Selected topics in Mathematics

Subject description

Functional analysis

Metric spaces (notion of distance, properties of matric spaces, examples of different metrics on vector spaces and on functional spaces). Normed vector spaces (notion of norm, relations between norms and metrics). Spaces with scalar product (Hilbert space). Bounded linear operators, matrices (contraction mapping principle and fixed point, spectral theory, eigenvalues and eigenvectors). Cryptography (RSA public-key).

Discrete mathematics
Graphs, basics. Paths any Cycles (Euler cycle, Hamiltonian cycle, the traveling salesman problem).

Flow in networks (maximum flow and minimum cut). Trees (decision trees, game trees).

Numerical solution of partial differential equations by the finite element method

Finite element method for second order boundary value problem. Variational (weak) form of the problem (appropriate functional spaces, equivalence of classical and variational form). Discretization (triangulation, bases with local support, matrix form notation). Numerical integration. Numerical solution using FeeFEM++ open source package.

The subject is taught in programs

Objectives and competences

Presentation of mathematical notions and methods which are frequently used in formulation and in solution of different problems which arise in electrical engineering. Deeper understanding of mathematical concepts and correct usage of mathematical methods are emphasized.

Teaching and learning methods

Lectures cover all three listed topics. Student choose one of the topic. Deeper understanding of the chosen topic and basic knowledge about the other two topics are required.

Expected study results

After successful completion of the course, students should be able to:

use new approaches to solve difficult problems in electrical engineering,
choose and use appropriate methods from functional analysis,
choose and use appropriate methods from discrete mathematics,
choose and use appropriate methods for solving partial differential equations,
critically evaluate the obtained results.

Basic sources and literature

M. Pedersen, Functional Analysis in Applied Mathematics and Engineering, Chapman & Hall/CRC, 1999.
J. T. Oden, L. Demkowicz, Applied Functional Analysis, CRC Press, 2010.
Richard Johnsonbaugh, Discrete Mathematics, Pearson, 2017.
R. Diestel, Graph Theory, Springer-Verlag, GTM 173, 3. izdaja, 2005.
J. M. Kleinberg, Éva Tardos, Algorithm design, Addison-Wesley, 2006.
B. S. Jovanović, E. Süli, Analysis of finite difference schemes, Springer, 2014.
J. N. Reddy, An Introduction to the Finite Element Method (Engineering Series), McGraw-Hill Education, 2005.