Topics requested for admission to the doctoral study at MI

Mathematical engineering

1. Functional analysis: Normed vector spaces, Banach spaces, Hilbert spaces, continuous versus bounded operators, operator norm, graph of an operator, closed operator, linear functionals, dual spaces, Riesz's lemma.
2. Functional analysis: Spectrum of a closed operator on a Banach space, resolvent, compact operators, spectrum of a compact operator operator, self-adjoint bounded operators on Hilbert space and their spectrum, spectral properties of self-adjoint compact operators.
3. Algebra: Group, subgroup, normal subgroup, factor group, homomorphisms of groups, kernel and image of a homomorphism, representation of a group.
4. Algebra: Bilinear and quadratic forms, real quadratic forms and law of inertia, scalar product, unitary spaces, orthogonal base, different types of square matrices (orthogonal, unitary, self-adjoint, normal), eigenvalues and eigenvectors, diagonalizable square matrices.
5. Numerical mathematics: The Crank and Nicolson method for solving a parabolic problem.
6. Numerical mathematics: The shooting method for solving a parabolic problem.
7. Probability and statistics: Random quantities and their distribution functions, probability density.
8. Probability and statistics: Methods of estimation of distribution parameters and their properties.
9. Differential equations: The existence and uniqueness of the solution to problems in normal form.
10. Differential equations: The method of variation of constants.