Mathematics
- Fourier series decomposition for square integrable functions (types of convergence).
- Relationship between symmetric and self-adjoint operators.
- Hahn-Banach theorem with examples of applications.
- Definition of a measurable space, Borel algebra and the construction of Lebesgue measure (in Rn).
- Lebesgue dominated convergence theorem.
- Green’s theorem and its consequences in the theory of vector fields.
- Cauchy integral theorem.
- Laurent series expansions of functions.
- Continuous functions in topological spaces. Convergence in topological spaces.
- Cartesian product of topological spaces. Tychonoff’s theorem.
- Transformation of a quadratic form to a diagonal form, Sylvester’s criterion and applications to conic sections.
- Computer representation of real numbers, float point arithmetic and condition number.
- Definition of numerical correctness (Kahan’s formula).
- Interpolation (Lagrange’s, Hermite’s) and approximation (in the mean square and uniform sense, Chebyshev’s alternance theorem).
- Numerical quadratures – simple and composite. Gauss theorem on the existence of a quadrature of maximal degree.
- Peano’s existence theorem. Definition of a sequence of Euler polylines and how the Arzela-Ascoli’s theorem is used in the proof of Peano’s theorem.
- Picard theorem on the existence and uniqueness of a solution of the initial value problem with globally Lipschitz continuous righ-hand side function. A usage of the Banach fixed point theorem in the proof of Picard theorem.
- Algorithmic aspects of König’s theorem and its applications.
- Applications of generating functions to recurrence relations.
- Truss stiffness and graph theory
- Independence of events, classes of events and random elements, definitions and criteria.
- Central limit theorem.
- Laws of large numbers.
- Sufficient statistics, definitions and their role in statistics.
- Formulation of the problem of testing statistical hypotheses.
- Reducing matrices to the Jordan form.