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Web application for teaching algebraic equations in complex numbers
Kneblík, Radovan ; Robová, Jarmila (advisor) ; Staněk, Jakub (referee)
This thesis deals with algebraic equations in the field of complex numbers and teaching of them in high school mathematics. The thesis is a web page, it includes the definition of complex numbers in various forms (form of ordered pair, algebraic, polar and exponential form) and operations with them, as well as conversions between forms. The main part of the thesis includes solving of linear, quadratic, binomial, trinomial, reciprocal and cubic equations in the field of complex numbers. The lecture is enhanced with interactive applets demonstrating given topic. The thesis also includes solved exercises, as well as exercises leading to sets of points satisfying a condition and two overall tests. Web page created within the thesis can be used by high school pupils and teachers. 1
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Series
Klosse, Patricie ; Staněk, Jakub (advisor) ; Rmoutil, Martin (referee)
This bachelor thesis focuses on the sums of series. The first part introduces the reader to the historical background and explains the basic theory of series. Then, the Cesaro summation method is introduced. Thethesis also contains demonstrations of individual sums of the series introduced in the paper. 1
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Complex numbers: definition and geometrical applications
Helus, Jiří ; Halas, Zdeněk (advisor) ; Staněk, Jakub (referee)
The thesis describes the introduction of complex numbers in teaching at secondary school, highlights problems that are associated with their introduction and mentions the possible use of complex numbers, espe- cially in geometry. The initial motivational considerations are followed by a brief introduction of complex numbers into the historical context. When introducing complex numbers, we take into account didactic aspects and draw attention to possible problems of interpretation. The chapter is sup- plemented with a reference to hyper complex numbers (quaternions, octets). Furthermore I show how it is possible to geometrically illustrate operations with complex numbers, namely addition, subtraction, multiplication and di- vision. The thesis also describes how Moivre's theorem can be interpreted by rotation. The following part is an analytical geometry built with complex numbers, focusing on point, line, circle, ellipse, and triangle. Next, we search for the square root of the complex number and the solution of the quadratic equation graphically. Finally, we prove Napoleon's theorem and existence of Feuerbach's circle using complex numbers. 1
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Studium vedení tepla metodami počítačového modelování
STANĚK, Jakub
This diploma thesis is concerned with a problem of the line heat simulation by different kinds of materials with computer - aided COMSOL Multiphysics. The thesis is composed of three thematic units. In the first part there is a line heat principle shortly described and his basic physical quantities which are necessary for correct defining of the simulation. The second part is concerned with the software, its function, auxiliary modulus and a description of single actions, which are essential for making of the model. In the last part there is the whole process of the multiphysical task creation that enables to simulate line heat in a specific material in reliance on time.
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Poker
Jelínek, Roman ; Staněk, Jakub (advisor) ; Slavík, Antonín (referee)
At the beginning of this thesis some necessary information about Texas Hold'em will be remembered - game rules, the probability of winning at the flop and turn. The next chapter is about calculating expected value and about successability of players in the long run. In the final chapter we will focus at basic strategies that players can choose. The reader will know which terms influence the choice of an adequate strategy and why. The main message is to show that only mathematical skills - that a good player should have - are not enough. More important are skills associated with the poker itself. Poker skills are mainly thought of: adjusting the strategy by position, working with the stack, choosing the right amount of bets, matching the opponent's typology, and so on. Overall, the thesis is mainly focused on didactics. There is an effort to bring poker to the wider community, with real examples. Poker players should be satisfied, because there are many useful tips to help them improve. Keywords: Texas Hold'em rules, the probability of winning, expected value, game strategy.
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Approach to secure the population knowledge of power outages in chosen states
STANĚK, Jakub
Today's globalized world is much more vulnerable than we are often willing to admit. We live in a society which is considerably dependent on a stable supply of electricity. Potential large power outage also called blackout may be a very dangerous situation for our community. Usually we don't realize all consequences there can be for our everyday life, people's health, the security and the economy of the country. Despite the efforts of the responsible authorities, the complications for the population can be very serious. The effects of such unfortunate event can be reduced in a variety of ways. An option is to support an informed society that can take care of itself in times of crisis. This text is focused on the current approach to raising and maintaining awareness of the people for masive power outage not only in the Czech Republic. The aim of the thesis is to compare the information environment of the Czech Republic with the situation in neighboring Germany. It is important to keep on mind that many important systems serving under normal conditions to inform the population will not work when such an event occurs. It can be necessary to proceed with alternative solutions and communication channels. In the education context, it's realisic to inform the majority of the population only about the most important measures and appropriate patterns of blackout behavior. With the restoration of energy supplies, the task of informing the population doesn't end. By selecting key information, this work could contribute to the society.
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Probability on high school
Kremlová, Štěpánka ; Staněk, Jakub (advisor) ; Rmoutil, Martin (referee)
The diploma thesis deals with probability at secondary schools. In the first chapter of the thesis we find a summary of the theory of probability in the range of Czech secondary school textbooks. The second chapter deals with the probability in curriculum documents. In the following chapters, two Czech secondary school textbooks are first compared, then we will focus on foreign textbooks. Finally, some examples are presented in detail. The work provides readers with a range of possible strategies for solving tasks, including graphical methods. 1
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Proofs
Hofman, Jakub ; Staněk, Jakub (advisor) ; Rmoutil, Martin (referee)
Title: Proofs Author: Jakub Hofman Department: Department of Mathematics Education Supervisor: RNDr. Jakub Staněk, Ph.D., Department of Mathematics Edu- cation Abstract: This bachelor thesis should clarify the concept of a mathematical proof, briefly describe the individual proof methods and provide examples of proved theorems from various branches of mathematics. Another objective of this thesis is to provide a high school pupil with a study material which with it structure corresponds to mathematical textbooks and lecture notes provided by universities. In the first part of the thesis the basic terms of logic are introduced to the pupil. These are terms a pupil can usually encounter in math classes at school. The knowledge of these terms is essential for understanding the principles of the individual proof methods. The main part of the thesis is focused on explaining the individual proof methods and their application in proving mathematical theorems with di- fficulty appropriate to a pupil of secondary education. The theorems are ordered by the difficulty of their proof; they do not follow from one another nor do they form a coherent mathematical theory as that is not the goal of this thesis. The thesis also contains a list of theorems which a pupil should be ca- pable of proving after being acquainted with...
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Combinatorial triangles
Proner, Matúš ; Slavík, Antonín (advisor) ; Staněk, Jakub (referee)
In combinatorics, there are several types of numbers which can be neatly arranged into triangular schemes. Important examples are binomial coefficients of the first kind, binomial coefficients of the second kind and the Lah numbers. The aim of the thesis is to introduce these numbers and derive their basic properties. Some interesting features are shown in relevant triangular schemes.
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Combinatorial sequences and divisibility
Michalik, Jindřich ; Slavík, Antonín (advisor) ; Staněk, Jakub (referee)
This work contains an overview of the results concerning number-theoretic pro- perties of some significant combinatorial sequences such as factorials, binomial coef- ficients, Fibonacci and Catalan numbers. These properties include parity, primality, prime power divisibility, coprimality etc. A substantial part of the text should be accessible to gifted high school students, the results are illustrated with examples. 1
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