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Course, academic year 2019/2020
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Entropy in Probability Dynamical Systems - NMTP569
Title in English: Entropie v pravděpodobnostních dynamických systémech
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: summer
E-Credits: 3
Hours per week, examination: summer s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: Mgr. Michal Kupsa
Class: M Mgr. PMSE
M Mgr. PMSE > Volitelné
Classification: Mathematics > Probability and Statistics
Annotation -
Last update: T_KPMS (16.05.2013)
The basic elements of ergodic theory are presented. We mainly focus on entropy and recurrence. Tight relationship between ergodic theory and the theory of finite-states stationary processes will be presented.
Aim of the course -
Last update: T_KPMS (16.05.2013)

To present the broadly used notion of entropy in the frame of the ergodic theory.

Course completion requirements - Czech
Last update: RNDr. Jitka Zichová, Dr. (19.04.2018)

Složení ústní zkoušky.

Literature - Czech
Last update: T_KPMS (16.05.2013)

K. Petersen: Ergodic Theory, Cambridge Univ. Press, 1983

P. Shields: The Ergodic Theory of Discrete Sample Path, Graduate Studies in Mathematics, AMS, 1996

Teaching methods -
Last update: T_KPMS (16.05.2013)

Lecture.

Requirements to the exam - Czech
Last update: RNDr. Jitka Zichová, Dr. (05.06.2019)

Zkouška je ústní. Požadavky ke zkoušce odpovídají sylabu předmětu v rozsahu, který byl prezentován na přednášce.

Syllabus -
Last update: T_KPMS (16.05.2013)

1. Probability (measure-theoretical) dynamical systems, finite-states stationary processes - definitions, examples, ergodicity, isomorfismus of probability dynamical systems, factorization

2. Entropy of the process, entropy of the system, strictly positive entropy - Kolmogorov property

3. Kolmogorov-Sinai theorem on generators, Shannon-McMillan-Breimann theorem

4. Recurrence, Ornstein-Weiss theorem, Lempel-Ziv algorithm for data compression

Entry requirements -
Last update: RNDr. Jitka Zichová, Dr. (05.06.2019)

Basics of mathematical analysis, measure tehory, Lebesgue integral. Basics of probability theory and linear algebra.

 
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