# Selected topics in Mathematics

## Course 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)

-wavelets

Discrete mathematics:
-graphs, basics.

-matchings and coverings in bipartite graphs, duality, Hall marriage condition, stable matchings

-flow problems in networks, maximum flow and minimum cut, Ford/Fulkerson theorem, duality, flow integrality

-linear programming, simplex method, primal and dual programs, applications

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

## 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.

## Learning and teaching 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.

## Intended learning outcomes

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.

## Reference nosilca

1. DOLINAR, Gregor, MOLNÁR, Lajos. Sequential endomorphisms of finite-dimensional Hilbert space effect algebras. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2012, vol. 45, no. 6, 065207 (11 str.).
2. DOLINAR, Gregor, KUZMA, Bojan, OBLAK, Polona. On maximal distances in a commuting graph. The electronic journal of linear algebra, ISSN 1081-3810, 2012, vol. 23, str. 243-256.
3. DOLINAR, Gregor, MOLNÁR, Lajos. Automorphisms for the logarithmic product of positive semidefinite operators. Linear and Multilinear Algebra, ISSN 0308-1087, 2013, vol. 61, no. 2, 161-169.
4. DOLINAR, Gregor, HOU, Jin Chuan, KUZMA, Bojan, QI, Xiaofei. Spectrum nonincreasing maps on matrices. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2013, vol. 438, iss. 8, str. 2504-3510
5. DOLINAR, Gregor, GUTERMAN, Aleksandr Emilevič, KUZMA, Bojan, OBLAK, Polona. Commuting graphs and extremal centralizers. Ars mathematica contemporanea, ISSN 1855-3966. [Tiskana izd.], 2014, vol. 7, no. 2, str. 453-459.

## Study materials

[1] M. Pedersen, Functional Analysis in Applied Mathematics and Engineering, Chapman & Hall/CRC, 1999.

[2] J. T. Oden, L. Demkowicz, Applied Functional Analysis, CRC Press, 2010.

[3] R. Diestel, Graph Theory, Springer-Verlag, GTM 173, 3. izdaja, 2005.
[4] J. M. Kleinberg, Éva Tardos, Algorithm design, Addison-Wesley, 2006.

[5] B. S. Jovanović, E. Süli, Analysis of finite difference schemes, Springer, 2014.

[6] J. N. Reddy, An Introduction to the Finite Element Method (Engineering Series), McGraw-Hill Education, 2005.

## Bodi na tekočem

Univerza v Ljubljani, Fakulteta za elektrotehniko, Tržaška cesta 25, 1000 Ljubljana