Selected chapters in mathematics

Lecturer:

Miroslav Dont (gar.)

Tutor:

Miroslav Dont (gar.)

Supervisor:

Department of Mathematics

Synopsis:

The first part is devoted to some problems of matrix analysis,

especially to propertries connected with eigenvalues and eigenvectors

of matices. That is, for example, similarity of matrices, spectral

decomposition and the singular value decomposition with applications.

In the second part notions of partial differential equations and

boundary value problems for partial differential equations are

explained. Some concrete boundary value problems are considered and

solved using Fourier method and using special function, in particular

Bessel and Legendre spherical functions.

Requirements:

Syllabus of lectures:

Part I - matrix analysis

1. Basic notions of linear algebra. Matrix algebra, matrix multiplication, block matrices.

2. Scalar multiplication, vector norm, Gramm-Schmidt ortogonalization.

3. Hermitian, unitary and real orthogonal matrices.

4. Eigenvalues and eigenvectors of a matrix. Similar matrices, diagonalizable matrices.

5. Spectral theorem for hermitian matrices. Schur triangularization theorem and normal matrices. Definite matrices.

6. Discrete Fourier transform, Fourier matrix, fast Fourier transform.

7. Singular value decomposition, least squares problem, Moore-Penrose pseudoinverse.

Part II - PDE and the Fourier method

8. The notion of PDE, equations of the second kind, wave equation,heat equation, Laplace an Poisson equations. Boundary value problems.

9. Vibrating string, vibrating rectangular membrane, the Fourier method.

10. Wave equation in cylindrical coordinates, Bessel equation and Bessel functions.

11. Fourier-Bessel and Dini-Bessel series. Solution of the problem of vibrating circular membrane.

12. Problems with spherical symmetry, Legendre equation, Legendre spherical functions.

13. Solutions of some boundary value problems using spherical

functions.

Syllabus of tutorials:

Part I

1. Spaces R^n and C^n, linear subspaces, bases and dimension.

2. Scalar multiplication, Gramm-Schmidt ortogonalization.

3. Matrix multiplication, block matrix multiplication, systems of linear equations.

4. Eigenvalues and eigenvectors of matrices.

5. Diagonalizable matrices.

6. Unitary and orthogonal diagonalization of hermitian and real symmetric matrices.

7. Positive definite and semidefinite matrices.

8. Singular value decomposition and the least squares problem.

Part II

9. PDE and some boundary value problems.

10. Fourier method for the vibrating string problem and for the one-dimensional heat equation.

11. Vibrations of rectangular membrane and the double Fourier series.

12. Vibrations of circular membrane, Bessel functions.

13. Boundary value problems with spherical symmetry and spherical functions.

Study Objective:

Study materials:

1. PDF soubory kap1 až kap4 dostupné pomocí ftp na math.feld.cvut.cz, adresář pub/dont/2009 (texty z teorie matic, česky).

2. D. C. Meyer: Matrix Analysis and Applied Linear Algebra, SIAM 2000.

3. M. Dont: Úvod do parciálních diferenciálních rovnic, Nakl. ČVUT, druhé přeprac. vydání 2008.

4. E. A. Gonzáles-Velasco: Fourier Analysis and Boundary Value Problems, Academic Press 1995.

Note:

Time-table for winter semester 2011/2012:

	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	roomT2:E1-103a Dont M. 12:45–14:15 (lecture parallel1) Dejvice haly Cvičebna
Fri	roomT2:A4-202b Dont M. 14:30–16:00 (lecture parallel1 parallel nr.101) Dejvice Učebna
Thu	roomT2:C4-364 Dont M. 12:45–14:15 (lecture parallel1) Dejvice Cvicebna
Fri

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: