Mathematics for Cybernetics
Code  Completion  Credits  Range  Language 

AE3M01MKI  Z,ZK  8  4P+2S 
 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 Ztransform 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. CauchyRiemann 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. Ztransform 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. CauchyRiemann 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. Ztransform 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 timetable 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)