### Selected Topics in Mathematics

Data is displayed for academic year: 2023./2024.

#### Lecturers

#### Lectures

#### Course Description

The subject qualifies stuents in deeper understanding of fundamental modern mathematical structures in the area of discrete mathematics, combinatorics, algorithm analysis, probability, numerical mathematics and mathematical modelling.

#### Study Programmes

##### University undergraduate

[FER3-HR] Computing - study

Courses for exceptionally successful students
(4. semester)
(6. semester)
Courses for exceptionally successful students
(4. semester)
(6. semester)
#### Learning Outcomes

- Solve mathematical problems by computer.
- Use mathematical tools in solving problems.
- Write own mathematical software.
- Define new mathematical structures.
- Compare with other colleagues in how fast can the stated problem be solved.

#### Forms of Teaching

**Lectures**4 hours of lectures a week. The lecturer is going to follow the lecture materials.

**Independent assignments**Each student will get a project in one of the three major areas: discrete mathematics, probability or mathematical modelling. The project includes analysis of a problem, studying as well as solving it using the computer.

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

Seminar/Project | 0 % | 25 % | 0 % | 25 % | ||

Mid Term Exam: Written | 0 % | 25 % | 0 % | |||

2. Mid Term Exam: Written | 0 % | 25 % | 0 % | |||

Final Exam: Written | 0 % | 25 % | ||||

Exam: Written | 0 % | 75 % |

#### Week by Week Schedule

- Introduction to the main topics and challenges. Student project presentations. Generation of all subsets of a given set. Different lexicographical orderings.
- Generation of all k-subsets, all functions, injective and surjective functions. Adding constraints and software solutions.
- Permutations. The successor function. Permutations without fixed elements. Integer partitions.
- Bell's numbers. Stirling numbers. Software solutions. Combinatorial designs.
- Introduction to mathematical modelling. Weak formulation of a mathematical model. Introduction to the finite element method.
- Implementation of the finite element method for the 1D problem. Methods of solving linear systems.
- Finite element method for a two-dimensional Poisson problem. GMSH and non-trivial domains.
- Midterm exam
- Weak formulation and finite element solvers for several models from practice.
- Parameter tables for designs. Construction strategies. Difference sets. Introduction to Monte Carlo simulations.
- Introduction to Markov chains. Transition probability matrix. Chapman-Kolmogorov Equations.
- Simulation of Markov chains. Classification of States.
- Limiting distribution. Stationary distribution. Ergodic theorem.
- Project work and presentation.
- Final exam

#### Literature

Donald L. Kreher, Douglas R. Stinson (1998.),

*Combinatorial Algorithms*, CRC Press
Hans Petter Langtangen, Anders Logg (2017.),

*Solving PDEs in Python*, Springer
Sheldon M. Ross (2007.),

*Introduction to Probability Models*, Academic Press#### For students

#### General

**ID**214698

Summer semester

**6**ECTS

**L0**English Level

**L1**e-Learning

**60**Lectures

**0**Seminar

**0**Exercises

**0**Laboratory exercises

**0**Project laboratory

**0**Physical education excercises

#### Grading System

**85**Excellent

**70**Very Good

**55**Good

**45**Sufficient