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- 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.

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