Chapters in higher mathematics

Course guarantor:

Pavel Pták

Lecturer:

Mirko Navara, Pavel Pták

Tutor:

Mirko Navara, Pavel Pták

Supervisor:

Department of Cybernetics

Synopsis:

The course consists of several deeper results in a few mathematical disciplines. The idea is to help a student to read, with a certain comfort, the monographs in given lines of applied mathematics. The contents of the course are fundamental results (principles) of nowadays mathematics. More specifically, the course concerns the Stone representation theorem for Boolean algebras (as applied in mathematical logics and probability theory), the Banach fixed-point theorem for complete metric spaces (as applied in numerical mathematics), the Tychonoff theorem on compact spaces (as applied in measure theory), the Riesz representation theorem for linear forms in a Hilbert space (as applied in the optimization theory), the Brower theorem for balls in Rn (as applied in linear algebra – the Perron theorem), the elements of category theory for a practical man, etc. The asset may be a certain encouragement in a student’s research.

Requirements:

Syllabus of lectures:

1. Introduction. Metric spaces

2. Connectedness and the curve connectedness in metric spaces

3. Compact metric spaces

4. Complete metric spaces and the Banach fixed-point theorem

5. Elementary proof of the Fundamental theorem of algebra

6. Lattices and Boolean algebras

7. The Stone representation for Boolean algebras

8. Extension of states on a Boolean algebra (the Tychonoff theorem)

9. Categories and morphisms

10. Normal and Hilbert spaces

11. The Riesz representation theorem for linear forms

12. The Sperner lemma

13. The Brower theorem on the fixed-points on the continuous mappings on ball in Rn

14. An application on Brower theorem: The Perron theorem on eigenvalues

Syllabus of tutorials:

Study Objective:

Study materials:

Mandatory bibliography:

Hoggar, S. G.:Mathematics for computer graphics. Cambridge University Press, Cambridge, 1992.

Rudin, W.: Functional analysis. Second edition. McGraw-Hill, Inc., New York, 1991.

Recommended bibliography:

Rudin, W.: The Principles of Mathematical Analysis 3rd Edition. McGraw-Hill Publishing Company, 2006

Note:

Further information:

https://cw.fel.cvut.cz/wiki/courses/XP33CHM

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Doctoral studies, daily studies (compulsory elective course)
• Doctoral studies, combined studies (compulsory elective course)