| Convex Geometry
Study of linear and convex envelopes, theorems of Radon and Helly and their application, polyhedrons and the solution of linear inequalities. |
| Boundary
Value Problems In the Space of Analytical Functions
The discipline consists of the following chapters: Boundary Value Problem of Riemann,
The Riemann - Hilbert Problem, index, singular integral equations, general solution
of elliptic homogeneous differential equations with constant coefficients and
various boundary value problems. |
| Real
and Complex Analysis
The discipline contains the following chapters: Theory of measure, measurable
functions, Lebesque Integral, analytic functions, residue theory, analytic continuity,
Fourier and Laplace transformations. |
| Discrete
Mathematics
The discipline contains: Theory of Boolean functions, Theory of graphs, Automata
theory, and Algebraic description of discrete circuits and issues of automated
design of chips. |
| Linear
Algebra
The discipline contains: Study of the system of linear equations, the linear,
Euclidean and unitary spaces, linear operators and their reduction to the canonical
form. Bilinear and quadratic forms. Groups, rings and fields. |
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| Mathematical Logic
The discipline contains: Statement calculus, Predicate Calculus and Formal Arithmetic. |
| Development
of Electronic Computers
The discipline contains: Principles of development and functioning of computing
machines, elements, data introduction, functional and structural organization
of computing machines, structure and operation of the processor and the memory,
development of contemporary computers. |
General
Algebra
The discipline contains: Theory of linear spaces, Theory and structure of linear
operators, quadratic forms, elements of Group Theory. |
| Complexity
of Algorithms
The discipline contains: Turing machines, various complexity criteria of algorithms,
undetermined Turing machines, polynominally calculable algorithms, elements of
polynominal reduction, NP-full problems. |
| Special
Functions
This course is an introduction into an extremely important sphere of applied mathematics.
Euler beta and gamma - functions, classical orthogonal polynomials, Bessel functions
are studied. Problems of approximation, interpolation, numeric integration are
investigated. Several problems of mathematical physics are being solved as well.
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Theory
of Algorithms
The discipline contains: Turing machines, partially recursive functions, solvable
and insolvable problems, recursively denumerable sets, elements of the reducibility
theory of bringing together. |
Probability
Theory and Mathematical Statistics
The discipline contains: Axiomatic construction of the probability theory, producing
function, characteristic function, central limit theorem of the probability theorem,
law of large numbers, pointwise evaluation of the unknown parameters, fiducial
evaluation of the unknown parameter, conception of the test of statistic hypotheses.
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| Theory
of Graphs
The discipline contains: planar graphs, dual graphs in respect to the given graph
and dual - in respect to Whitney graph Eulerian and Hamiltonian graphs. The theorem
of Kuratowski - Pontrjagin is proved. The Ramsay theorem is introduced along with
its application in some issues of geometry, as well as the coloration of the graph
and the problem of 4 colors. |
| Theory
of Numbers
Study the Euclid algorithms, theorems of Euler and Fermat in the ring of whole
numbers, theory of comparisons, resolution of comparisons with one unknown value
and indefinite equations. Investigation of second-degree equations with one unknown,
indexes and their application. |
| Equations
of Mathematical Physics
The discipline contains: Classification of second-degree differential equations,
classical methods of resolution of the model boundary value problems (the method
of separation of variables, the method of the theory of the potential), formulation
of correct boundary value problems for various types of equations. |
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Functional
Analysis
The course acquaints with classical metric, Banach and Hilbert spaces, as well
as the operators acting in those spaces. The purpose is to introduce the ideas
connected with the conceptions, approaches and methods of contemporary continuous
mathematics to the students. Through specially selected problems it is envisaged
to impart the experience and skills of practical application of the acquired knowledge
in the students. This subject may serve as a basis for further more specialized
disciplines. |
Mathematical
Backgrounds of Digital Processing of Signals
The following problems are investigated: discrete and fast Fourier transformations,
problem of spectral analysis, development of optimal filter, the problem of noise
elimination. |
Numerical
Methods
This course investigates the most known and effective methods of approximation
of functions and resolution of operator equations: Theory of interpolation and
related to it issues of numerical differentiation and integration (quadrature
formulas of Newton - Cotes and Gauss), mean square and uniform approximation of
functions, the method of successive approximations, the methods of simple iteration
and Seydel's for the resolution of linear systems, Newton's method and the method
of descent, methods of Runge- Kutta to solve the Cauchy problem for ordinary differential
equations. |
Numerical
Methods of Resolution of Partial Differential Equations
This course investigates the classical boundary problems for partial differential
equations: Cauchy problem for hyperbolic equations, first boundary problem for
elliptic and parabolic equations. The issues pertaining to correctness of those
boundary problems are studied as well along with their numeric resolution using
the grid method and methods of finite elements. The resolution of difference equations,
to which the numerical solution of boundary problems is brought is also included
in the studies. |
