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Evolutionary computing
Popelka, Jan ; Smékal, Zdeněk (referee) ; Karásek, Jan (advisor)
The aim of this Bachelor's Thesis was to get acquainted with the Evolutionary Optimization Techniques, mainly with the Genetic Algorithm and Genetic Programming. It was subsequently described the role of optimization problem TSP solved using Genetic Algorithms and other Chapter solving Symbolic Regression using Genetic Programming. This optimalization problems were created in the programming JAVA and there are solved practical part of the thesis.
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Comparison of genomes by synteny block analysis
Pavel, Tomáš ; Škutková, Helena (referee) ; Maděránková, Denisa (advisor)
The theoretical part of this bachelor thesis is aimed at basics of genetics. Term of gene and mutation are introduced in this section. There are gene and chromosome mutations mentioned and described. Following section is devoted to comparative genomics and especially to synteny. There is described what the synteny actually is and how the synteny arises. The end of the theoretical part of this thesis is about the evolution and there are described ways of sorting permutation vectors. The practical part of this bachelor thesis includes description of developed software. Output of this software is a dot-plot which shows detected synteny blocks. Indexes of these blocks are listed in GUI. The second important output is number of permutation steps. This number determines evolutionary distance between two analysed DNA sequences. The very last section is aimed at analyse of synthetic and real DNA sequences.
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Mixing cards and convergence of Markov chains
Drašnar, Jan ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee)
This thesis presents mixing of a deck of cards as a random walk on the group of permutations. Perfectly shuffled deck of cards is defined as uniform distribution on this group. For analysis of the distance between the uniform distribution and the current distribution of the Markov chain generated by the shuffling quite general methods are used that can be applied to many other problems - i.e. strong stacionary time, coupling and transformation to an inverse distribution. In the last chapter the riffle shuffle is studied and a rather well-known fact is proved that seven or eight shuffles should be enough to shuffle a deck of 52 cards.
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Rubik's cube and related puzzles
Chalupa, Radek ; Slavík, Antonín (advisor) ; Halas, Zdeněk (referee)
Title: Rubik's cube and related puzzles Author: Radek Chalupa Department: Department of Mathematics Education Supervisor: doc. RNDr. Antonín Slavík, Ph.D., Department of Mathematics Edu- cation Abstract: This thesis deals with the Rubik's Cube from the viewpoint of mathema- tics. We look into the mathematical rules concerning this famous puzzle and other similar puzzles. Using mathematical tools, we try to answer the question of which Rubik's Cube scrambles can possibly be solved. We learn about various problems preventing us from successfully solving the puzzle. We demonstrate the solution to these problems in a syste- matic way, separately and using examples. We also find out how many different possible Rubik Cube scrambles exist. Keywords: Rubik's cube, solvability, permutations, orientations 1
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Pattern-avoiding permutation classes
Opler, Michal ; Jelínek, Vít (advisor) ; Klazar, Martin (referee)
For a permutation π, the major index of π is the sum of all indices i such that πi > πi+1. In this thesis, we study the distribution of the major index over pattern-avoiding permutations of length n. We focus on the number Mm n (Π) of permutations of length n with major index m and avoiding the set of patterns Π. First, we are able to show that for a singleton set Π = {σ} other than some trivial cases, the values Mm n (Π) are monotonic in the sense that Mm n (Π) ≤ Mm n+1(Π). Our main result is a study of the asymptotic behaviour of Mm n (Π) as n goes to infinity. We prove that for every fixed m, Π and n large enough, Mm n (Π) is equal to a polynomial in n and moreover, we are able to determine the degrees of these polynomials for many sets of patterns. 1
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Combinatorics in mathematics competitions
Kadeřábek, Václav ; Jančařík, Antonín (advisor) ; Zhouf, Jaroslav (referee)
This work analyzes the possibilities of division of combinatorial problems that occur in mathematical competitions. It contains presentation of Combinatorics taught at secondary schools. It shows the differences between solving problems in schools and math competitions. Using graphs and tables, it demonstrates an unbalanced distribution of combinatorial problems. In conclusion, it offers some types of examples that are missing in competitions, or are there in insufficient numbers.
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Mixing cards and convergence of Markov chains
Drašnar, Jan ; Prokešová, Michaela (advisor) ; Beneš, Viktor (referee)
This thesis presents mixing of a deck of cards as a random walk on the group of permutations. Perfectly shuffled deck of cards is defined as uniform distribution on this group. For analysis of the distance between the uniform distribution and the current distribution of the Markov chain generated by the shuffling quite general methods are used that can be applied to many other problems - i.e. strong stacionary time, coupling and transformation to an inverse distribution. In the last chapter the riffle shuffle is studied and a rather well-known fact is proved that seven or eight shuffles should be enough to shuffle a deck of 52 cards.
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Comparison of genomes by synteny block analysis
Pavel, Tomáš ; Škutková, Helena (referee) ; Maděránková, Denisa (advisor)
The theoretical part of this bachelor thesis is aimed at basics of genetics. Term of gene and mutation are introduced in this section. There are gene and chromosome mutations mentioned and described. Following section is devoted to comparative genomics and especially to synteny. There is described what the synteny actually is and how the synteny arises. The end of the theoretical part of this thesis is about the evolution and there are described ways of sorting permutation vectors. The practical part of this bachelor thesis includes description of developed software. Output of this software is a dot-plot which shows detected synteny blocks. Indexes of these blocks are listed in GUI. The second important output is number of permutation steps. This number determines evolutionary distance between two analysed DNA sequences. The very last section is aimed at analyse of synthetic and real DNA sequences.
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Evolutionary computing
Popelka, Jan ; Smékal, Zdeněk (referee) ; Karásek, Jan (advisor)
The aim of this Bachelor's Thesis was to get acquainted with the Evolutionary Optimization Techniques, mainly with the Genetic Algorithm and Genetic Programming. It was subsequently described the role of optimization problem TSP solved using Genetic Algorithms and other Chapter solving Symbolic Regression using Genetic Programming. This optimalization problems were created in the programming JAVA and there are solved practical part of the thesis.
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