharov-Sorbi approach. Syntactic and algorithmic criteria for the computability of numberings in the classical case of families of computably enumerable sets. The general approach of Goncharov-Sorbi 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. Goncharov-Sorbi theorem on minimal pairs of arithmetic numberings. The concept of the Badayev-Lemp 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 non-discrete families with the trivial Rogers semilattice. Families without minimal computable numberings.
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Boundary value problems for differential equations in partial derivatives
- Type of control: [RK1+MT+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).
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Methods for solving extremal problems
- Type of control: [RK1+MT+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, natural-science and general scientific information contextually, leading it to the problem-problem 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, Newton-Kantorovich method, integral functionals method.
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Mathematical foundations of optimal control
- Type of control: [RK1+MT+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.
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The Qualitative Theory of Differential Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the discipline is to study the basic concepts of qualitative theories of differential equations; explain the properties of solutions to a system of differential equations (singular points, classification of integral curves and trajectories, classification of singular points, etc.); – 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
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The Method of Compactness and Monotony for Nonlinear Problems of Mathematical Physics
- Type of control: [RK1+MT+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 non-linearity prevails throughout the course. Here we consider the following modern methods for studying initial-boundary 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 initial-boundary value problems; - Identify weak and strong generalized solutions for non-linear 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.
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Iterative methods for solving nonlinear equations and their applications
- Type of control: [RK1+MT+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 functional-theoretical 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 two-point boundary value problem for ordinary differential equations.
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Computable Functions
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The goal of the discipline: to form the ability to determine the computability of various functions. The content of the discipline is aimed at studying the computability of a function, 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.
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Ideals and Diversity
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The goal of the discipline: to form the ability to transform the basic concepts of commutative algebra and geometry from abstract theoretical to concretely computable. to form reproductive-activity components for working with polynomials and affine space; monomial ideal; Buchberger's algorithm. to acquaint with the basic algebraic structures such as a group, a ring, a field, which have applications in various branches of modern science and technology.
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Boundary Value Problems for Ordinary Differential Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: We study boundary value problems for ordinary differential equations of arbitrary order with a small parameter at the highest derivative. Asymptotic expansions of solutions with a work degree of accuracy in a small parameter will be obtained.
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Multidimensional Complex Analysis
- Type of control: [RK1+MT+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.
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Nikolsky-Besov Spaces and Their Applications To Boundary Value Problems for Generalized Analytic Functions
- Type of control: [RK1+MT+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.
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Direct and Inverse Problems for Nonclassical Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The goal of the discipline is to develop the ability to: –To master the methods of simplifying the formulation of the studied inverse problems, technologies and tools used to solve inverse problems; -classify typical methods in the subject area using physical principles; - Analyze and formulate typical direct and inverse problems of natural science.
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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.
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The Theory of Finite Fields
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the discipline: to form the ability to use the elements of the theory of finite fields in mathematics and technology. The content of the discipline is aimed at studying the theory of groups and fields, finite and infinite groups
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Elements of theory of numberings
- Type of control: [RK1+MT+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 category-theoretic approaches in numbering theory, learning how to work with sub-objects and main sub-objects in the category of numbered sets.
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Inverse Problems in Hydrodynamics
- Type of control: [RK1+MT+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 non-linear Navier-Stokes equations. Inverse problems of heat convection, magnetic hydrodynamics. Inverse problems for non-Newtonian fluids.
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Second Order Elliptic Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Second-order elliptic equations are one of the most beautiful and sought-after 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 in-depth 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 elliptic-type 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 Navier-Stokes equation. Namely, General information about linear functionals and linear bounded operators in Hilbert spaces. Compact sets. Completely continuous operators. Linear equations in Hilbert space. Self-adjoint 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.
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Optimal control of systems with partial derivatives
- Type of control: [RK1+MT+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, natural-science and general scientific information contextually, leading it to the problem-problem 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 first-order optimality conditions.
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Number-theoretic Methods in Approximate Analysis and Their Applications
- Type of control: [RK1+MT+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 computer-implemented 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 number-theoretic methods in the approximate analysis of NM. Korobov, as well as listeners are offered new unsolved problems.
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General algebra
- 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.
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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.
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Approximations for Functions of Several Variables
- Type of control: [RK1+MT+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 1-3 formulas. In the course, an estimate is given of the best best approximation by particular best approximations for a function of several variables.
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AppIied Statistic
- Type of control: [RK1+MT+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; Blake-Scholes 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; Blake-Scholes model; Markowitz theory; types and properties of options; stochastic models of price dynamics.
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Sums of Independent Random Variables
- Type of control: [RK1+MT+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.
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Statistics of Random Processes
- Type of control: [RK1+MT+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.
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Application of Approximate Calculations To the Problems About Eigenvalues
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: A survey of papers devoted to approximate calculation of eigenvalues and eigenfunctions of differential operators of Sturm-Liouville 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.
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Reducibility and completeness
- Type of control: [RK1+MT+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 category-theoretic approaches in numbering theory, learning how to work with sub-objects and main sub-objects in the category of numbered sets. Category-theoretic 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 m-universality for computable numberings.
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Stochastic Differential Equations
- Type of control: [RK1+MT+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: one-dimensional 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.
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Singularly Perturbed Integro-differential Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Aim: to study the basic problems of the theory of initial and boundary problems for singularly perturbed integro-differential 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 integro-differential equations and clarify the scope of these methods; - the skills of mathematical modeling of applied problems described by singularly perturbed integro-differential 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 self-education and self-development. Initial and boundary value problems with an initial jump for singularly perturbed integro-differential 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 integro-differential equations. Estimation of the difference between solutions of singularly perturbed and unperturbed problems. Uniform asymptotic expansion of solutions of initial and boundary value problems.
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Singularly perturbed differential equation with piecewise constant argument
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Aim: to study the basic problems of the theory of initial and boundary problems for singularly perturbed differential equations with piecewise-constant 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 piecewise-constant argument and clarify the scope of these methods; - the skills of mathematical modeling of applied problems described by singularly perturbed differential equations with piecewise-constant 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 self-education and self-development. Initial and boundary value problems for singularly perturbed integro-differential equations with piecewise-constant argument. Initial and boundary functions of singularly perturbed homogeneous differential equations with piecewise-constant argument. Analytical formula and asymptotic estimate of the solution of initial and boundary value problems for singularly perturbed linear integro-differential equations with piecewise-constant argument. Estimation of the difference between solutions of singularly perturbed and unperturbed problems. Uniform asymptotic expansion of solutions of initial and boundary value problems.
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Theoretical and computational problems of mathematical physics
- 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. 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.
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Theory of boundary value problems of optimal control
- Type of control: [RK1+MT+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.
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Theory of martingales
- Type of control: [RK1+MT+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 sigma-algebra; 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.
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The Theory of Identification of the Boundary Conditions and Its Applications
- Type of control: [RK1+MT+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.
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Evolution Equations of the Second Order
- Type of control: [RK1+MT+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.
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Theory of Statistical Estimation
- Type of control: [RK1+MT+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.
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Theory of stability of dynamic systems
- 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. 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.
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The theory of stability regulated systems
- Type of control: [RK1+MT+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 right-hand side contains non-linear 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, non-uniqueness 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 one-dimensional 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.
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The Theory of the Navier-Stokes Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the course: The purpose of the discipline "Theory of Navier-Stokes Equations" is the study by undergraduates of generalized solutions of boundary and initial boundary-value problems for the Navier-Stokes equations. The problems of solvability and stability of solutions, boundary value problems for the Navier-Stokes equation are investigated. The method of a priori estimates, the Leray-Schauder 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 Navier-Stokes equation; - Calculate tasks (Obtaining a priori estimates in Hölder and Sobolev spaces. Solvability of initial-boundary problems) using modern methods of functional analysis; - To prove solvability of boundary value problems of the Stokes equation, initial boundary value problems of the Navier-Stokes equation and various applied problems using the method of a priori estimates, the Leray-Schauder method, the Faedo-Galerkin method .; - Solve theoretical and applied problems of physics, mechanics, etc .; - Describe the solution of the linear and non-linear problem of the Navier-Stokes 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 Navier-Stokes 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.
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Pedagogical
- 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.
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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.