MAs01 Selected Topics in Mathematics 1

Faculty of Education
Autumn 2019
Extent and Intensity
0/0/3. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
Mgr. Helena Durnová, Ph.D. (lecturer)
RNDr. Břetislav Fajmon, Ph.D. (lecturer)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Guaranteed by
Mgr. Helena Durnová, Ph.D.
Department of Mathematics – Faculty of Education
Supplier department: Department of Mathematics – Faculty of Education
Prerequisites
fundamenal subjects of the Bachelor's study program for teachers of grades 6 to 9.
Course Enrolment Limitations
The course is only offered to the students of the study fields the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The aim of this course is the enhancement of students' knowledge in algebra and probability theory
Learning outcomes
After the completion of the course the students will a) be able to present some applications of group theory on some geometric constructions using a ruler and compass; b) constuct a code to cipher a message; c) know and be able to present some applications of matrix theory: eigenvalues in solving a system of linear equations; the predator-prey theory and asymptotic behaviour of a solution of a system of differential exuations; d) be able to use descriptive statistics and fundamental types of random variables.
Syllabus
  • 1. Group action on a set: (homomorphism of group G into a group of permutations on a set X). Stabilizer, orbit, Burnside lemma and its applications. Some applications of group action in the theory of groups.
  • 2. Field extentsion. Definition of field extension, its degree, multiplicativity of the degree. Algebraic vs. transcendental elements, minimum polynomial of the algebraic element and its properties. Applications: impossibility of some geometric constructions with ruler and compass, classical problems of the antiquity.
  • 3. Coding and encrypting. Motivation of coding: detection and correction of mistakes emerging during the signal transmission. Polynomial codes over Z_2. Construction of polynomial code for a given number. Motivation of encryption:sending a message through a channel that may be intercepted. Requirement of a signature. Solution: encryption algorithm RS, Rabin encryption system, Diffie and Helman systems of key exchange, ElGamal encryption algorithm.
  • 4. Matrices and their applications. Matrix of a linear transformation of a finite-dimensional vector space, similar matrices, eigenvalues / eigenvectors, Jordan canonical form of a square matrix, characteristic and minimum polynomial of a square matrix, their mutual relationship, Perron and Frobenius theory of primitive matrices. Application: iteration processes, population models (predator-prey,age distribution of a population - Leslie matrix), Markov processes.
  • 5. Descriptive statistics: variable characteristics, arithmetic mean, median and other quartiles, mean value, standard deviation, variance, their calculation and meaning.
  • 6. Random variable: definition, probability function, probability density, distribution function. Basic distribution of random variables (uniform - discrete and continuous, alternative, geometric, binomial, exponential, normal) and their properties. Law of large numbers. Central limit theorem.
Literature
  • BUDÍKOVÁ, Marie. Statistika. 1. vyd. Brno: Masarykova univerzita v Brně, 2004, 186 s. ISBN 8021034114. info
  • BUDÍKOVÁ, Marie, Štěpán MIKOLÁŠ and Pavel OSECKÝ. Teorie pravděpodobnosti a matematická statistika. Sbírka příkladů. (Probability Theory and Mathematical Statistics. Collection of Tasks.). 3rd ed. Brno: Masarykova univerzita, 2004, 127 pp. ISBN 80-210-3313-4. info
  • ADÁMEK, Jiří. Kódování. 1. vyd. Praha: Státní nakladatelství technické literatury, 1989, 191 s. URL info
Teaching methods
Theoretical lectures. Homework.
Assessment methods
Written and oral exam.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: in blocks.
Information on the extent and intensity of the course: 36 hodin.
The course is also listed under the following terms Autumn 2017, Autumn 2018, autumn 2020, Autumn 2022, Autumn 2023.
  • Enrolment Statistics (Autumn 2019, recent)
  • Permalink: https://is.muni.cz/course/ped/autumn2019/MAs01