|
|
Hausdirff metric and its application in fractals
Roháľ, Branislav Ján ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
Title: Hausdorff metric and its application in fractals Author: Branislav Ján Roháľ Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Miroslav Hušek, DrSc., Department of Mathematical Analysis Abstract: In this thesis we focus on the themes naturally connected with the con- cept of a fractal. In the first part of the thesis we pay attention to Banach fixed point theorem and to the Hausdorff metric which are later used when studying self-similar sets. There are included parts on the Hausdorff, similarity, and box- counting dimension, too. In the second part of the thesis the new approaches to fractal dimension and some their properties are refered. We introduce generaliza- tion of this concept for any space admitting a fractal structure and for a distance space where also the "size" of sets on each level of fractal structure is considered. In the last chapter the contribution of new approache is demonstrated, - this enables defining the notion needed and counting fractal dimension where it was not possible under the classical approaches, too. Application to the domain of words and counting of dimensions of a language generated by a regular expresion are presented. Keywords: Hausdorff metric, Banach fixed point theorem, self-similar set, Hausdorff dimension, fractal dimension
|
|
| |
|
| |
|
|
Open problems in Continuum thory
Seifrt, Jan ; Spurný, Jiří (referee) ; Pyrih, Pavel (advisor)
ISYixev prace: Otevfene problemy teorie konlinm Autor: Jan Seifrt Katcdra (ustav): Katedra matematicke analyzy Vedouci ba.kaliirske prace: Doc. RNDr. Pavel Pyrih, CSc. e-mail vedouciho: poliodai6.gniail.com Anstrakt: PiYdlo/ena pracc sc /a))yva v/tahrin iiia^iicli('l\ycb a koiH-rnr pn- riodickych bodu v jislycli koiiipaktiiu-li .souvislych mno/iuach. Teziste prace s])ociva v in)di-()l)in''in ro/boru dvou ])ul)liku\>uiycli vyslcdkii (motor a null- comb). Fungovam Irclito ])fikladu JL- /achycono na fade pomocnydi obraxkii. Pia(.:c obsabnjc polrubnc definicr a /aktadni 1 vr/cm' Inv, dukaxu. V praci JHOU dale doka'/ana i dalsi tvr/oni / dam'1 problcmatiky. vii shiva: dcndril., ill]I' vla.st.imsr, ina^iiutickr body a mill-comb Tillc: Open problems in Continuum thmry Author: Jan Scifrt Department.; Department of Matheinal ical Analysis Supervisor: Doc. HNDr. Pavel Pyrili, CSc. Supervisor's e-mail address: Abstract: In t lit1 present work \ve study the relation between non-wandering ami eventually-periodic- points in certain compact conned ed sets. The goal of the work consists of detailed study of two published results (engine and null-comb). How these examples work is demonstrated by a. sequence of fi- gures. The work contain all needed definitions a.nd lacts wit.lumt proofs. In the work are proved some other...
|
|
| |
|
|
Fundaments of IT thinking
Jelínek, Jakub ; Pyrih, Pavel (advisor) ; Obdržálek, David (referee)
In the propounded work author presents results of his looking for important informatics thinking features, its fundaments. He divides these fundaments into two categories - static and dynamic. Static fundaments represent basic building material that the informatics world is built from. As static fundaments are considered information, time, room and money. Dynamic fundaments express basic "active factors" which are using static fundaments and are transforming them, creating so more complicated elements of informatics. As dynamic fundaments are considered abstraction, iteration and recursion, metasyntactic variables, universality, simplicity and inspiration.
|
|
|
Aplikace Baireovy věty
Peprníková, Ľubica ; Pyrih, Pavel (referee) ; Simon, Petr (advisor)
The aim of this work is to show, having three di®erent spaces and a set of elements with some common property in each one of them that the given set is the set of typical elements in that space. First we will show that a typical continuous function deffined on the interval [0; 1] is a nowhere differentiable one. Then we will show that a typical compact set in R2 is a discontinuum. And lastly, we will show that a typical planar continuum is an indecomposable one. A valuable tool will be the Baire theorem, the use of which will ensure, besides the density, also the fact that the given set is a countable intersection of open sets.
|
|
| |
|
| |
|
|
Extension of mappings into Banach spaces
Novotný, Vojtěch ; Pyrih, Pavel (referee) ; Hušek, Miroslav (advisor)
This diploma thesis deals with extending continuous and uniformly continuous mappings. It studies Lebesgue's and Tietze's work in metric spaces through Urysohn's theorem in normal topological spaces, Kat etovs' papers about uniformly continuous functions up to Dugundji's theorem and relationship between continuous extending of pseudometrics and mappings. It connects the articles of nineteen mathematicians of the twentieth century, presents plenty of theorems in more general form and shows that they could be formulated earlier or proved in another way.
|
guest ::
