# Examination topics in the discipline Mathematics

• 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.