1. Basic concepts, money, capital and securities.

2. Simple and compound interest rate, discounting.

3. Investments, cash lows and its measures, time value of money.

4. Assets and liabilities, insurance.

5. Bonds, options, futures, and forwards.

6. Exchange rates, inlation, indices.

7. Portfolio optimization - classical model.

8. Postoptimization, risk, funds.

The knowledge of Calculus and Linear Algebra together with probabilistic and statistical methods (including time series) as well as optimisation techniques within the framework of SOP and SO2 courses is required.

1. Dupačová,J. et al.: Stochastic Models for Economics and Finance, Kluwer, 2003.

The course presents basic financial models. It focuses on main concepts and computational methods.

Several lectures are especially developed to make students familiar with optimization models.

The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible low: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous lows will be also studied, namely the pressure-correction method and the spectral element method. Students ought to realize that only the knowledge of substantial physical and mathematical aspects of particular types of lows enables them to choose an effective numerical method and an appropriate software product. The development of individual semester assignement constitutes an important experience enabling to verify how the subject matter was managed.

2. Constitutive relations, thermodynamic state equations, Navier-Stokes and Euler equations, initial and boundary conditions.

Evolution partial differential equations, functional analysis, numerical methods for partial differential equations.

R.J. LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.

E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999.

S.V. Patankar: Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980.

J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002.

M.O. Deville, P.F. Fischer, E.H. Mund: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambdrige, 2002.

A. Quarteroni, A. Valli: Numerical Approximatipon of Partial Differential Equations. Springer- Verlag, Berlin, 1994

Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible luids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible lows: Navier-Stokes equations, pressure-correction method, spectral element method.

The aim of the course is to explain basic ideas and results of the optimal control theory, demonstrate the utilized techniques and apply these results to solving practical variational problems.

1. The scheme of variational problems and basic task of optimal control theory.

2. Maximum principle.

3. Time-optimal control of an uniform motion.

4. Time-optimal control of a simple harmonic motion.

The aim of the course is to explain basic ideas and results of the optimal control theory, demonstrate the utilized techniques and apply these results to solving practical variational problems.

• Pontrjagin, L. S. - Boltjanskij, V. G. - Gamkrelidze, R. V. - Miščenko, E. F.: Matematičeskaja teorija optimalnych procesov, Moskva, 1961.
• Lee, E. B. - Markus L.: Foundations of optimal control theory, New York, 1967.

The aim of the course is to provide students an overview of modern methods applied for solving boundary value problems for differential equations based on function spaces and functional analysis including construction of the approximate

1. Motivation. Overview of selected means of functional analysis.

2. Lebesgue spaces, generalized functions, description of the boundary.

3. Sobolev spaces, different approaches, properties. Imbedding and trace theorems, dual spaces.

4. Weak formulation of the linear elliptic equations.

5. Lax-Mildgam lemma, existence and uniqueness of the solutions.

6. Variational formulation, construction of approximate solutions.

7. Linear and nonlinear problems, various nonlinearities. Nemytskiy operators.

8. Weak and variational formulations of the nonlinear equations.

9. Monotonne operator theory and its applications.

10. Application of the methods to the selected equations of mathematical physics.

11. Introduction to Stochastic Differential Equations. Brown motion.

12. Ito integral and Ito formula. Solution of the Stochastic differential equations.

13. Reserve.

Differential and integral calculus of one and more real variables, ordinary and partial differential equations, functional analysis, function spaces, probability theory.

• S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980.
• K. Rektorys: Variational Methods in Mathematics, Science and Engineering, Dordrecht, D. Reidel Publ. Comp., 1980.
• J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. B. Oksendal: Stochastic Differential Equations, Springer, Berlin 2000.

The course yields overview of modern methods for solving differential equations based on functional analysis. It deals with the following topics: Survey of spaces of functions with integrable derivatives. Linear elliptic equations: the weak and variational formulation of boundary value problems, existence and uniqueness of the solution, approximate solutions and their convergence. Characteristics of the nonlinear problems. Weak and variational formulation of the nonlinear coercive problems, existence of the solution. Application to the selected nonlinear equations of mathematical physics. Introduction to stochastic differential equations.

The aim of the course is to show the students possibility of a unified perspective on seemingly different mathematical subjects.

1. Sets and classes

2. Mathematical structures

3. Isomorphisms

4. Fibres

5. Subobjects

Students are expected to know the mathematics taught within the bachelor's study programme and the graph theory taught in the master's study

•  Jiří Adámek, Theory of Mathematical Structures, D. Reidel Publ. Company, Dordrecht, 1983.
• A.Adámek, H.Herrlich. G.E.Strecker: Abstract and Concrete Categories, John Willey & Sons, New York, 1990

The course will familiarise students with basic concepts and results of the theory of mathematical structures. A number of examples of concrete structures will be used to demonstrate the exposition.