Fourier series decomposition for square integrable functions (types of convergence).
Relationship between symmetric and self-adjoint operators.
Provide examples of applications formulation Hahn-Banach theorem.
Definition of the measurable space, Borel σ-algebra and the construction of Lebesgue measure (in Rn).
Lebesgue dominated convergence theorem.
Differentiation in the complex domain. The Cauchy-Riemann equations.
The Cauchy integral theorem.
Laurent series expansions of functions.
Definitions of continuous functions in topological spaces. Homeomorphisms.
Cartesian product of topological spaces. Tychonoff topology.
Definition of a net (Moore–Smith sequence). Convergence in topological spaces.
Computer representation of real numbers, float point arithmetic and condition number. Provide definition of numerical correcteness (Kahan’s formula).
Interpolation (Lagrange, Hermite) and approximation (in the mean square and uniform sense, Chebyshev’s alternance theorem).
Numerical quadratures – simple and composite. Gauss theorem on quadrature of maximal degree.
Peano’s existence theorem. Provide definition of a sequence of Euler polylines and discuss how the Arzela-Ascoli’s theorem is used in the proof of Peano’s theorem
Picard theorem on existence and uniqueness of solution of initial-value problem with globally Lipschitz continuous righ-hand side function. Discuss how the Banach fixed point theorem is used in the proof of Picard theorem.
Algorithmic aspects of König’s theorem and its application within the assignment problem.
Applications of generating functions to recurrence relations on the example of Fibonacci sequence.
Combinatorial configurations – constructions and applications in statistics and sports.
System of linear equations – Cramer’s theorem, Kronecker-Capelli theorem and theorem on consistent systems of equations.
Exponentiation of a matrix – the method of diagonalization and diagonalizability of a matrix.
Transformation of a quadratic forms to a diagonal form, Sylvester’s criterion and applications to conic sections.