# Subject description - XP33CHM

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XP33CHM Chapters in higher mathematics
Roles:S Extent of teaching:2P
Department:13133 Language of teaching:EN
Guarantors:Pták P. Completion:ZK
Lecturers:Pták P. Credits:4
Tutors:Pták P. Semester:

Anotation:

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.

Content:

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.

Course outlines:

 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

Exercises outline:

Literature:

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

Requirements:

Webpage:

https://moodle.fel.cvut.cz/courses/XP33CHM

Subject is included into these academic programs:

 Program Branch Role Recommended semester DOKP Common courses S – DOKK Common courses S –

 Page updated 5.10.2022 17:50:42, semester: L/2021-2, L/2022-3, Z/2024-5, Z/2022-3, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)