1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - drugiego stopnia z tyt. magistra
5. Combinatorial Analysis

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Combinatorial Analysis - opis przedmiotu

Cel przedmiotu

Introducing students to basic definitions, theorems and methods of combinatorial analysis and examples of applications of them.

Wymagania wstępne

Zapisz zmiany

Completed courses of mathematical analysis, linear algebra and discrete mathematics.

Zakres tematyczny

Lecture

1. The binomial coefficients (2 h)
2. Rook polynomials (2 h)
3. Latin squares (2 h)
4. Van der Waerden’s Theorem, Schur’s Theorem (2 h)
5. Map-colourings, Four – Colour Theorem (3 h)
6. Minimax theorems (4 h)
7. Combinatorial designs (2 h)
8. Principles of encoding and decoding, perfect codes, Hamming code, Hadamard matrices, construction of error correcting codes (5 h)
9. Sperner's Lemma (3 h)
10. Minkowski's theorem, Radon's theorem, Helly's theorem, Tverberg's theorem (5 h)

Class

1.  Proving combinatorial identities (2 h)
2. Applications of rook polynomials (3 h)
3. Making latin squares; proving properties of latin squares (3 h)
4. Applications of van der Waerden’s and Schur’s Theorems (2 h)
5. Test (2 h)
6. Applications of Four - Colour Theorem and minimax theorems (4 h)
7. Proving properties of combinatorial designs; applications of combinatorial designs (3 h)
8. Constructing of perfect codes (3 h)
9. Solving problems using Sperner's lemma and the basic theorems of combinatorial geometry learned during the lecture (6 h)
10. Test (2 h)

Metody kształcenia

Traditional lecture, discussion exercises, work in groups.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The grade for the course is the arithmetic average of the grade for the exercises and the grade for the exam. The condition for taking the exam is a positive grade from the exercises. The condition for passing the course is a positive grade in the exam.

Literatura podstawowa

1. V. Bryant, Aspects of Combinatorics. A wide-ranging introduction, Cambridge University Press, 1993.
3. J. Matoušek, Lectures on Discrete Geometry, Springer, New York, 2002.

Literatura uzupełniająca

2. R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics - A foundation for computer science, Oren, 1994.
 

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 21-02-2024 22:26)