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Circuits and matchings in graphs
Tesař, Karel ; Pangrác, Ondřej (advisor) ; Šámal, Robert (referee)
O grafu řekneme, že je k-linkovaný, pokud pro každých k dvojic jeho vrchol· existují navzájem disjunktní cesty, které dané dvojice spojují. Existuje vztah mezi k-linkovaností a vrcholovou souvislostí grafu. V této práci hledáme vztah mezi vrcholovou souvislostí grafu a vlastností, že každých k jeho disjunktních hran leží na společné kružnici. Tento problém se dá řešit pomocí k-linkovanosti. Naším cílem je dosáhnout lepších odhad· na souvislost, resp. jiných postačujících podmínek než těch, které jsou známe pro k-linkovanost. 1
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Algebraic curves in history and school
Fabián, Tomáš ; Kvasz, Ladislav (advisor) ; Vondrová, Naďa (referee)
TITLE: Agebraic Curves in History and School AUTHOR: Bc. Tomáš Fabián DEPARTMENT: The Department of mathematics and teaching of mathematics SUPERVISOR: prof. RNDr. Ladislav Kvasz, Dr. ABSTRACT: The thesis includes a series of exercises for senior high school students and the first year of university students. In these exercises, students will increase their knowledge about conics, especially how to draw them. Furthermore, students can learn about two unfamiliar curves: Conchoid and Quadratrix. All these curves are afterwards used for solving other problems - some Apollonius's problems, Three impossible constructions etc. Most of the construction is done in GeoGebra software. All the tasks are designed for students to learn how to work with this software. The subject discussed is put into historical context, and therefore the exercises are provided with historical commentary. The thesis also includes didactic notes, important or interesting solutions of exercises, possible issues, mistakes and another relevant notes. KEYWORDS: conic, circle, ellipse, parabola, hyperbole, conchoid, quadratrix, trisecting an angle, squaring the circle, rectification of the circle, doubling a cube, Apollonius's problem, GeoGebra
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Mascheroni Construction
Kaprasová, Monika ; Jančařík, Antonín (advisor) ; Zamboj, Michal (referee)
This Bachelor's thesis presents basic knowledge about constructions with a compass. It's divided into a historical part which indicates how thinking about constructions with a compass evolved. The next part are Mascheroni's constructions. Three Mascheroni's main problems and other two are listed here. Then there is compiled the basic of the proof of Mascheroni's theorem. This thesis includes series of several constructional tasks with compass only. These tasks are solved in GeoGebra program. Each construction contains a proof of its rightness. The proofs are made as simply as possible so that for anyone who shows his interest is the thesis suitable. Because of it this thesis can serve students for their self-study as teachers for an enrichment of teaching geometry.
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Mascheroni Construction
Kaprasová, Monika ; Jančařík, Antonín (advisor) ; Zamboj, Michal (referee)
This Bachelor's thesis presents basic knowledge about constructions with a compass. It's divided into a historical part which indicates how thinking about constructions with a compass evolved. The next part are Mascheroni's constructions. Three Mascheroni's main problems and other two are listed here. Then there is compiled the basic of the proof of Mascheroni's theorem. This thesis includes series of several constructional tasks with compass only. These tasks are solved in GeoGebra program. Each construction contains a proof of its rightness. The proofs are made as simply as possible so that for anyone who shows his interest is the thesis suitable. Because of it this thesis can serve students for their self-study as teachers for an enrichment of teaching geometry.
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Rotation Number on a Circle
Bíma, Jan ; Vejnar, Benjamin (advisor) ; Pražák, Dalibor (referee)
We apply the dynamical method to obtain structural results concerning certain classes of one-dimensional maps. The notion central to the work is that of a rotation number on a circle; we relate the rotation modulus to periodicity of an orientation-preserving circle homeomorphism and generalize the concept to continuous degree-1 circle maps. We investigate the asymptotic orbit behaviour of circle homeomorphisms with irrational rotation number and develop the Poincaré Classification Theorem which establishes topo- logical (semi-)conjugacy of a circle homeomorphism with an irrational rotation number to a rotation with the same rotation number. 1
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Visual inspection of axial bearing
Sýkora, Vojtěch ; Richter, Miloslav (referee) ; Janáková, Ilona (advisor)
This thesis is about visual control and measurement of bearings by using image sensors and it is about providing appropriate conditions for this measurement. It describes the selection of suitable hardware to solve this case. A large part of the thesis is the design and creation of my own light. Furthermore, algorithms for processing the acquired images of bearings are proposed. The result of the processing is the determination of the type of bearing in the image and the finding of possible defects.
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An Unusual Approach to Circular Inversion
Šebek, Jakub ; Škorpilová, Martina (advisor) ; Boček, Leo (referee)
This bachelor thesis aims to present the topic of circular inversion in a closer way to the non-standard knowledge of high school students actively competing in Mathematical Olympiads. The first chapter describes the topic of antiparallel lines, a relatively common knowledge among such students. The second chapter introduces an antiparallel mapping which is actually a circular inversion, but deduced solely from the properties of antiparallel lines. We consider this way of introduction to be original and closer to the principle of solving more complex olympiad problems using circular inversion. In the following two chapters the topics of power of a point and cross ratio are described and their connection to antiparallel map is shown. In those chapters the circular inversion itself is also introduced and many of its properties are proven. In the last chapter, we solve the Problem of Apollonius and prove the Feuerbach's theorem using inversion. 1
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Mascheroni constructions
Hatschbachová, Jana ; Hromadová, Jana (advisor) ; Škorpilová, Martina (referee)
This bachelor's thesis concerns with the topic of Mascheroni constructions. These are constructions done with compass alone. The thesis contains three chapters, the first is devoted to a brief summary of the history of Mascheroni constructions. The second chapter presents Mohr-Mascheroni theorem and its proof. The third chapter contains a description of several basic Mascheroni constructions including proofs. Part of this work is also a GeoGebra book - Mascheroni constructions, which is divided into two parts. In the first part, there are stepped solutions for all constructions mentioned in the third chapter. The second part contains interactive exercises for each construction.
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Advanced Topics in Plane Geometry
Hajmová, Kateřina ; Štěpánová, Martina (advisor) ; Moravcová, Vlasta (referee)
The aim of this thesis is to introduce a series of knowledge from advanced planimetry, which can be proved using the knowledge of high school geometry. Selected theorems deal with characteristic of squares with a common vertex (Finsler-Hadwiger theorem, Theorem of Four Squares, Bottema's theorem), significant points of plane entities (Gergonne point theorem, Švrček point theorem, Simson's Theorem, Miquel's theorem), Feuerbach's circle and its relation to the Euler's line. In this thesis, there is also mentioned Reim's theorem, Napoleon's theorem, and Thebault's theorem. This thesis contains a lot of illustrations created in the Geogebra Mathematical Software, which are available online in interactive form.
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