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Perfect tilings of simplices
Safernová, Zuzana ; Cibulka, Josef (referee) ; Matoušek, Jiří (advisor)
In the present work we study the problem of k-reptile d-dimensional simplices. A simplex is called a k-reptile if it can be tiled in k simplices with disjoint interiors that are all congruent and similar to S. The only k-reptile simplices that are known for d 3 have k = md, where m 2. We also show an idea of Matoušek's proof of nonexistence of 2-reptile simplices of dimensions d 3. We correct a mistake in the proof. Then we give several geometric observations for k = 2. At the end we prove that there is no 3-reptile simplex for d = 3.
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Ramseyovské otázky v euklidovském prostoru
Cibulka, Josef ; Černý, Jakub (referee) ; Valtr, Pavel (advisor)
One of the problems in Euclidean Ramsey theory is to determine the chromatic number of the Euclidean space. The chromatic number of a space is the minimum number of colors with which the whole space can be colored so that no two points of the same color are at unit distance. We prove that the chromatic number of the six-dimensional real space is at least 11 and that the chromatic number of the seven-dimensional rational space is at least 15. In addition we give a new proof of the lower bound 9 for the chromatic number of the five-dimensional real space. We also simplify the proof of the lower bound 7 for the four-dimensional real space. It is known that the chromatic number of the n-dimensional real space grows exponentially in n. We show some of its subspaces, in which the growth is slower than exponential. We also summarize previous results for normed spaces in general and for some interesting non-Euclidean spaces.
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Comparing chess strategies
Kacz, Kristián ; Babilon, Robert (referee) ; Cibulka, Josef (advisor)
The aim of this work is to provide an overview of approaches in computer chess. It designs and implements a chess engine for multiplayer network chess program ChessNet. Within the engine implements several search algorithms like Negamax, Alpha-beta, Negascout and points to their weaknesses. Adds a possibility to the ChessNet environment to compare chess engines. Compares implemented algorithms in terms of time complexity. Shows several factors wich we have to take into consideration during the design of evaluation function of game states. Implements some such functions and compares them in terms of success against each other.
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Ditchers
Slabý, David ; Cibulka, Josef (referee) ; Lidický, Bernard (advisor)
The aim of this work is to combine the playability and the idea of the legendary game Tunneler with the option of programming arti cial intelligence for computer players. Writing scripts and their use in the game is separated from the game itself, so the author of a script only has to know the scripting language and a few interface functions to successfully create an intelligent robot. At the same time the game is suffciently attractive to casual players, it is easy to operate and has some additional features, such as multiple weapons, maps and robot types. Another important feature is the option of playing over LAN.
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Strategies for generalized Reversi
Tupec, Radek ; Lidický, Bernard (referee) ; Cibulka, Josef (advisor)
The aim of this work is to design generalized Reversi (also known as Othello) and implement the application for editing and testing strategies of artificial intelligence. In the beginning of the work full rules of the game and solved problems are presented. After that follows detailed description of implemented strategies, programmer and user manual. At the conclusion of the work there is the report about using the application and possible extensions of the application.
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Samodlážditelné simplexy
Safernová, Zuzana ; Cibulka, Josef (referee) ; Matoušek, Jiří (advisor)
In the present work we study tetrahedral k-reptiles. A d-dimensional simplex is called a k-reptile if it can be tiled in k simplices with disjoint interiors that are all congruent and similar to S. For d = 2, triangular k-reptiles exist for many values of k and they have been completely characterized. On the other hand, the only simplicial k-reptiles that are known for d 3 have k = md, where m 2 (Hill simplices). We prove that for d = 3, tetrahedral k-reptiles exist only for k = m3. This partially confirms the Hertel's conjecture, asserting that the only tetrahedral k-reptiles are the Hill tetrahedra. We conjecture that k = md is necessary condition for existence of d-dimensional simplicial k-reptiles, d > 3.
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Ramseyovské otázky v euklidovském prostoru
Cibulka, Josef
One of the problems in Euclidean Ramsey theory is to determine the chromatic number of the Euclidean space. The chromatic number of a space is the minimum number of colors with which the whole space can be colored so that no two points of the same color are at unit distance. We prove that the chromatic number of the six-dimensional real space is at least 11 and that the chromatic number of the seven-dimensional rational space is at least 15. In addition we give a new proof of the lower bound 9 for the chromatic number of the five-dimensional real space. We also simplify the proof of the lower bound 7 for the four-dimensional real space. It is known that the chromatic number of the n-dimensional real space grows exponentially in n. We show some of its subspaces, in which the growth is slower than exponential. We also summarize previous results for normed spaces in general and for some interesting non-Euclidean spaces.
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