# Courses

Here you can find brief description of courses.

**Mathematical analysis**

Mathematical analysis I: function of single variable. Annual course of studies. First semester: 72 hours of lectures, 72 hours of seminar studies. Second semester: 68 hours of lectures, 68 hours of seminar studies. Control forms: control works, oral examination, examination each semester. The course serves as essential component of qualification of mathematicians, mechanics, statisticians. Contains the following parts: elementary theory of sets and of real numbers, limit of a sequence, limit of a function in a point, continuous functions, derivative and its application, indefinite and definite integral, series and functional series, functions of a bounded variation, Stieltjes integral.

Mathematical analysis II: function of a several variables. Annual course of studies. Third semester: 72 hours of lectures, 72 hours of seminar studies. Fourth semester: 68 hours of lectures, 68 hours of seminar studies. Control forms: control works, oral examination, examination each semester. Serves as essential component of a qualification of mathematicians, mechanics, statisticians. Contains the following parts: metric spaces, differential calculus of a real and vector-valued functions of several variables, infinite integrals, integrals with parameters, integrals on varieties and Stokes theorem, Fourier series, Fourier integral.

**Measure and Integral Theory**

The discipline “Measure and Integral Theory” is an obligatory course for students of specialities “Mathematics” and “Statistics”. It is given in 5th semester within 3 credits (according to European Credit-Transfer System ECTS) and covers 198 hours of study, including 72 lecture hours, 36 hours of practical classes, and 90 hours of self-work. The semester is finished with a test and exam.

**Functional analysis and integral equations**

"Functional analysis and integral equations" is a one of the basic mathematical discipline for the students studying mathematics and statistics. The basis properties of linear operators in Banach and Hilbert spaces are studied. In particular the spectral theory of linear operators is considered. The abstract results on operator theory of compact operators are applied to the Fredholm theory for integral equations. The basic notions of distribution theory are also considered. The course has a numerous applications in Mathematical Physics, Stochastics, Numerical Methods, Calculus of Variations, etc.

**The spectral theory of linear operators.**

The basics of the spectral theory of linear operators are studied in the course. In particular the spectral theorem for bounded self-adjoint operators is proved. The the basics of the theory of unbounded operators in Hilbert space are also studied. More over the elements of the perturbation theory of linear operators is considered. In particular the Weyl theorem on the invariance of the essential spectrum under compact perturbations is proved.

**Approximation theory**

"Approximation theory" is basic course for students of magistrature who study mathematics. General questions of approximation theory (existence, uniqueness and characterization of element of best approximation) are studied. Especial attention is for uniform approximation of comtinuous functions by polynomials, direct and inverse theorems of approximation theory. It is used in the modern course of numerical methods.

**Measurement Error Regression Models (DFC)**

The discipline “Measurement Error Regression Models” is a special program of choice for students of specialities “Mathematics” and “Statistics”. It is given in 8th semester within 1 credit (according to European Credit-Transfer System ECTS) and covers 32 hours of study, including 32 lecture hours. The semester is finished with an exam.

**Harmonic Analysis**

The main properties of Fourier transform and Fourier series on Euclidian spaces, elements of the theory of harmonic and subharmonic functions, and basis of interpolation theory are studied in the course. In particular, formulae of inverse Fourier transform as generalized Abel and Dirichlet integrals are proved, connection with functions convolution is established. During the studying of Fourier series the absolute convergence of the coefficients series is studied, Poisson formula is obtained. Riesz convexity theorem in interpolation theory is proved.

**Stochastic analysis with elements of financial mathematics**

The basis of martingale theory, Ito integral, and stochastic differential equations are studied in the course. The Black – Scholes formula of European option price is established. The statements on limit existence of martingale sequence and on characterization of uniformly integrable martingale are proved. The construction of Ito integral is given, main properties are established. Theorem on existence and uniqueness of solution of stochastic differential equation is proved in the course. Self-financed strategies on financial market in continuous time are considered, Black – Scholes formula is obtained.

**Spline-Functions in Statistics**

The discipline “Spline-Functions in Statistics” is a special discipline for Ms students of specialities “Mathematics” and “Statistics”. It is given in 1st semester within 1 credit (according to European Credit-Transfer System ECTS) and covers 36 hours of study, including 36 lecture hours. The semester is finished with an exam.

**Modern systems of programming**

The course includes algorithms of data processing on computer, computer architecture, structure programming on Pascal in the integrated medium Borland Pascal for Windows, structure programming on Fortran in the integrated medium Compaq Visual Fortran 6.x, fundamentals of operation in Microsoft Office.

**Informatics, programming, computer**

The course includes main stages of solving typical problems of mechanics and computer mathematics by means of programming on algorithmic language Fortran and in software medium Matlab, as well as studying tools of graphical and table representation of computational results.

**Strength of materials with elements of civil engineering**

The course of lectures and seminars includes main hypothesis, problems and objects of strength of materials, types of stress-strain states, methods for calculation of values of deformations and stresses in structure elements under temperature and force factors.

**Fundamentals of continuum media**

The course includes basic hypothesis, problems and objects of mechanics of continuum media, main mathematical statements of problems, methods of investigation of kinematical and dynamical characteristics of objects, directions of practical application of results in engineering practice, basic problems of dynamics of ideal liquid.

**Theoretical hydromechanics**

The course includes basic hypothesis, problems and objects of mechanics of viscous liquid and theory of wave motion of continuum media, basic mathematical statements of problems, methods of investigation of kinematical and dynamical characteristics of objects, directions of practical application of results in engineering practice.

**Theory of elasticity**

The course includes basic hypothesis, statement of main problems of theory of elasticity (one-dimensional, two-dimensional, spatial), types of stresses-strained states, methods of determination of stresses-strained state of structure elements under temperature and force loadings.

**Nonlinear oscillations and waves**

Course includes description of main problems and objects of oscillation and wave systems, basic mathematical statements of problems, methods of investigation of kinematical and dynamical characteristics of elastic and liquid objects, directions of practical application of results in engineering practice.

**Fundamentals of theory of strength, deformation and material damage**

The course includes basic hypothesis of theory of strength of materials and structures, mathematical modeling of behavior of materials and structures, methods of computation of magnitudes of deformations and stresses in structure elements under temperature and force loadings.

**Vital activity safety**

The course includes main legislative, legal and regulatory decisions, instructions, laws and enactments, which regulate labour relations between employer and worker during peacetime or wartime, as well as during realization of emergency activities.

**Experimental methods of mechanics of deformable bodies**

Course includes classification of experimental methods of mechanics, determination of mechanical characteristics of rigid bodies, verification of basic theorems of strength of materials, testing equipment and modern methods of investigation of stressed-strained state of structure elements.

**Theory of optimal planning and processing of experiment**

Course includes basic notions of theory of strength of materials and structures, mathematical modeling of behavior of materials and structures, theory of probability and elements of statistical processing of experimental data,

**Inhomogeneous theory of elasticity**

Within the framework of the course basic mathematical models of theory of elasticity if inhomogeneous bodies, methods of solving of problems of theory of elasticity of inhomogeneous bodies are studied. We study basic notions of theory of elasticity of inhomogeneous bodies, in particular, notions of thermosensitive body, stress, the Duhamel-Newmann law, etc.

**Pedagogy and methods of teaching of mathematics at school**

History of pedagogy, main technique of solving problems of elementary mathematics, methods of teaching mathematics at school, notions of pedagogy (didactics, forms of teaching, etc.) are studied.

**Dynamics of bounded volume of liquid**

The course includes basic methods of investigation of kinematical and dynamical characteristics of objects, which include to their content reservoirs with liquid, pipelines, hypothesis, mathematical statements of typical problems, directions of practical application of results in engineering practice.

**Modern problems of continuum mechanics**

The course includes problems, theoretical background of creation of new promising composite materials and methods of prediction and optimization of mechanical behavior of composites, including nanomaterials, supplementary sections of failure mechanics, in particular, limiting strength of metals and alloys and their influence on methods of determination of limiting values of deformations and stresses in structure elements, created from new promising composite materials, under action of thermal and force factors.

**Mathematical models in continuum mechanics**

The course includes description of main problems and objects of continuum mechanics, basic mathematical statements of problems, methods for investigation of kinematical and dynamical characteristics of objects, directions of practical application of results in engineering practice. Mainly the course is based on variational methods of mathematical physics and methods of nonlinear mechanics.

**Modern models of turbulence and numerical hydrodynamics**

The course includes studying main modern models of turbulence and numerical grid algorithms, which are aimed at solving problems of numerical hydrodynamics on computer.

**Theory of plasticity**

The course deals with studying basic mathematical models of plasticity in rigid bodies, methods of solving problems of plasticity, basic notions of theory of plasticity, namely, notion of loading surface, loading trajectory, stress, plastic deformation, etc.

**Numerical-analytical methods in continuum mechanics**

The course considers main numerical-analytical methods for solving basic problems of mechanics. By the example of problems of dynamics of liquid with a free surface it is shown specificity of application of methods for solving problems of algebra, calculation of quadratures, solving of transcendent equations, integration of systems of differential equations and reduction of problems of continuum mechanics to discrete models on the basis of variational methods of mathematical physics.

**Oscillation of liquid in reservoirs**

Methods of construction of applied models of dynamics of reservoirs with liquid with a free surface are under investigation. Range of application of different types of models and their properties, which are manifested for different problems and types of motion disturbance, are shown.