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.