Applied Mathematics

Course guarantor:

Karel Roubík

Lecturer:

Ondřej Fišer, Jiří Hozman, Karel Roubík, Martin Rožánek

Tutor:

Ondřej Fišer

Supervisor:

Department of Biomedical Technology

Synopsis:

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

Requirements:

Active participation in the exercises; excused absence from a maximum of 2 exercises. A total of 2 tests are written during the semester, from which a total of 100 points can be obtained. The tests are based on questions and exercises based on the topics covered and practiced. Participation in the tests is not compulsory. A total of 10 bonus points can be earned for solving homework problems during the semester. A minimum of 50 points is required for successful completion of the course. The assessment 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. Nonlinear systems and their linearization.

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

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

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

11. Wavelet transform (wavelets). Hilbert transform, signal envelope.

12. 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. Processes and differential equations of 2nd order.

3. Description and response of linear systems.

4. Convolutions. Convolution theorem. Edge detection in medical image.

5. Fourier transform, demonstration by examples.

6. Hilbert transform, signal envelope, correlation function, autocorrelation function.

Study Objective:

Study materials:

ASMAR, Nakhlé H. a Loukas. GRAFAKOS. Complex Analysis with Applications [online]. Cham: Springer International Publishing, 2018. Undergraduate Texts in Mathematics [cit. 2019-03-06]. Dostupné z: <https://doi.org/10.1007/978-3-319-94063-2>. ISBN 9783319940632.

CICOGNA, Giampaolo. Exercises and Problems in Mathematical Methods of Physics [online]. Cham: Springer International Publishing, 2018. Undergraduate Lecture Notes in Physics [cit. 2019-03-06]. Dostupné z: <http://dx.doi.org/10.1007/978-3-319-76165-7>. ISBN 9783319761657.

Note:

Time-table for winter semester 2025/2026:

	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	roomKL:B-435 Fišer O. 12:00–13:50 EVEN WEEK (lecture parallel1 parallel nr.1) Kladno FBMI roomKL:B-230 Roubík K. Fišer O. 16:00–17:50 (lecture parallel1) Kladno FBMI roomKL:B-230 Rožánek M. Hozman J. 16:00–17:50 (lecture parallel2) Kladno FBMI
Wed
Thu
Fri

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Biomedical Engineering (compulsory course)