Lecturer(s)


Knobloch Roman, Mgr. Ph.D.

Soudský Filip, RNDr. Ph.D.

Course content

Lectures: 1. Motivational problems 2. Aproximational methods for onedimensional problems  Taylor polynomial, Newton tangential method 3. Introduction to Banach and Hilbert spaces 4. Function spaces $\mathcal{C}^k$ and spaces of Holder continuous functions 5. Fixed point theorems and their applications  proof of Picard's theorem 6. Approximative solution of differential equations 7. Dynamical systems, stability of solution 8. Dynamical systems, optimal regulation 9. Introduction to game theory, pure and mixed strategies, games with full and partial information 10. More on fixed point theorems (Brouwer's theorem) 11. Nash's theorem on existence of equilibrium point. 12. Some examples continuous games and existence of optimal strategy 13. Pursuit games 14. Some applications of game theory in problems fo economy biology and other disciplines Exercises: Practice topics according to lectures.

Learning activities and teaching methods

Monological explanation (lecture, presentation,briefing), Written assignment presentation and defence
 Class attendance
 56 hours per semester
 Preparation for credit
 15 hours per semester
 Semestral paper
 20 hours per semester

Learning outcomes

The subject focuses on solution of optimalization problems by means of classical and modern methods of optimalization. The initial part is devoted to numerical solutions of certain problems, for which the analytical methods fail. Namely finding the root of equation f(x)=0, approximative evaluation of functions values and approximative solution of ordinary differential equations. In the following part we shall give a brief introduction to game theory, we shall recall the classical result on existence of equilibria of the game (Nash theorem) and we will discuss the methods of finding the optimal strategy of the game. In the end we apply the theory in solving several problems motivated in biology and economy.
Knowledge of fundamentals of numerical mathematics.

Prerequisites

Passing of mathematical lectures of first four semestrs.

Assessment methods and criteria

Oral exam, Written exam
Credit: Working out a semestral work. Exam: Written.

Recommended literature


Border, Kim C. Fixed point theorems with applications to economics and game theory. Cambridge university press, 1985. ISBN 9780521388085.

Hans Peters. Game Theory A MultiLeveled Approach. 2008. ISBN 9783540692904.

Martin Chvoj. Pokročilá teorie her. 2013. ISBN 9788024746203.

Petr Habala, Petr Hájek, Václav Ziezler. Introduction to Banach spaces. Matfyzpress, 1996. ISBN 8085863146.

Rufus Isac. Differential Games. New YorkJohn Willey and sons, 1964. ISBN 0486406822.

Simon, D.:. Evolutionary Optimization Algorithms. New Jersey, John Wiley, 2013. ISBN 9780470937419.
