Selected Mathematical Methods

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Applied Mathematics

Synopsis:

We start reviewing geometric properties of linear spaces with inner product. Next, we introduce and analyze the discrete Fourier transform (DFT) and its fast implementation (FFT).

Further we deal with differential calculus of functions involving multiple variables. We present methods for the localization of extreme values of functions. For this purposes, we study normed linear spaces and quadratic forms. In addition, we introduce the least square method.

The last part of the course is devoted to optimization and duality. The linear programming and the Simplex method is analyzed in more detail.

Requirements:

The fundamental knowledge of mathematical analysis and linear algerbra is required as they are given in BI-ZMA and BI-LIN.

Syllabus of lectures:

1. Complex numbers, complex function of complex variable, exponential function.

2. Fourier series.

3. Hilbert spaces of finite dimension, unitary matrices.

4. Discrete Fourier transformation (DFT) and Fast Fourier transform (FFT).

5. Basic objects from theory of multivariate functions.

6. (Constrained) extrema of multivariate functions.

7. General optimization problem.

8. Weak and strong duality.

9. Linear programming (introduction, formulation).

10. Linear programming (problem in standard form).

11. SIMPLEX algorithm.

12. Examples and applications of Linear programming.

Syllabus of tutorials:

1. Complex numbers, complex function of complex variable, exponential function.

2. Fourier series.

3. Hilbert spaces of finite dimension, unitary matrices.

4. Discrete Fourier transformation (DFT) and Fast Fourier transform (FFT).

5. Basic objects from theory of multivariate functions.

6. (Constrained) extrema of multivariate functions.

7. General optimization problem.

8. Weak and strong duality.

9. Linear programming (introduction, formulation).

10. Linear programming (problem in standard form).

11. SIMPLEX algorithm.

12. Examples and applications of Linear programming.

Study Objective:

The goal of the course is to improve student's mathematical skills and to present classical mathematical methods with applications in IT.

Study materials:

Howard Karloff: Linear Programming.

O. Julius Smith: Mathematics of the Discrete Fourier Transform with Audio Applications.

J.Kopáček: Matematika nejen pro fyziky II (lecture notes in czech).

Note:

Further information:

https://courses.fit.cvut.cz/BI-VMM/

No time-table has been prepared for this course

The course is a part of the following study plans:

• Bachelor program Informatics, unspecified branch, in Czech, 2015-2020 (elective course)
• Bachelor branch Security and Information Technology, in Czech, 2015-2020 (elective course)
• Bachelor branch Computer Science, in Czech, 2015-2020 (elective course)
• Bachelor branch Computer Engineering, in Czech, 2015-2020 (elective course)
• Bachelor branch Information Systems and Management, in Czech, 2015-2020 (elective course)
• Bachelor branch Web and Software Engineering, spec. Software Engineering, in Czech, 2015-2020 (elective course)
• Bachelor branch Web and Software Engineering, spec. Web Engineering, in Czech, 2015-2020 (elective course)
• Bachelor branch Web and Software Engineering, spec. Computer Graphics, in Czech, 2015-2020 (elective course)
• Master specialization Computer Science, in Czech, 2018-2019 (elective course)
• Bachelor branch Knowledge Engineering, in Czech, 2018-2020 (elective course)
• Master specialization Computer Security, in Czech, 2020 (elective course)
• Master specialization Design and Programming of Embedded Systems, in Czech, 2020 (elective course)
• Master specialization Computer Systems and Networks, in Czech, 202 (elective course)
• Master specialization Management Informatics, in Czech, 2020 (elective course)
• Master specialization Software Engineering, in Czech, 2020 (elective course)
• Master specialization System Programming, in Czech, version from 2020 (elective course)
• Master specialization Web Engineering, in Czech, 2020 (elective course)
• Master specialization Knowledge Engineering, in Czech, 2020 (elective course)
• Master specialization Computer Science, in Czech, 2020 (elective course)
• Mgr. programme, for the phase of study without specialisation, ver. for 2020 and higher (elective course)
• Bachelor specialization Information Security, in Czech, 2021 (elective course)
• Bachelor specialization Management Informatics, in Czech, 2021 (elective course)
• Bachelor specialization Computer Graphics, in Czech, 2021 (elective course)
• Bachelor specialization Computer Engineering, in Czech, 2021 (elective course)
• Bachelor program, unspecified specialization, in Czech, 2021 (elective course)
• Bachelor specialization Web Engineering, in Czech, 2021 (elective course)
• Bachelor specialization Artificial Intelligence, in Czech, 2021 (elective course)
• Bachelor specialization Computer Science, in Czech, 2021 (elective course)
• Bachelor specialization Software Engineering, in Czech, 2021 (elective course)
• Bachelor specialization Computer Systems and Virtualization, in Czech, 2021 (elective course)
• Bachelor specialization Computer Networks and Internet, in Czech, 2021 (elective course)
• Study plan for Ukrainian refugees (elective course)
• Master specialization System Programming, in Czech, version from 2023 (elective course)
• Master specialization Computer Science, in Czech, 2023 (elective course)