Mathematics for Cybernetics

Lecturer:

Tutor:

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/hamhalte/A3M01MKI.htm

No time-table has been prepared for this course

The course is a part of the following study plans:

• Cybernetics and Robotics - Robotics (compulsory course in the program)
• Cybernetics and Robotics - Senzors and Instrumention (compulsory course in the program)
• Cybernetics and Robotics - Systems and Control (compulsory course in the program)
• Open Informatics - Artificial Intelligence (elective course)
• Open Informatics - Computer Engineering (elective course)
• Open Informatics - Computer Vision and Image Processing (elective course)
• Open Informatics - Computer Graphics and Interaction (elective course)
• Open Informatics - Software Engineering (elective course)
• Communications, Multimedia and Electronics - Wireless Communication (elective course)
• Communications, Multimedia and Electronics - Multimedia Technology (elective course)
• Communications, Multimedia and Electronics - Electronics (elective course)
• Communications, Multimedia and Electronics - Networks of Electronic Communication (elective course)
• Electrical Engineering, Power Engineering and Management - Technological Systems (elective course)
• Electrical Engineering, Power Engineering and Management - Electrical Machines, Apparatus and Drives (elective course)
• Electrical Engineering, Power Engineering and Management - Electrical Power Engineering (elective course)
• Electrical Engineering, Power Engineering and Management - Economy and Management of Power Eng. (elective course)
• Electrical Engineering, Power Engineering and Management - Economy and Management of Electrical Eng. (elective course)
• Cybernetics and Robotics - Air and Space Systems (compulsory course in the program)
• Communications, Multimedia and Electronics - Communication Systems (elective course)
• Open Informatics - New - Software Engineering (elective course)