### DisCont mathematics 1

#### Course Description

Selected topics of discrete mathematics and mathematical analysis, with emphasis on solving complex examples and tasks, based on algorithmic approach.

#### General Competencies

The course enables students to a deeper understanding of the basic modern mathematical structures, mainlz in the field of discrete mathematics, combinatorics, number theory and analysis of algorithms.

#### Learning Outcomes

1. Understand the principles and analysis of complex algorithms.
2. Apply the technique of recursive relations, in various situations.
3. Apply complex terchniques of calculations of finite sums.
4. Analyse the complexity of an algortihm.
5. Undersand the connection between various mathematical structures.
6. Use the technique of generating functions in various situations.
7. Understand the principles of cyphers and coding.
8. Analyze the principles of sorting and searching algorithms.

#### Forms of Teaching

Lectures

Lectures are organized through two cycles. The first cycle consists of 7 weeks of classes and mid-term exam, a second cycle of 6 weeks of classes and final exam. Classes are conducted through a total of 15 weeks with a weekly load of 4 hours.

Exams

Mid-term exam in the 8th week of classes and final exam in the 15th week of classes.

Consultations

Consultations are held one hour weekly according to arrangement with students.

Continuous Assessment Exam
Homeworks 0 % 20 % 0 % 20 %
Class participation 0 % 2 % 0 % 10 %
Seminar/Project 0 % 20 % 0 % 20 %
Mid Term Exam: Written 0 % 40 % 0 %
Final Exam: Written 0 % 40 %
Exam: Written 0 % 80 %

#### Week by Week Schedule

1. Introductory example - The Tower of Hanoi
2. Finite sums
3. Binomial coefficients. Combinatorial identities
4. Walks on integer latices. Generating functions.
5. Binomial series. Polynomial formulae. Increasing and decreasing factoriels. Finite differences.
6. Recursions. Sequences given by recursive formulas. Examples.
7. Fibonacci numbers.
8. Exams
9. Euler and Stirling numbers. Sum of powers. Bernoulli numbers.
10. Elementary inequalities.
11. Means. Inequality between means. Symmetric functions.
12. Euclid's algorithm, divisibility. Relatively prime numbers. Congruences.
13. Prime numbers. Fermat's and Wilson's Theorem. Applications.
14. Basic search and sorting algorithms and their complexity.
15. Exam.

#### Study Programmes

Computing (study)
Courses for exceptionally successful students (4. semester)
Electrical Engineering and Information Technology (study)
Courses for exceptionally successful students (4. semester)
Electrical Engineering and Information Technology and Computing (study)
Courses for exceptionally successful students (2. semester)

#### Literature

M. Aigner (2007.), A Course in Enumeration, Springer
R. Graham, D.E. Knuth, O. Patashnik (2004.), Concrete Mathematics, 2ed, Addison-Wesley
M.W. Baldoni, C. Ciliberto, G.M.P. Cattane (2009.), Elementary Number Theory, Cryptography and Codes, Springer
J. Herman, R. Kučera, J. Šimša (2000.), Equations and Inequalities, Springer
N. Ya. Vilenkin (1971.), Combinatorics, Academic Press

#### General

ID 90094
Summer semester
6 ECTS
L0 English Level
L1 e-Learning
60 Lectures