REQUIREMENTS FOR THE COMPREHENSIVE EXAMINATION  Bachelor level (study fields: Mathematics  Applied Mathematics, Mathematical Methods in Economics, Applied Mathematics for Crises Management) 1. Matrices and determinants (matrix operations, properties of determinants, rank of matrix, eigenvalues of matrix, Jordan normal form of a square matrix, examples). 2. Vector spaces, linear maps (linear dependence, bases, subspaces, representing linear map with respect to a basis, examples of vector spaces and linear maps). 3. Scalar product (bilinear and quadratic forms, inner product spaces, angle of subspaces, orthogonality, examples of inner product spaces, orthogonal matrices). 4. Linear algebraic equations (homogeneous and nonhomogeneous systems, solution methods, inetrative methods and computer aided solutions). 5. Polynomials (methods of root finding, numerical solution of algebraic equations on computer). 6. Sequences and series (of numbers and functions, convergence criteria). 7. Functions of one or several real variables (coninuity and limit, basic theorems on continuity, uniform continuity, Lipschitz condition). 8. Derivatives and differentials (definition and basic properties, directional and partial derivatives, derivatives and differentials of higher order). 9. Extrema of functions of one or several real variables, constrained extrema. 10. Taylor polynomial and Taylor series (in one or several real variables, Taylor remainder, Taylor series of functions of one complex variable). 11. Elementary functions (trigonometric functions, exponential function, the logarithm in the real and complex domain). 12. Riemann integral of a function of one or several real variables (definition and basic properties, contour integrals). 13. Computation of integrals (relationship betweem integral and primitive, integration by parts, change of variable in the integral, integrating rational functions, integrals that can be reduced to integration of rational functions, Fubini's theorem, numerical integration). 14. Implicit function theorem (solvin functional equations involving one or several unknon functions). 15. Ordinary differential equations of first order (separation of variables, method of successive approximations, approximate solution methods, linear equations). 16. Ordinary differential equations of higher order, systems of ordinary differential equations (properties of solution sets, solving equations with constant coefficients). 17. Approximation and interpolation (least squares method, spline approximation). 18. Basic properties of functions of complex variable (continuity and limit, derivative with respect to a complex variable, CauchyRiemann equations). 19. Contour integrals and primitives of functions of a complex variable. 20. Holomorphic functions (definition, basic properties, singularities). 21. Basics of probability theory (probability, dependent and independent phenomena, conditional probability). 22. Random variables (basic characteristics, relationships between random variables, law of large numbers). 23. Basics of mathematical statistics (basic notions, estimation theory). 24. Testing statistical hypothesis (examples of applications).


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