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Cavalieri's principle
Kreslová, Iva ; Halas, Zdeněk (advisor) ; Štěpánová, Martina (referee)
The Bachelor thesis deals with the development of key ideas important for formal formulating of Cavalieri's principle, its proof in general form and using Cavalieri's principle in determining the area of plane figures and volumes of solids. Determining of area and volumes using Cavalieri's principle is associated with the derivation of the well-known formulae for calculating area and volume.
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On the interior of a minimal convex polygon
Šplíchal, Ondřej ; Valtr, Pavel (advisor) ; Rataj, Jan (referee)
Zvolme konečnou množinu bod· P v rovině v obecné poloze, tj. žádné 3 body neleží na přímce. Konvexní n-úhelník je minimální, pokud v jeho konvexním obalu neleží jiný konvexní n-úhelník s vrcholy v P. Erd®s a Szekeres (1935) ukázali, že pro každé n ≥ 3 existuje minimální číslo ES(n) takové, že mezi libovolnými ES(n) body v rovině v obecné poloze lze vybrat n bod·, které tvoří vrcholy konvexního n-úhelníku. Z jejich tvrzení vyplývá, že v topologic- kém vnitřku minimálního konvexního n-úhelníku m·že ležet jen omezený po- čet bod· P pro libovolnou volbu P. Označíme maximální takový počet jako mci(n). V práci ukážeme horní odhad mci(n) ≤ ES(n) − n a spodní odhad 2n−3 − n + 2 ≤ mci(n) pro n ≥ 3.
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Jordan Curve Theorem
Dudák, Jan ; Vejnar, Benjamin (advisor) ; Kurka, Ondřej (referee)
Title: The Jordan Curve Theorem Author: Jan Dudák Department: Department of Mathematical Analysis Supervisor: Mgr. Benjamin Vejnar, Ph.D., Department of Mathematical Analysis Abstract: The crucial part of this work is the proof of the Jordan curve theorem. To this end, the work starts by introducing necessary notions (e.g. a curve or an arc) and showing their basic properties. Further on, the Brouwer fixed point theorem is proved (in the 2-dimensional case) as well as some of its corollaries which are then used (together with several other proven assertions) in the proof of the Jordan curve theorem. The last chapter of this work briefly informs about possible generalisations of the Jordan curve theorem, referencing to appropriate bibliography. Keywords: Jordan curve, arc, plane, connected component 1
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Extremal Polyominoes
Steffanová, Veronika ; Valtr, Pavel (advisor) ; Cibulka, Josef (referee)
Title: Extremal Polyominoes Author: Veronika Steffanová Department: Department of Applied Mathematics Supervisor: Doc. RNDr. Pavel Valtr, Dr. Abstract: The thesis is focused on polyominoes and other planar figures consisting of regular polygons, namely polyiamonds and polyhexes. We study the basic geometrical properties: the perimeter, the convex hull and the bounding rectangle/hexagon. We maximise and minimise these parameters and for the fixed size of the polyomino, denoted by n. We compute the extremal values of a chosen parameter and then we try to enumerate all polyominoes of the size n, which has the extremal property. Some of the problems were solved by other authors. We summarise their results. Some of the problems were solved by us, namely the maximal bounding rectan- gle/hexagon and maximal convex hull of polyiamonds. There are still sev- eral topics which remain open. We summarise the literature and offer our observations for the following scientists. Keywords: Polyomino, convex hull, extremal questions, plane 1
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Viditelnostní grafy
Král, Karel ; Valtr, Pavel (advisor) ; Balko, Martin (referee)
In the thesis we study visibility graphs focusing on the Big Line Big Clique conjecture. For a given finite point set P in real plane we say that two points see each other if and only if the open line segment between them contains no point from P. Points from P are vertices of the visibility graph, and two points are connected by an edge if and only if they see each other. Kára et al. conjectured that for every finite big enough point set there are at least ℓ collinear points, or the clique number of its visibility graph is at least k. In the thesis we generalize the conjecture, and thus provide an alternative proof for k = ℓ = 4. We also review related known results. We strengthen an observation about occurrence of a Hamiltonian cycle in visibility graphs. We characterize the asymptotic behavior of the edge chromatic number of visibility graphs. We show that for given n, ℓ, k the original conjecture is decidable by a computer. We also provide computer experiments both for the generalized and for the original conjecture. 1
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Design installation of motors ROTAX for aircraft SKYLEADER 400
Rakušan, Miroslav ; Katrňák, Tomáš (referee) ; Mališ, Michal (advisor)
The bacheloir thesis is focused to calculation forces in engine mount, design and construction of engine mount for engines Rotax 912iS and 914UL, which will be instalated to ultralight plane Skyleader 400. In the first part is short search about planes, which use the engines. Then the engine mount is calculate according to rule UL2-I.part. After the MKP analysis is calculation of screws, which hold the engine mount onto the firewall. In the end of the thesis is installation both of engines in the nose of the plane and engine mount test draft according to rule UL2-I.part.
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Grid representations of graphs and the chromatic number
Balko, Martin ; Valtr, Pavel (advisor) ; Kratochvíl, Jan (referee)
Grid Representations and the Chromatic Number Martin Balko August 2, 2012 Department: Department of Applied Mathematics Supervisor: doc. RNDr. Pavel Valtr Dr. Supervisor's email address: valtr@kam.mff.cuni.cz Abstract In the thesis we study grid drawings of graphs and their connections with graph colorings. A grid drawing of a graph maps vertices to distinct points of the grid Zd and edges to line segments that avoid grid points representing other vertices. We show that a graph G is qd -colorable, d, q ≥ 2, if and only if there is a grid drawing of G in Zd in which no line segment intersects more than q grid points. Second, we study grid drawings with bounded number of columns, introducing some new NP- complete problems. We also show a sharp lower bound on the area of plane grid drawings of balanced complete k-partite graphs, proving a conjecture of David R. Wood. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures of D. Flores Pe˝naloza and F. J. Zaragoza Martinez. Keywords: graph representations, grid, chromatic number, plane
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Technological stage of the gross superstructure of the apartment house in Prague - Dablice
Kucsera, Adam ; Kopecký, Lukáš (referee) ; Šlanhof, Jiří (advisor)
The work adresses the technological stage of the gross superstructure of the apartment house in Prague – Dablice. The house consist of reinforced concrete structure on the ground floor, three other floors in a brick form of pramic blocks Porotherm, ceilings above the second and third floor are composed of Spiroll prefabricated ceiling panel and roof structure is made up of gang nail trusses, thanks to this composition of house, it was necessary to address how to storage materials on the site, and also the arrival, departure and parking of vehicles. Technological regulativ is focused on concreting of the reinforced concrete monolithic framework.
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Fast Implementation of Geometric Algorithms
Krba, Martin ; Jošth, Radovan (referee) ; Havel, Jiří (advisor)
One of the most often used calculations in computer graphics is detection of collision between ray of light and triangle on the surface of object in 3D space. This frequent usage is motivation to search for the best methods for this calculations. This thesis contains explanation of basics of this topic and also testing of methods with real input data to better and accurate comparison.
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Information System for Robotic Arm Workcell at FIT BUT
Fekiač, Filip ; Samek, Jan (referee) ; Luža, Radim (advisor)
The aim of this work is to create an information system that will allow exclusive access to the workplace robotic arm by making a reservation timer. The information system provides the possibility of planning for the future bookings. His project was implemented in PHP using relational database systems supported by PDO library. Testing focused on the detection of errors in mass access a large number of users and determine the size of the necessary system resources. The result of the process is a web application that has a defined interface to the server robotic arm.
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