Selected Mathematical Methods

Garant předmětu:

Tomáš Kalvoda

Přednášející:

Cvičící:

Předmět zajišťuje:

katedra aplikované matematiky

Anotace:

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.

Požadavky:

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

Osnova přednášek:

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.

Osnova cvičení:

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.

Cíle studia:

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

Studijní materiály:

Howard Karloff: Linear Programming.

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

Poznámka:

Information about the course and courseware are available at https://courses.fit.cvut.cz/BIE-VMM/

Další informace:

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

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