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Title: Solved Problems in Optics Author: Michaela Jungová Department: Department of Physics Education Supervisor: RNDr. Marie Snětinová, Ph.D., Department of Physics Education Abstract: Twenty problems in optics were created for electronic collection of solved problems in physics in my thesis. These problems contain detailed structured solutions with solved hints and pictures as well. You can find the problems in the chapter "Geometrical optics" on the website: http://reseneulohy.cz and on the enclosed CD. One of the chapters in my thesis deals with description of the problems. The thesis also engages in thesis engages in the way how to classify the problems using different standards. A proposal of a problem classification, that has been used in during study of both elementary and high school physics textbooks and collections, is presented in the thesis. The goals of this classification were two: to get the picture of school curriculum connected to optical problems and to propose suitable structure of the chapter Optics in the electronic collection, which is in the aboved mentioned electronic collection. Keywords: solved problems, classification of physics problems, electronic collection, optics

This work deals with application problems regarding to geometry issues. A deep search has been performed throughout existing excercise books from various periods and of various difficulties. Then, 10 new application problems have been devised and solved, concerning miscellaneous mathematics education stages and geometry branches.

Neste trabalho nós estudamos alguns sistemas de spins e vértices exatamente solúveis em duas dimensões. A solubilidade exata está ligada ao fato de existirem soluções não triviais das equações de fatorização, o que nos permite obter a energia livre no limite termodinâmico. Introduzimos e resolvemos pelo método de espalhamento inverso, um modelo de dez vértices assimétrico com dois e três estados nas ligações. Obtemos o diagrama de fases e mostramos que o sistema exibe uma transição de fase de primeira ordem. Analisamos um modelo de oito vértices de férmions livres e propomos uma nova relação funcional que nos permite calcular a energia livre por vértice. Mostramos que este sistema de vértices corresponde ao modelo de Ising na rede Union Jack. Apresentamos um método de solução de modelos de spin em redes triangulares a partir da solução do mesmo modelo na rede quadrada. O método se aplica sempre que o modelo de spins envolver interação de primeiros vizinhos e satisfizer a relação triângulo-estrela. Estendemos para a rede triangular, as soluções autoduais de Fateev e Zamolodchikov para a rede quadrada, de modelos de spin com simetria Z(N). Analisamos as conjecturas existentes sobre a criticalidade do modelo de Potts definido na rede de Kagomé. Baseados na simetria e nas degenerescências dessa rede conjecturamos uma expressão para a sua linha crítica. / We study some spin and vertex systems which are exactly solvable in two dimensions. The exact solubility is connected to the existence of non trivial solutions of the factorization equations which allow us to determine the free energy in the thermodynamic limit. We introduce and solve by the inverse scattering method, a ten vertex model with two and three states on the links. We get the phase diagram of the system and show that it exhibits a first order phase transition. Analysing a free fermion eight vertex model, we propose a new functional relation which permit us to get the free energy per vertex. We also show that this system is equivalent to the Ising model in a Union Jack lattice. We present a method to solve spin models on triangular lattices from the known solution of the same model on square lattices. The method applies whenever the model involves first neighbours interactions and satisfies the star triangle relation. We extend to the triangular lattice the self dual solutions of Fateev and Zamolodchikov for Z(N) invariant spin systems. We also analyse the conjectures made before for the critical Potts model on a Kagomé lattice. Based on symmetry and on the collapses of this lattice we conjecture an expression for their critical line.

This thesis is a follow-up to several bachelor and diploma theses which were dedicated to creating solved problems for Electronic Collection of Solved Problems in Electromagnetism. The first goal of this thesis was to make a short survey about electronic resources in electromagnetism, especially those which contain solved problems and provide open access to their contents. The second goal was to make a small collection of solved problems in this area which would be suitable for undergraduate students and which would fill in chapters with little amount of problems in the Electronic Collection. This Electronic Collection is openly accesible on the website of Department of Physics Education. Total of 30 solved problems have been made in this thesis, including hints, detailed solutions and suitable pictures. Methodical comments are also available for almost all problems.

Title: Book of problems of kinematics of mass point Author: Jana Moltašová Department: Department of Physics Education Supervisor: RNDr. Dana Mandíková, CSc. Supervisor's e-mail address: Dana.Mandikova@mff.cuni.cz Abstract: This thesis is a follow-up to the bachelor thesis titled Book of problems of kinematics of mass point for students of Physics Education. This work is a part of wider project which aim is to create an electronic collection of problems completed with detailed solutions and structured hints. Within the frame of this work was made part of collection with forty problems, which theme is kinematics of mass point. Most of problems were chosen from already existing collections and textbooks. Each problem contains task, system of hints and resulting solutions of hints, detailed solution and answer. At the beginning of the work are given aims of the work and its structure. In the next part of work is mentioned a survey of curriculum which is instructed in lower and upper secondary school and university in brief, which relates to this theme. The next part is a survey and brief characterization of collections of problems with this theme. The fourth part of the work is devoted to the electronic collection of problems, its history, the technical elaboration of problems, there is described the...

2 Title: Interactive GeoGebra applets for Collection of Solved Problems in Physics Author: Bc. Ondřej Broža Department: Department of Physics Education Supervisor: RNDr. Zdeňka Koupilová, Ph.D., Department of Physics Education Abstract: The aim of the diploma thesis was to focused on designing of interactive applets and desiging of solution to physical problems in the field of electricity and magnetism for Electronic Collection of Solved Problems. The thesis is also focused on the useability of the interactive applets during the physics education. Within this thesis was totaly created 22 interactive applets, a description and questions for working with them and 13 solutions to physical problems. Another parts of the thesis are: manual how to design interactive applets in the GeoGebra system, manual how to integrate these interactive applets into Collection of Solved Problems and library of partial construction in GeoGebra. The last part of the thesis is about verification of usability of these interactive applets in physics education, which was performed via observation of the students during the work with the interactive applets and via feedback questionnaire. Keywords: physics, solved problems, GeoGebra, applets, electricity and magnetism

[pt] Este trabalho possui o intuito de desmistificar a dificuldade encontrada por professores e alunos no ensino e aprendizagem do tópico análise combinatória. A razão que motivou este trabalho foi o fato de que boa parte dos professores de matemática do ensino médio e últimas séries do ensino fundamental consideram a Análise Combinatória como algo complicado de ser ensinado; além da questão das dificuldades de entendimento por parte dos alunos que são induzidos à memorização de fórmulas e a aplicação das mesmas à resolução dos exercícios para compreenderem tal conteúdo. Inicialmente apresentaremos alguns conceitos que servirão como auxílio para que o professor possa trabalhar nas atividades propostas a serem desenvolvidas juntamente com os alunos. E ao longo do trabalho iremos falar de alguns tópicos abordados pela análise combinatória sem, inicialmente, mencionarmos fórmulas que servem apenas para serem memorizadas. O mais importante é fazer o aluno trabalhar um problema sugerido através do roteiro e dos conceitos que serão propostos e ao final de alguns exercícios, quando tal aluno tiver entendido tal conceito, ser anunciado a ele que acabou de aprender e entender o conceito em questão, ao invés de memorizar um determinado exercício ou outro, pois sabemos que desta forma, quando o aluno deparar-se com um novo problema, não será capaz de solucioná-lo. Dessa maneira, elaborou-se um roteiro na solução dos exercícios, ou seja, uma forma do professor trabalhar qualquer atividade proposta que envolva problemas de contagem em sala de aula. Enfim, buscou-se com esse trabalho, apresentar aos docentes, estratégias eficientes que podem ser utilizadas para o ensino de combinatória e ajudar os alunos a compreenderem melhor os problemas de contagem utilizando o raciocínio lógico e de contagem. / [en] This work has the intent to explain the difficulties found by teachers and student on teaching and learning combinatorics. The motivation of this work was the fact that most of the Mathematics Teachers of High School consider combinatorics as something complicated to be taught; contributing as well the fact that students are led to memorize the formulas and apply it on exercises so they can understand the subject. Initially we will show some concepts that will help the Teachers to work together with the students on the proposed activities. During the work, we will talk about Combinatorics topics without mentioning formulas that needs memorization only. The most important thing is to make the student work on a suggested problem following a guide and concepts shown and after finishing a few exercises, when the student will show the understanding of the concepts, the Teacher will tell him that he learned that concept, instead of memorizing a specific exercise. Because we know that not doing this, when this student faces a new problem, he will not be able to solve it. Thus it was elaborated a guide to solve exercises and that means a way that the Teacher can work with any proposed activity that has counting in it. Finally, it was sought with this work to show the scholars some efficient strategies that can be used on teaching Combinatorics and help the students to understand better the problems about counting, using logical reasoning and logical counting.

Neste trabalho nós estudamos alguns sistemas de spins e vértices exatamente solúveis em duas dimensões. A solubilidade exata está ligada ao fato de existirem soluções não triviais das equações de fatorização, o que nos permite obter a energia livre no limite termodinâmico. Introduzimos e resolvemos pelo método de espalhamento inverso, um modelo de dez vértices assimétrico com dois e três estados nas ligações. Obtemos o diagrama de fases e mostramos que o sistema exibe uma transição de fase de primeira ordem. Analisamos um modelo de oito vértices de férmions livres e propomos uma nova relação funcional que nos permite calcular a energia livre por vértice. Mostramos que este sistema de vértices corresponde ao modelo de Ising na rede Union Jack. Apresentamos um método de solução de modelos de spin em redes triangulares a partir da solução do mesmo modelo na rede quadrada. O método se aplica sempre que o modelo de spins envolver interação de primeiros vizinhos e satisfizer a relação triângulo-estrela. Estendemos para a rede triangular, as soluções autoduais de Fateev e Zamolodchikov para a rede quadrada, de modelos de spin com simetria Z(N). Analisamos as conjecturas existentes sobre a criticalidade do modelo de Potts definido na rede de Kagomé. Baseados na simetria e nas degenerescências dessa rede conjecturamos uma expressão para a sua linha crítica. / We study some spin and vertex systems which are exactly solvable in two dimensions. The exact solubility is connected to the existence of non trivial solutions of the factorization equations which allow us to determine the free energy in the thermodynamic limit. We introduce and solve by the inverse scattering method, a ten vertex model with two and three states on the links. We get the phase diagram of the system and show that it exhibits a first order phase transition. Analysing a free fermion eight vertex model, we propose a new functional relation which permit us to get the free energy per vertex. We also show that this system is equivalent to the Ising model in a Union Jack lattice. We present a method to solve spin models on triangular lattices from the known solution of the same model on square lattices. The method applies whenever the model involves first neighbours interactions and satisfies the star triangle relation. We extend to the triangular lattice the self dual solutions of Fateev and Zamolodchikov for Z(N) invariant spin systems. We also analyse the conjectures made before for the critical Potts model on a Kagomé lattice. Based on symmetry and on the collapses of this lattice we conjecture an expression for their critical line.

The main aim of this thesis was to create a set of problems in Quantum Mechanics, which would be included in a collection of solved problems in Physics available online on the website http://www.fyzikalniulohy.cz. As a result, thirty nine problems were completed, covering seven thematic areas partially of high school and mostly of introductory college Quantum Mechanics course. In general, the collection is mainly designed for college students as an exercise in basic quantum-mechanical problems and also for high school students, who have a desire to deal with some special problems. A detailed description of content, structure, and technical execution of this collection can be found in chapter 1 of this thesis. Another task was to review existing online collections of applets dealing with Quantum Mechanics. A selection of them is introduced in chapter 2 of this thesis.