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Computational complexity of combinatorial problems in specific graph classes
Masařík, Tomáš ; Fiala, Jiří (advisor)
The topic of this diploma thesis is the edge distance labeling problem with specified parametres p, q and λ. We found a dychotomy for p = 2 and q = 1. So the problem is polynomial if λ ≤ 4 and it is NP-complete for λ > 4. The boundary is shifted by one prior to the vertex distance labeling problem, which has already been solved. Polynomial cases are characterized as some special paths and cycles with a few additional vertices. To show NP-completeness we use a well-known NP-complete problem of Monotone not all equal 3-SAT. That section has four parts: One for odd λ, one for even λ and two more reductions for λ = 5 and λ = 6. 1
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Tests of knowledge and skills on websites and their use in combinatorics
Hamáček, Jan ; Robová, Jarmila (advisor) ; Slavík, Antonín (referee)
The goal of this thesis is creation of testing system that works on web pages. Created test system with test excercises serves as instrument to self education of students. Another goal is creating combinatorics tests. This thesis contains summary of basic concepts of didactic tests and properties of created tests. It contains also detailed instructions on how to create test using testing system and short description of possibilities to extend testing system. Last part contains examples of generated combinatorics tests and results of trying the system with students of high school. 1
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Computational complexity of combinatorial problems in specific graph classes
Masařík, Tomáš ; Fiala, Jiří (advisor) ; Dvořák, Zdeněk (referee)
The topic of this diploma thesis is the edge distance labeling problem with specified parametres p, q and λ. We found a dychotomy for p = 2 and q = 1. So the problem is polynomial if λ ≤ 4 and it is NP-complete for λ > 4. The boundary is shifted by one prior to the vertex distance labeling problem, which has already been solved. Polynomial cases are characterized as some special paths and cycles with a few additional vertices. To show NP-completeness we use a well-known NP-complete problem of Monotone not all equal 3-SAT. That section has four parts: One for odd λ, one for even λ and two more reductions for λ = 5 and λ = 6. 1
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Educational environment additive polygons and polyhedrons
Sukniak, Anna ; Jirotková, Darina (advisor) ; Vondrová, Naďa (referee)
Title: Educational environment additive polygons and polyhedrons Summary: The main intention of the work is to introduce a new mathematical educational environment that would be especially attractive for pupils in the grades 6. -9., but also in the secondary schools, universities or primary schools The work consists of six parts. In the introduction are mentioned the reasons that led me to choose this topic. The second chapter describes the theoretical basis of the work. The third section describes in detail the environment of additive polygons, both its aspects - mathematical and educational one. Analogously, as it is in the third chapter, is processed the fourth chapter that is dedicated to the environment of additive polyhedrons. The fifth chapter is devoted to the linking of the environment of additive polygons and polyhedrons into the linear algebra. In conclusion are provided further opportunities of work with this environment.
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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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Mathematics on the chess board
Šperl, Jiří ; Jančařík, Antonín (advisor) ; Pilous, Derek (referee)
TITTLE: Mathematics on the chessboard AUTHOR: Jiří Šperl DEPARTMENT: The Department of mathematics and the teaching of mathematics SUPERVISOR: RNDr. Antonín Jančařík, Ph.D. ABSTRACT: The main subject of my thesis is mathematical problems on the chessboard using chess pieces. The work aims to demonstrate how a secondary school student would approach and solve several typical mathematical tasks of this nature. Consequently, it outlines ways to incorporate chessboard mathematical problems and exercises in mathematical classes. Moreover, the thesis includes a compact collection of solved problems on the chessboard that can serve as an inspiring source of unconventional mathematical tasks in conventional mathematical education. My own mathematical research forms a major part of the thesis. The research was conducted as a series of tests in three school classes. In order to achieve a high de- gree of objectivity classes of students with different specializations were selected to take part in the tests. The participating classes were also of different age groups. The theoretical part of the thesis takes a look at the past of the subject and presents several interesting historical problems concerning the mathematics on the chess- board. Last but not least, the thesis contains a discussion of solutions of the...
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The use of logic in IT security
Švarný, Petr ; Řepa, Václav (advisor) ; Mařík, Vladimír (referee)
This thesis studies the use of dynamic epistemic logics for the sake of information privacy. The core of the work is the synthesis of three approaches: security logics from A. Hommersom, plausibility frames and communication logic from A. Baltag and S. Smets, and studies concerning the so called Russian cards protocol. Thereafter we present a program, made in the NetLogo environment, in order to demonstrate the workings of the basic ideas.
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KAM-DIMATIA Series 2004-685 and ITI Series 2004-206. Two algorithms for general list matrix partitions
Sgall, Jiří ; Feder, T. ; Hell, P. ; Králď, D.
List matrix partitions are restricted binary list constraint satisfaction problems which generalize list homomorphisms and many graph partition problems arising, e.g., in the study of perfect graphs. Most of the existing algorithms apply to concrete small matrices, i.e., to partitions problems, provide algorithms for their solution, and discuss their implications.
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