Mathematics for Cybernetics

Lecturer:

Jan Hamhalter (gar.)

Tutor:

Jan Hamhalter (gar.), Martin Bohata

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:

Time-table for winter semester 2011/2012:

	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
Mon
Tue	roomT2:C3-135 Hamhalter J. 12:45–14:15 (lecture parallel1) Dejvice PosluchárnaroomT2:C3-51 Hamhalter J. 16:15–17:45 (lecture parallel1 parallel nr.104) Dejvice PosluchárnaroomT2:C3-51 18:00–19:30 (lecture parallel1 parallel nr.101) Dejvice Posluchárna
Fri
Thu	roomT2:C3-340 Hamhalter J. 12:45–14:15 (lecture parallel1) Dejvice PosluchárnaroomT2:C4-78 Bohata M. 14:30–16:00 (lecture parallel1 parallel nr.103) Dejvice PosluchárnaroomT2:C4-78 Bohata M. 16:15–17:45 (lecture parallel1 parallel nr.102) Dejvice Posluchárna
Fri

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• Kybernetika a robotika - Robotika (compulsory course in the program)
• Kybernetika a robotika - Senzory a přístrojová technika (compulsory course in the program)
• Kybernetika a robotika - Systémy a řízení (compulsory course in the program)
• Kybernetika a robotika - Letecké a kosmické systémy (compulsory course in the program)