CZECH TECHNICAL UNIVERSITY IN PRAGUE
STUDY PLANS
2021/2022

# Mathematics for Cybernetics

Code Completion Credits Range Language
A3M01MKI Z,ZK 8 4P+2S Czech
Lecturer:
Jan Hamhalter (guarantor)
Tutor:
Veronika Sobotíková, Jan Hamhalter (guarantor)
Supervisor:
Department of Mathematics
Synopsis:

The goal is to explain basic principles of complex analysis and its applications. Fourier transform, Laplace transform and Z-transform are treated in complex field. Finally random processes (stacinary, markovian, spectral density) are treated.

Requirements:
Syllabus of lectures:

1. Complex plane. Functions of compex variables. Elementary functions.

2. Cauchy-Riemann conditions. Holomorphy.

3. Curve integral. Cauchy theorem and Cauchy integral formula.

4. Expanding a function into power series. Laurent series.

5. Expanding a function into Laurent series.

6. Resudie. Residue therorem.

7. Fourier transform.

8. Laplace transform. Computing the inverse trasform by residue method.

9. Z-transform and its applications.

10. Continuous random processes and time series - autocovariance, stacionarity.

11. Basic examples - Poisson processes, gaussian processes, Wiener proces, white noice.

12. Spectral density of the stacionary process and its expression by means of Fourier transform. Spectral decomposition of moving averages.

13. Markov chains with continuous time and general state space.

Syllabus of tutorials:

1. Complex plane. Functions of compex variables. Elementary functions.

2. Cauchy-Riemann conditions. Holomorphy.

3. Curve integral. Cauchy theorem and Cauchy integral formula.

4. Expanding a function into power series. Laurent series.

5. Expanding a function into Laurent series.

6. Resudie. Residue theroem

7. Fourier transform

8. Laplace transform. Computing the inverse trasform by residue method.

9. Z-transform and its applications.

10. Continuous random processes and time series - autocovariance, stacionarity.

11. Basic examples - Poisson processes, gaussian processes, Wiener proces, white noice.

12. Spectral density of the stacionary process and its expression by means of Fourier transform. Spectral decomposition of moving averages.

13. Markov chains with continuous time and general state space.

Study Objective:
Study materials:

[1] S.Lang. Complex Analysis, Springer, 1993.

[2] L.Debnath: Integral Transforms and Their Applications, 1995, CRC Press, Inc.

[3] Joel L. Shiff: The Laplace Transform, Theory and Applications, 1999, Springer Verlag.

Note:
Further information:
http://math.feld.cvut.cz/veronika/vyuka/b3b01kat.htm
Time-table for winter semester 2021/2022:
 06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00 roomT2:D3-209Hamhalter J.11:00–12:30(lecture parallel1)DejviceT2:D3-209roomT2:C3-52Hamhalter J.14:30–16:00(lecture parallel1parallel nr.101)DejviceT2:C3-52 roomT2:C3-340Hamhalter J.11:00–12:30(lecture parallel1)DejviceT2:C3-340
Time-table for summer semester 2021/2022:
Time-table is not available yet
The course is a part of the following study plans:
Data valid to 2022-07-05
For updated information see http://bilakniha.cvut.cz/en/predmet12562104.html