Selected Mathematical Methods

Enrollement in the course requires an successful completion of the following courses:

Linear Algebra (BI-LIN)
Elements of Calculus (BI-ZMA)

Lecturer:

Tomáš Kalvoda (guarantor)

Tutor:

Tomáš Kalvoda (guarantor)

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/

Time-table for winter semester 2020/2021:

Time-table is not available yet

Time-table for summer semester 2020/2021:

Time-table is not available yet

The course is a part of the following study plans:

• Bc. Programme Informatics, in Czech, Version 2015 to 2020 (elective course)
• Bc. Branch Security and Information Technology, in Czech, Version 2015 to 2020 (elective course)
• Bc. Branch Computer Science, in Czech, Version 2015 to 2020 (elective course)
• Bc. Branch Computer Engineering, in Czech, Version 2015 to 2020 (elective course)
• Bachelor Branch Information Systems and Management, in Czech, Version 2015 to 2020 (elective course)
• Bachelor Branch Knowledge Engineering, in Czech, Version 2015, 2016 and 2017 (elective course)
• Bachelor Branch WSI, Specialization Software Engineering, in Czech, Version 2015 to 2020 (elective course)
• Bachelor Branch, Specialization Web Engineering, in Czech, Version 2015 to 2020 (elective course)
• Bachelor Branch WSI, Specialization Computer Grafics, in Czech, Version 2015 to 2020 (elective course)
• Specialization Computer Science, Presented in Czech, Version 2018 to 2019 (elective course)
• Bachelor Branch Knowledge Engineering, in Czech, Version 2018 to 2020 (elective course)
• Computer Security, Presented in Czech, Version 2020 (elective course)
• Design and Programming of Embedded Systems, Presented in Czech, Version 2020 (elective course)
• Computer Systems and Networks, Presented in Czech, Version 2020 (elective course)
• Management Informatics, Presented in Czech, Version 2020 (elective course)
• Software Engineering, Presented in Czech, Version 2020 (elective course)
• System Programming, Presented in Czech, Version 2020 (elective course)
• Web Engineering, Presented in Czech, Version 2020 (elective course)
• Knowledge Engineering, Presented in Czech, Version 2020 (elective course)
• Specialization Computer Science, Presented in Czech, Version 2020 (elective course)
• Master's Program Informatics, Plan for students without Specialization, in Czech, 2020 (elective course)