Applied Mathematics

Garant předmětu:

Martin Rožánek

Lecturer:

Ondřej Fišer, Jiří Hozman, Jakub Ráfl, Karel Roubík, Martin Rožánek

Tutor:

Ondřej Fišer, Karel Roubík, Martin Rožánek

Supervisor:

Department of Biomedical Technology

Synopsis:

The course deals with the practical applications of mathematics and its demonstration with examples from the field of biomedical engineering.

Requirements:

Active participation in exercises; excused absence from a maximum of 2 exercises. During the semester, a total of 2 tests are written, from which it is possible to obtain a total of 100 points. The tests are based on questions and problems related to lectured and practiced topics. Participation in the tests is not mandatory. During the semester, you can get a total of 5 bonus points for solving homework. To successfully complete the course, it is necessary to obtain at least 50 points. The rating is based on the ECTS scale.

Syllabus of lectures:

1. Exponential processes - theory and examples.

2. Complex numbers - description and calculations with complex numbers, orthogonal and orthonormal functions.

3. Processes and differential equations of 1st order.

4. 2nd order differential equations: Undamped oscillations.

5. Events and 2nd order differential equations: Damped oscillations.

6. Numerical solution of differential equations.

7. Description and response of linear systems.

8. Non-linear systems and their linearization.

9. Fourier series, Fourier transform, images of common signals.

10. Integral transforms, 2D Fourier transform from different points of view.

11. Convolution theorem - description of convolution and relation to Fourier transform, time and frequency domain.

12. Wavelet transform (wavelets).

13. Hilbert transform, signal envelope. Other integral transforms and their properties

14. Stochastic processes and signals, their description. White and coloured noise.

Syllabus of tutorials:

1. Exponential processes. Complex numbers.

2. Processes and differential equations of 1st order.

3. Processes and 2nd order differential equations.

4. Events and 2nd order differential equations (continued).

5. Description and response of linear systems.

6. Integral transformations. FFT, DFT, wavelet transform.

7. Convolution. Convolution theorem.

Study Objective:

Study materials:

Mandatory::

1.ROWELL, D. Review of First- and Second-Order System Response. [online] https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=11&ved=2ahUKEwjalt6-uPXhAhVBElAKHZvaDrYQFjAKegQIBhAC&url=http%3A%2F%2Fweb.mit.edu%2F2.151%2Fwww%2FHandouts%2FFirstSecondOrder.pdf&usg=AOvVaw3QQpc9YaM-QDW1r9UqhyxD. Massachusetts Institute of Technology (cit. 29. 4. 2019).

2.JAMES, J. F. A student's guide to Fourier transforms: with applications in physics and engineering. 3rd ed. Cambridge: Cambridge University Press, 2011. ISBN 9780521176835.

3.ADDISON, Paul S. The illustrated wavelet transform handbook: introductory theory and applications in science, engineering, medicine and finance. Second edition. Boca Raton: CRC Press, Taylor & Francis Group, 2017. ISBN 978-1-4822-5132-6.

4.Kak, A.C., Slaney, M.: Principles of Computerized Tomographic Imaging. USA [online] http://www.slaney.org/pct/pct-toc.html Citováno: 27.2.2019 - (zejména Kap. 2 věnovaná FT)

5.GUSTAVII, Bjorn. How to write and illustrate scientific papers [online]. 2nd ed. Cambridge, UK: Cambridge University Press, 2008 [cit. 2018-12-21]. Dostupné z: <http://search.ebscohost.com/login.aspx?direct=true&scope =site&db=nlebk&AN=224505>. ISBN 9780511394638.

Recommended:

1.DAY, Robert A. a Barbara GASTEL. How to write and publish a scientific paper. Eighth edition. Cambridge, United Kingdom: Cambridge University Press, 2017. 326 stran. ISBN 978-1-316-64043-2.

2.Poularikas, A. D.: Transforms and Applications Handbook. 3rd Ed. Boca Raton: CRC Press, 2010. 911 p. ISBN 978-1-4200-6652-4.

Other study materials::

1. Gerla, V., Hozman, J., Pop, M.: MIPS 2.0 - Microscopy Image Processing Software. Česká republika [online] https://www.instaluj.cz/mips 2000-2019. Poslední aktualizace: 26.5.2005. Citováno: 27.2.2019 (výukový software umožňující ilustraci dílčích výsledků při použití 2D Fourierovy transformace - na vytvoření se podílel i jeden z vyučujících předmětu).

1. Khutoriansky, E.: Fourier Series and Fourier Transform. Educational video. [online] https://www.youtube.com/watch?v=r18Gi8lSkfM 2015-2019. Poslední aktualizace: 6.9.2015. Citováno: 27.2.2019.

Note:

Time-table for winter semester 2024/2025:

	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	roomKL:B-433 Fišer O. 10:00–11:50 EVEN WEEK (lecture parallel1 parallel nr.1) Kladno FBMI Lab. KI a MZT
Tue	roomKL:B-433 Roubík K. Rožánek M. 08:00–09:50 (lecture parallel1) Kladno FBMI Lab. KI a MZT roomKL:B-433 Ráfl J. Hozman J. 08:00–09:50 (lecture parallel2) Kladno FBMI Lab. KI a MZT
Wed
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Prospectus - magisterský (!)
• Biomedical and Clinical Engineering (compulsory course)