PhD courses
Speciality Code:
8D05401
Speciality Name:
Mathematics
Faculty:
Mechanics and Mathematics
Qualification:
- Scientific and pedagogical direction - Doctor of Philosophy (PhD)
- Model of graduating student
- Mandatory disciplines
- Elective disciplines
- Professional
ON 1.To apply innovative educational technologies and methods for teaching special subjects in universities; to develop assessment tools, methodological instructions;
ON 2. To obtain new results based on the conducted research and analytical work in relevant fields of science and apply these results to solve practical problems in the form of participation in research projects and tenders, conference presentations, publishing articles in journals with nonzero impact factor;
ON 3. To develop new mathematical methods for solving extremal problems and boundary value problems for nonlinear differential equations and mathematical physics;
ON 4. Competently use linguistic and cultural linguistic knowledge for communication in a multilingual and multicultural Kazakh society and in the international arena;
ON 5. Apply mathematical and computer modeling knowledge and skills, use modern programming, as well as modern software packages for solving problems in the field of insurance and financial risks;
ON 6. To construct different mathematical and economic models of the object under study based on the principles and tools of mathematical methods for making management decisions in the field of forecasting in the financial and insurance sectors;
ON 7. To conduct research on the numbering of specific classical objects, as well as finding analogs of the numbering theory problems;
ON 8. Planning stages an application research process using mathematical and statistical methods;
ON 9. To expand a theory of insurance risk assessment and improve the tools of this theory, which is a system of mathematical models and statistical methods;
ON 10. Synthesize new and complex ideas, hypotheses, techniques based on the results of research work;
ON 11. To create mathematical and sequential models of mathematical physics and filtration processes;
ON 12. To conduct a dialogue on topics in the competent field with equal status, as well as with the broad scientific community and society.
ON 2. To obtain new results based on the conducted research and analytical work in relevant fields of science and apply these results to solve practical problems in the form of participation in research projects and tenders, conference presentations, publishing articles in journals with nonzero impact factor;
ON 3. To develop new mathematical methods for solving extremal problems and boundary value problems for nonlinear differential equations and mathematical physics;
ON 4. Competently use linguistic and cultural linguistic knowledge for communication in a multilingual and multicultural Kazakh society and in the international arena;
ON 5. Apply mathematical and computer modeling knowledge and skills, use modern programming, as well as modern software packages for solving problems in the field of insurance and financial risks;
ON 6. To construct different mathematical and economic models of the object under study based on the principles and tools of mathematical methods for making management decisions in the field of forecasting in the financial and insurance sectors;
ON 7. To conduct research on the numbering of specific classical objects, as well as finding analogs of the numbering theory problems;
ON 8. Planning stages an application research process using mathematical and statistical methods;
ON 9. To expand a theory of insurance risk assessment and improve the tools of this theory, which is a system of mathematical models and statistical methods;
ON 10. Synthesize new and complex ideas, hypotheses, techniques based on the results of research work;
ON 11. To create mathematical and sequential models of mathematical physics and filtration processes;
ON 12. To conduct a dialogue on topics in the competent field with equal status, as well as with the broad scientific community and society.
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Scientific Research methods
- Number of credits: 3
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The method of scientific research as a way of knowing objective reality, which is a certain sequence of actions, techniques and operations to achieve the goals of their scientific research. The concept of a method, methodology and methodology of scientific research. Classification of research methods. General, general scientific and special research methods. Universal and private research methods. Empirical, empirically - theoretical, theoretical, quantitative, qualitative and other research methods. The approximate structure of the article in mathematics: Keywords; Introduction; Literature review; Research methods; Results and discussion; Conclusion Bibliography
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Academic writing
- Number of credits: 2
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: In the course, doctoral students will be acquainted with the following topics related to the search for information in scientific data bases, analysis and summarizing, working with genres academic writing: -Main genres of academic writing.. -Scientific databases. -The structure of the academic community. -Orientation in the academic space. -Reference in scientific and scientific-technical information environment. -Features of the analytical review. -Review and types of reviews. -Report of a scientific event. -Features of editing in scientific publication.
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Actual problems of mathematics
- Number of credits: 5
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the course is to familiarize doctoral students with modern problems and the achievements of mathematics. In the course of studying the course, form the abilities of doctoral students: -Contextualize the mathematical foundations of the theory of functional spaces. - Demonstrate knowledge in the field of Banach algebras, matrix algebras, quaternions, group rings, radicals, subalgebras, nilpotent algebras, ideals. - To identify the similarities and differences of the basic methods of researching the fundamental areas of mathematics. - Apply the methods of the theory of generalized functions to solve specific practical problems. - Know and understand the general theory of differentiation as a connection between the theory of equations and the theory of extremum; - Critically evaluate the current state of theories of differential equations and extremum;
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PhD thesis writing and defence
- Number of credits: 12
- Type of control: Докторская диссертация
- Description: The main purpose of "PhD thesis writing and defence": of a doctoral dissertation is the formation of the doctoral students' ability to disclose the content of research work for the defense of the thesis. During the study of course, doctoral student's should be competent in: 1. to substantiate the content of new scientifically grounded theoretical and experimental results that allow to solve a theoretical or applied problem or are a major achievement in the development of specific scientific directions; 2. explain the assessment of the completeness of the solutions to the tasks assigned, according to the specifics of the professional sphere of activity; 3. they can analyze alternative solutions for solving research and practical problems and assess the prospects for implementing these options; 4. apply the skills of writing scientific texts and presenting them in the form of scientific publications and presentations. 5. to plan and structure the scientific search, clearly highlight the research problem, develop a plan / program and methods for its study, formalize, in accordance with the requirements of the State Educational Establishment, the scientific and qualification work in the form of a thesis for a scientific degree Doctor of Doctor of Philosophy (PhD) on specialty «8D07502 – Standardization and certification (by industry)». During the study of the discipline doctoral student will learn following aspects: Registration of documents for presentation of the thesis for defense. Information card of the dissertation and registration-registration card (in the format Visio 2003). Extract from the minutes of the meeting of the institution, in which the preliminary defense of the thesis took place. Cover letter to the Higher Attestation Commission. Expert conclusion on the possibility of publishing the author's abstract. Expert opinion on the possibility of publishing a dissertation. Minutes of the meeting of the counting commission. Bulletin for voting. A shorthand record of the meeting of the dissertational council. List of scientific papers. Response of the official opponent. A review of the leading organization. The recall of the scientific adviser.
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Mathematical Analysis on Manifolds
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: To familiarize operators of exterior differential forms on smooth manifolds and integrate them. Tasks of the discipline is to study smooth oriented manifolds, to acquaint with the calculus of exterior differential forms, to teach the integration of exterior differential forms along smooth manifolds. In the course of studying the course to form the ability of doctoral students: - Describe the basic concepts and definitions of the theory of entire functions of many variables; - Apply operators of external differential forms on smooth manifolds; - Describe k-forms, replacing variables in k-forms; - Apply multiple integrals, integrals of the k-form along the manifold; - Describe multiple integrals, integrals of a k-form along a manifold; - Apply the principles of the general theorem of Stokes; - Build metric spaces and their completion; - Implement a variational method for solving integrals of a k-form along a manifold; Summary Map, atlas, smooth manifold, orientation of the manifold, tangent spaces; K-forms, operations on k-forms, replacement of variables in k-forms, external differential k-forms; Multiple integrals, integrals of the k-form along the manifold, and the general Stokes theorem.
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Mathematical Models of Nonequilibrium Filtration Processes
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: We consider a process non stationary filtrational flow of uniform droplet-compressible monophase fluid in isotropic weakly-deformable porous environment. There are a various models to describe this process. The most popular is a model of classical elastic regime. But this model describes only non-stationary “equilibrium” filtration. During the study of course, PhD students should be competent in: – The basic laws of the development of science and technology; - Know the basic concepts and methods of non-equilibrium filtration processes; - Know about algorithmic modeling techniques; - Build models for practical tasks; – Work in a team, be able to prove the correctness of your method of choosing a task; – Apply methods of modeling and analysis in solving engineering problems; - Solve the main tasks of nonequilibrium filtration processes;– To possess tools for solving mathematical problems in their subject area; – Possess skills in mathematical formalization of problem statements, skills in solving typical problems, skills in critical perception of information; The course content is aimed at studying the fundamentals of mathematical models of non-equilibrium filtration processes. Expand the knowledge of PhD doctoral students in the issue of filtration modeling of groundwater, oil, which are closely related to the tasks of water supply and sanitation, as well as with the works of waterworks. To form the ability to develop models in the theory of filtering for solving applied problems. As a result of studying the discipline of PhD, doctoral students will learn the basic tools and capabilities of filtering models, modeling technologies and methods for creating custom applications; will be able to program applied tasks, choose the optimal algorithm for the implementation of applied tasks; will have an idea about the methods of constructing the algorithm, about how to optimize the program. As a result of the study (passing, hearing) this course PhD doctoral student will know the physical and mathematical models of nonequilibrium relaxation filtering. And will know how to numerically solve the mathematical model Monte Carlo methods and implement them on a personal computer
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The Mathematical Theory of Viscous Incompressible Fluid
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The aim of the course "Mathematical problems of the dynamics of a viscous incompressible fluids" is the study of the basic theory and the main problems of a viscous incompressible fluids and methods for solving them. In the course of studying the course to form the ability of doctoral students: - Know and understand the basic theory of the dynamics of a viscous incompressible fluids for to independent research scientific works on the theory of Newtonian fluids; - To create and explore mathematical models of the dynamics of Newtonian and non-Newtonian fluids; - Apply the methods for solving problems of a viscous incompressible fluids to solve other applied problems of natural science; - Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To teach special courses on the theory of incompressible viscous liquids in universities; The course content is aimed to studying the basic theory of the dynamics of a viscous incompressible fluids, the models and the main problems of the dynamics of a viscous incompressible fluids, the basic equations of continuum mechanics; statements of boundary and initial boundary problems and methods for solving them.
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Computable Numberings in the Arithmetical Hierarchy
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The course aims to show that arithmetical and hyperarithmetical hierarchies provide a natural measure of the complexity of sets of natural numbers that occur both in mathematics itself and in its applications, measured by the complexity of their description in the first-order language. The course tasks to study the Tarski-Kuratovsky algorithm; to evaluate the algorithmic complexity of arithmetic sets; to evaluate the algorithmic complexity of sets relative to the hyperarithmetical hierarchy. In the course of the course, the students will be able to formulate doctoral dissertation: - Describe the basic criteria for complete and predictable numbering. - Implement the facts and problems of the numeracy theory. - Realization of the Tarko-Kurat algorithm. - Аlgorithmic algorithmic algebraic estimation. - To analyze the theory of numerical quantities and open problems in the field, theorem on decomposition. - Theorems on the numerical dimensional Rogers dimensions of the theorem. - Тo evaluate the arithmetic multiplicity of the relative hyperarithmic hierarchy. Summary. Syntactic, algorithmic and structural approaches to the definition of arithmetic hierarchy classes; Algorithmic approach to the definition of the hyperarithmetical hierarchy; The concept of a computable infinite formula and a syntactic approach to the definition of the hyperarithmetical hierarchy; Equivalences of different approaches to the definition of arithmetical and hyperarithmetical hierarchies; The Post theorem and the hierarchy theorem.
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The Category of Numbered Sets
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose of the discipline is to present a theory-categorical approach for the analysis of facts and results on the numbering of concrete classical objects, as well as finding analogues of problems of number theory in topology and other areas of mathematics. From the point of view of category theory, the main problems of number theory are considered. Tasks of the discipline are to study the problems of number theory in topology and other fields of mathematics; During of studying the course to form the ability of doctoral students: – Explain the key concepts of the theory of algorithms (such as computable functions, recursive and recursively enumerable sets, and computable reducibility of sets); – Use the knowledge gained to solve typical problems (such as building a Turing machine for computable functions, identifying the algorithmic complexity of certain sets, etc.); – Use modern methods of step-by-step designs for solving some typical tasks; – Use obtained knowledge to solve the problems of diploma or other scientific works; – Work in a team, reasonably defend the correct choice of the problem. Summary. Numbered sets and morphisms of numbered sets. Classes of liftings of numbered sets: principal, wn- and n-subobjects, retracts and e-subobjects and their characterizations. Definitions and criteria for complete and pre-complete numberings. Basic facts and problems of number theory.
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Statistical estimation methods and mathematical forecasting methods
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the discipline is to familiarize with mathematical methods of demographic forecasting and assessment of the risks of morbidity and mortality. The tasks of the discipline are to study the basic methods of demographic forecasting with a view to their further application in insurance practice. In the course of the course, the students will be able to formulate doctoral dissertation: - The purpose of the discipline is to teach students the principles of risk management in insurance, decision-making in risk situations for various types of insurance schemes, methods of formalization and solving problems of determining financial characteristics. - Quantitative characteristics of the financial stability of the insurer; - Basic types and properties of insurance contracts; - Formalize the setting of decision-making tasks under risk; - Analyze the sources of risk, use adequate risk measures for their description; Abstract: Theory of population. Constituents of population movements. Equation of demographic balance. Demographic indicators. Mathematical foundations of demography. Methods for measuring mortality and fertility. Definitions and methods of measuring migration. Demographic Methods for Determining the Population. Steady and stable population. Economic factors and population movement. Demographic projections.
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The methods of statistical evaluation of insurance premiums and reserves, taking into account the quality of data
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the discipline is to familiarize with the methods of statistical evaluation of insurance premiums and reserves taking into account the quality of the data. The task of the discipline is to study the basic methods of calculating premiums and reserves in insurance with a view to their further application in insurance practice. In the course of studying the course to form the ability of doctoral students: - Describe the mathematical methods of demographic prediction; - Apply the general theory of the pseudonym process to the value of risk of death and slaughter; - Describe mathematical models of demographic prediction; - Describe the theories of narrowing; - Select the driving motions of the cemetery; - Describe solubility in the demographic balance; - Formulation of the problem of the existence and independence of stochastic differential equations and systems equilibrium; - Methods of measurement of mortality and mortality; Abstract: The models of Bühlmann and Bühlmann - Straub. The Hahemeister model. Hierarchical estimation model with regard to data quality. Nonlinear models and alternative accuracy criteria. The relationship between estimates based on data quality and linear models. Prediction based on data quality and linear models. The problem of multicollinearity in the Hahemeister model. Practical aspects of the assessment, taking into account the quality of the data. Evaluations of James - Stein. The features of premium assessment taking into account the quality of the data. Features of the estimation of reserves taking into account the quality of the data.
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Models of random environment theory
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: In the course, we will consider some mathematical problems in the theory of random media: the infinitesimal (generating) operator of a random process; The method of random trajectories for solving parabolic equations; Solution of the Cauchy problem for the heat equation in a random environment perturbed by the "white noise" process; Finding the distributions of various additive functional from the Wiener process; In the course of the course, the students will be able to formulate doctoral dissertation: - 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; - 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. The equation of the temperature field in a random flow model with an update; The equation of the temperature field in short-correlated models of random currents; Multiscale random flow; Momentary equations of the temperature field in multiscale flows; Linear model of the hydromagnetic dynamo and products of random matrices; Evolution of the magnetic field in Markov linear models
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Definability and Computability
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose of the discipline– Familiarization with the notion of enumerability of sets in an arithmetic hierarchy is equivalent in the broadest sense of their natural definability in arithmetic. Tasks of the discipline-Consists in a new approach to the proof of Gödel's incompleteness theorem, based on the systematic use of formulas with bounded quantifiers and the application of the Gandhi fixed point theorem. In the course of studying the course to form the ability of doctoral students: - Describe the main classes of formulas with restricted quantifiers; - Select the technology, library or tool needed for parallel computing; - Describe the explanation of the formulas with limited quantifiers; - Know the term and the nature of the formula in this interpretation; - Formation of semantic effects and equivalents; - Implementation of delta zero restrictions; - analysis of sigma-reflex and sigma-restraint principles; - Implement Gandhi operator; - Use the classes of sub-entities of the numbered set; - configuration of operating system utilities and applications; - understanding the rules of wn- and n-subobjects, retrieval and e-subobjects and their description; Summary: Classes of formulas with bounded quantifiers and their interpretation; Basic concepts of computability theory; Theorem on Sigma-definability of the truth of Sigma-formulas, Church's theorem on the undecidability of arithmetic, Gödel's incompleteness theorem. The meaning of the term and the truth of the formula in a given interpretation; Semantic consequences and equivalences; Principles Delta-zero boundedness, the principles of Sigma-reflection and Sigma-boundedness; The Gandhi operator.
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Sequential Models of Mathematical Physics
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The aim of the course is the familiarization of doctoral student-mathematicians with methods of justification of determining of mathematical models, the forms of its solutions and justification of its solving. Studying this course, it is necessary to form the ability of doctoral students: - Know and understand the different forms of solutions of problems of mathematical physics; - Justify the methods for constructing mathematical models of physical processes; - Apply and justify the methods of practical solving of mathematical physics problems; - critically evaluate the current state of mathematical physics; - To teach special courses in mathematical physics. Defining of mathematical models and methods of its justification. Constructive methods of justification of its convergence. Classical, generalized and sequential forms of solutions of mathematical physics problems and methods of proving its existence
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Singularly Perturbed Partial Differential Equations of Hyperbolic Type
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The aim of the course "Singularly Perturbed Partial Differential Equations of Hyperbolic Type" is the study of the basic of the asymptotic theory of singularly perturbed partial differential equations of hyperbolic type and methods for solving such equations. During of studying the course to form the ability of doctoral students: - To know and understand the basic of the theory of singularly perturbed partial differential equations of a hyperbolic type for independent research on singularly perturbed partial differential equations of other types; - To create and research mathematical models of processes occurring in the real world and leading to singularly perturbed equations; - Apply methods for solving singularly perturbed partial differential equations to solving other applied problems of natural science; - Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To teach special courses on the theory of singularly perturbed partial differential equations in universities. The singularly perturbed Cauchy problem with an initial jump for second order hyperbolic equations that degenerate into a first order equation. Singularly perturbed mixed boundary value problems with initial jumps for second order hyperbolic equations that degenerate into first order differential equations. The phenomenon of the angular boundary layer. Construction of the asymptotics of solutions. Estimates of the remainder terms.
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Bonus-malus system and its application in insurance
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: It describes the main difficulties in the practical solution of optimal control problems. The existence and uniqueness of the optimal control, the sufficiency of optimality, correctness, extreme challenges, the bifurcation of extremals. In the course of the course, the students will be able to formulate doctoral dissertation: Contextualize mathematical methods for modeling Bonus-Malus systems. - Apply the theory of modeling Bonus-Malus systems. - Describe the effectiveness and bonus hunger. - Describe system solutions with multiple events. - Formulate conditions for the existence and uniqueness of solutions of stochastic differential equations and systems of equations. - Perform actuarial analysis of Bonus-Malus systems. - Summarize the results of the integration of the Bonus-Malus system and mathematical methods to support management decision-making under uncertainty of the economic environment. - Recommend the obtained knowledge and methods when solving research problems Analysis of the difficulties identified in the practical solutions of optimal control problems. The presentation is held at an extremely simple, but far from trivial examples, which are modeled on various kinds of "emergency situations" that may occur in the numerical solution of optimal control.
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System Nonlinear Differential Equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The aim of the course is to study the basic of the theory of systems of nonlinear differential and integro-differential equations with a small parameter with the highest derivative and methods for solving such equations. During of studying the course to form the ability of doctoral students: - To know and understand the basic of the theory of nonlinear differential and integro-differential systems with a small parameter for independent research on self-research in this area; - To create and research mathematical models of processes occurring in the real world and leading to differential and integro-differential systems with a small parameter; - Apply methods for solving differential and integro-differential systems with a small parameter to solving other applied problems of natural science; - Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To teach special courses on the theory of ordinary differential and integro-differential systems with a small parameter in universities. Autonomous systems of nonlinear ordinary differential equations with a small parameter. Non-autonomous systems of nonlinear ordinary differential equations with a small parameter with the highest derivative. Systems of nonlinear integro-differential equations with a small parameter with the highest derivative. Construction of the asymptotic expansion of solutions by the zone integration method. Calculation of initial solution jumps and integral terms. Proof of asymptotic expansion of the solution.
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Random parabolic equations and systems of equations
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The course will describe probabilistic statistical methods for analyzing the asymptotic behavior of solutions and questions of averaging parabolic equations and systems of partial differential equations. In the course of studying the course to form the ability of doctoral students: In the course of the course, the students will be able to formulate doctoral dissertation: - Formulation of conditions of existence and identity of stochastic differential equations and systems equilibrium. - Provide an infinitely-active (proizvodychiy) operator with an exponential process. - Provide analysis of the trajectory pathways for parabolic equilibrium. - Exit the Approach of Applied Mathematical Approaches to Trajectories of Constant Processes. - Provide analyze solution for Cauchy solution for thermal conductivity in the period of time. - Decide the solution of the Cauchy problem for heat dissipation in the instantaneous process, the "collapsible process"; The main method used here is the so-called random trajectory method, which allows us to write out the solutions of the equations in question in the form of conditional mathematical expectation along the trajectories of a solution related in a certain way to the initial equation of the stochastic differential equation (or the system of the equation).
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Modern problems of stochastic analysis
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The course is devoted to the presentation of some important sections of the general theory of modern stochastic analysis and its applications. In this case, considerable attention will be paid to the presentation of the basic concepts, facts and results relating to the general theory of random processes and the theory of martingales, which constitute an important part of the theory of stochastic calculus. In the course of studying the course to form the ability of doctoral students: Describe mathematical expectations of relative sigma-algebras and their properties. - Apply the general theoretical concepts. - Describe the evident and mathematical basis of the trajectory. - Select mathematical models for stochastic differential equations. - Describe the choice of stochastic differential equilibrium and system equilibrium. - Formulation of conditions of existence and identity of stochastic differential equations and systems equilibrium. - Provide basic algorithm for parallel machining. - Maintain analyze stochastic differential equations and systems equilibrium. - Exit the Approach of Applied Mathematical Approaches to Trajectories of Constant Processes. - Use the apparatus of mathematical expectations on trajectories of random processes; Knowledge: Basic concepts and the most important fundamental results of the general theory of random 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. Ability: to distinguish the most important classes of the theory of random processes; Prove martingale and semimartingale equalities and inequalities; Apply the Ito formula to solve standard problems; Write out probabilistic solutions of equations of parabolic type.
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Modern Problems of the Theory of Mathematical Physics
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the course "Modern Problems of the Theory of Mathematical Physics" is the study of the theory and the main nonlinear problems of mathematical physics and methods for their solution. During of studying the course to form the ability of doctoral students: - To know and understand the basics of the theory of functional analysis and the equations of mathematical physics for independent research on nonlinear equations; - Create and explore mathematical models of fluid mechanics, the theory of plasticity and continuum mechanics; - Apply methods for solving a non-linear problem to solving other applied problems of natural science; - critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To carry out the teaching of special courses on the theory of mathematical physics in universities. The course content is aimed at studying the nonlinear equations of mathematical physics, the basic equations of continuum mechanics; statements of boundary and initial boundary problems and methods for their solution.
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Spectral theory of operators and analytical methods research of differential operators
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose of the discipline is to familiarize with methods of the theory of linear operators and spectral expansions connected with differential operators. Task of the discipline is to acquaint with the main results of the spectral theory of linear operators. To study the analytical methods of investigating the spectral decomposition generated by differential operators. During studying the course to form the ability of doctoral students: - To know and understand the basic of the theory of nonlinear differential and integro-differential systems with a small parameter for independent research on self-research in this area; - To create and research mathematical models of processes occurring in the real world and leading to differential and integro-differential systems with a small parameter; - Apply methods for solving differential and integro-differential systems with a small parameter to solving other applied problems of natural science; - Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To teach special courses on the theory of ordinary differential and integro-differential systems with a small parameter in universities. Summary Linear operators in Hilbert spaces. Spectral theory of operators in Hilbert spaces. Differential operators in function spaces and their spectral decompositions.
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Embedding theorems and the theory of function spaces
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: Purpose of the discipline is to familiarize us with the functional spaces of Sobolev and Nikol'skii and the corresponding imbedding theorems. Tasks of the discipline is to introduce the functional spaces of Sobolev and Nikol'skii and the basic inequalities between their norms, and, as a consequence, to obtain corresponding embedding theorems. During of studying the course to form the ability of doctoral students: - Aims and objectives of the discipline: the main goal of the course is to learn the basics of the modern theory of functional spaces and its applications to the problems of modern mathematical and functional analysis. - The study of basic integral inequalities and their application. Teaching students the basics of approximation theory with the help of differentiable functions. - A circle of questions united under the name of "theorems Embedding for differentiable functions ", is devoted to the following general problem: how, knowing differential Properties of functions in one metric, set their properties Summary The space of summable functions, the Sobolev spaces, the Nikol'skii space. Integraline qualities of the embedding theorem and their application.
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The theory of generalized Lyapunov Exponents
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the course: generalization of the theory of Lyapunov's indicators and the formation of students' labor skills. As a result of mastering the discipline, the student can: - Explain the basic concepts of the theory of generalized Lyapunov exponents, understand the statements and proofs of the main theorems; - Classify the main classes of linear systems of differential equations; - Apply knowledge of the theory of generalized exponents in other areas of mathematics, such as, for example, differential equations, methods of mathematical physics; - Own methods for calculating generalized indicators of a system of differential equations; - Correctly choose methods for solving generalized indicators of linear systems of differential equations; - Apply the theory of generalized Lyapunov exponents to solve practical problems of studying various processes and phenomena and interpreting the results obtained.
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ControIIability Theory for Dynamical Systems
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the course “ControIIability Theory for Dynamical Systems” is the study of the theory and main problems of controllability of dynamic systems and methods for their solution. During of studying the course to form the ability of doctoral students: - Know and understand the basics of theories of controllability of dynamic systems for independent research and development to solve a number of boundary problems of controllability and optimal control; - Investigate the necessary and sufficient conditions for the existence of a solution of linear integral equations; - Apply methods for solving the problem of dynamic systems to solve other applied problems of natural science; - Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; - To teach special courses on the theory of controllability of dynamic systems in universities; The course content is aimed at studying the fundamentals of controllability of dynamic systems, the basic integral equations; statements of boundary and initial boundary problems and methods for their solution; Controllability of systems described by linear differential equations; controllability of linear systems with constant coefficients; controllability of the processes described by integro-differential equations.
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Theory of Extremal Problems in Banach Space
- Type of control: [RK1+MT+RK2+Exam] (100)
- Description: The purpose of the discipline is to familiarize methods of solving extreme problems in Banach spaces During the study of course, students should be competent in: • calculation of functional gradients defined on the set of solutions of ordinary differential equations, parabolic equation, hyperbolic equation; • formulation optimality conditions for a variety of extreme tasks and control their performance; • choosing modern computational optimization methods and apply them in solving applied problems; • using mathematical programming techniques and develop new programs to optimize computing processes and production planning; • analyze mathematical models of control processes and justify the correct choice of the method of solving problems (analytical, numerical, analytical-numerical) The content of the course is aimed at solving extreme problems on sets generated by control actions from given sets of Hilbert spaces. Algorithms for determining gradients of objective functionals, Lipschitz conditions for gradients, and optimality conditions for various extreme problems are studied.
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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.