Algebra and Calculus in Applications

Course guarantor:

Lubomíra Dvořáková, Edita Pelantová

Lecturer:

Lubomíra Dvořáková, Edita Pelantová

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

We illustrate methods based on combination of (CONtinuous) calculus and discrete (disCRETE) structures, so

calles concrete mathematics. Theorems are motivated by problems from informatics and they are illustrated on problems

from informatics.

Requirements:

Active participation, solving homework problems, presenting a topic selected from the list, preparing a professional text about an application of mathematics.

Syllabus of lectures:

1. Difference calculus on sequences

2. Matrix calculus for solving of systems of linear difference equations

3. Words without forbidden subchains

4. Generating functions: ordinary, exponential, Dirichlets function

5. Methods of use of generating functions

6. Applications of generating functions to difference equations

7. Hilberts metric and its applications to linear maps

8. Joint spectral radius

9. Nonlinear recurrence relations

10. Asymptotic growth of solutions of recurrence relations

Syllabus of tutorials:

Study Objective:

Study materials:

Povinná literatura:

[1] P. Cull, M. Flahive, R. Robson, Difference Equations: From Rabbits to Chaos, Springer Verlag (2005)

[2] R. L. Graham; D. E. Knuth; O. Patashnik, Concrete Mathematics. A foundation for computer science, 27. dotisk

(2013)

Doporučená literatura:

[3] J. Jungers: The Joint Spectral Radius: Theory and Applications, Springer Verlag (2009)

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (elective course)
• Matematické inženýrství - Matematická fyzika (elective course)
• Matematické inženýrství - Matematická informatika (PS)
• Matematické inženýrství - Matematické modelování (elective course)
• Mathematical Engineering - Mathematical Informatics (PS)