<p>In statistical inference of the distribution-free straight-line regression problem, two common transformations, rank transformation and sign transformation, are used to construct the test statistics. When shall we need to use the transformations and which transformation is more efficient are two common questions met by researchers. In this thesis, we will discuss the comparison of the efficiencies of the statistics before and after the rank transformation or the sign transformation in both theoretical and practical ways. Simulation is also used to compare the efficiencies of the statistics under different distributions. Some recommendations about when to use transformations and which one to choose are put forward associated with the conclusion drawn from the research work we have done.</p>

In statistical inference of the distribution-free straight-line regression problem, two common transformations, rank transformation and sign transformation, are used to construct the test statistics. When shall we need to use the transformations and which transformation is more efficient are two common questions met by researchers. In this thesis, we will discuss the comparison of the efficiencies of the statistics before and after the rank transformation or the sign transformation in both theoretical and practical ways. Simulation is also used to compare the efficiencies of the statistics under different distributions. Some recommendations about when to use transformations and which one to choose are put forward associated with the conclusion drawn from the research work we have done.

This thesis consists of two parts: digital geometry and discrete optimization. In the first part we study the structure of digital straight line segments. We also study digital curves from a combinatorial point of view. In Paper I we study the straightness in the 8-connected plane and in the Khalimsky plane by considering vertical distances and unions of two segments. We show that we can investigate the straightness of Khalimsky arcs by using our knowledge from the 8-connected plane. In Paper II we determine the number of Khalimsky-continuous functions with 2, 3 and 4 points in their codomain. These enumerations yield examples of known sequences as well as new ones. We also study the asymptotic behavior of each of them. In Paper III we study the number of Khalimsky-continuous functions with codomain Z and N. This gives us examples of Schröder and Delannoy numbers. As a byproduct we get some relations between these numbers. In Paper IV we study the number of Khalimsky-continuous functions between two points in a rectangle. Using a generating function we get a recurrence formula yielding this numbers.   In the second part we study an analogue of discrete convexity, namely lateral convexity. In Paper V we define by means of difference operators the class of lateral convexity. The functions have plus infinity in their codomain. For the real-valued functions we need to check the difference operators for a smaller number of points. We study the relation between this class and integral convexity. In Paper VI we study the marginal function of real-valued functions in this class and its generalization. We show that for two points with a certain distance we have a Lipschitz property for the points where the infimum is attained. We show that if a function is in this class, the marginal function is also in the same class. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Submitted. Paper 5: Manuscript. Paper 6: Manuscript.

Visão é um processo que envolve uma grande quantidade de informações, as quais precisam ser otimizadas de alguma forma para propiciar um processamento eficiente. Grande parte das informações visuais estão contidas nos contornos de uma imagem e uma grande redução no volume dos dados pode ser conseguida com a análise dos contornos. Além dos contornos, a detecção de segmentos de reta é o próximo passo na compressão das informações visuais. A detecção de retas ocorre no sistema visual humano, e também no de outros seres vivos. Entre os invertebrados terrestres, o melhor sistema de visão é o das aranhas da família Salticidae e este apresenta características que facilitam a detecção de retas. Este trabalho propõe um novo método de detecção de retas, baseado no sistema visual das aranhas saltadoras, que aborda este problema através de um enfoque inédito, por otimização. O método realiza a busca por retas através de janelas lineares. Para isso, a detecção de retas será feita em um espaço de parâmetros, com a utilização do algoritmo de maximização de funções \"Downhill Simplex\". O método desenvolvido leva em consideração a natureza discreta da imagem e do espaço de parâmetros utilizado, e este trabalho inclui um estudo detalhado destes espaços discretos. O método incorpora, para lidar adequadamente com as peculiaridades do problema, características como \"Simulated Annealing\" e largura adaptativa. O desempenho do método depende de um conjunto de parâmetros cujo comportamento é de difícil previsão, e a escolha de um conjunto foi realizada através de um algoritmo genético. O trabalho envolve também a construção de um protótipo para a realização de testes utilizando o método desenvolvido. Os resultados foram analisados quanto a precisão na detecção de retas, ao tempo de processamento e a movimentação das janelas lineares, relacionada aos esforços na busca por retas. / Vision is a process that involves a large amount of information that need to be somehow optimized to allow efficient processing. Most of the visual information is contained in the contours of an image and a considerable reduction in the amount of data can be achieved by fmding and processing these contours. The next step to further compress the visual data is to fmd straight segments, and represent the contours in terms of these entities. Straight-line segment detection is performed by the human visual system, as well as by other creatures. Among terrestrial invertebrates, the best visual system is that of the Salticidae family of spiders, also known as jumping spiders. This visual system presents some characteristics that facilitate the detection of straight-lines. The present work proposes a new method for straight-line detection, based on the visual system of the jumping spiders, using linear windows. This method approach the straight-line detection problem through an optimization point of view yet unexplored in literature. The detection will be accomplished in a parameter space, using the \"Downhill Simplex\" maximization algorithm. The method considers the discrete nature of both the image and the parameter spaces, and this work includes a detailed analysis of these discrete spaces. The method also incorporates, to adequately deal with the specific characteristics of the problem, resources such as \"Simulated Annealing\" and adaptive width of the linear windows. The performance of the method depends on a set of parameters, which behavior is hard to predict, and the choice of an adequate set was made using a genetic algorithm. The work also involves the project and construction of a prototype, to evaluate the proposed method. Results were analyzed regarding their precision, processing time and the movements of the linear windows, related to the effort made to detect the straight lines.

Lärare som undervisar i matematik förväntas kunna mer avancerad matematik än vad de undervisar om. Men formell matematikkunskap anses inte vara tillräckligt för att lärare ska kunna undervisa så att ämnesinnehållet blir begripligt för eleverna, de behöver även pedagogical content knowledge (PCK). Begreppet belyser en speciell form av ämneskunskap för undervisning och skiljer sig från den matematikkunskap som används av andra välutbildade vuxna. Det har föreslagits att olika arrangemang av kollegialt och praktikbaserat lärande kan utveckla lärares PCK. Ett exempel på ett sådant arrangemang är learning study. Den här avhandlingen handlar om den kunskap om lärande och undervisning i matematik som studiens lärare utvecklar då de deltar i learning studies och utforskar sin praktik utifrån ett variationsteoretiskt perspektiv. Det yttersta syftet med en learning study är att utveckla elevernas lärande om specifika lärandeobjekt, genom att undersöka vad som kan vara kritiskt för elevernas lärande. I ett samarbetsprojekt med fyra högstadielärare genomfördes två learning studies i matematik, under ett år. Lärargruppen undersökte vad eleverna behöver lära för att de ska förstå i) varför en kvot kan vara större än talet i täljaren och ii) olika representationer av konstanterna k och m i räta linjens ekvation. Under learning study-arrangemangets olika steg samlades studiens empiri in och denna består av filmade lektioner, inspelade möten där lärargruppen planerade och analyserade undervisning och elevers lärande, skriftliga elevtest samt elevintervjuer. Studien har en variationsteoretisk utgångspunkt, vilket innebär att lärande förklaras ske när en person ser något på ett nytt och mer kvalitativt sätt, genom att personen urskiljer aspekter som han/hon inte tidigare har urskilt. Studien visar de två lärandeobjektens kritiska aspekter samt hur de kritiska aspekterna gradvis förändrades och specificerades. Förändringen var ett resultat av att lärargruppen fick syn på avgörande detaljer om på vilket sätt eleverna förstod ämnesinnehållet samt hur skilda sätt att förstå kunde användas i undervisningen för att utveckla elevernas lärande. Där av titeln att få syn på avgörande skillnader. Denna form av utvecklad kunskap om lärandeobjektet kan ses som ett bidrag om PCK och vad det kan vara. / It is a common view that teachers need more than formal content knowledge to teach and to make the content comprehensible to others. They also need pedagogical content knowledge, or PCK (Shulman, 1986). It has been suggested that different teacher collaboration approaches may support teachers’ development of PCK (Chapman, 2013, Davis &amp; Renert, 2014; Steele &amp; Rogers, 2012). This thesis aims to provide insights into the kind of knowledge about teaching and learning mathematics that teachers develop through their participation in a specific collaboration approach called learning study. Four teachers of mathematics and their 74 students (aged 15−16 years) participated in two learning studies over the course of one year. The foremost aim of a learning study is to enhance student learning about specific objects of learning and to identify what is critical for the students’ learning (Marton &amp; Tsui, 2004). The objects of learningin the two learning studies were to understand that dividing with a denominator between 0 and 1 gives a quotient larger than the numerator and to understand different representations of the constants b and m in the equation of the straight line. During the two learning studies data were collected from 8 video-recorded lessons, 2 written student tests, student interviews, and 14 audio-recorded sessions in which the teachers and I (PhD student) planned, analysed and revised teaching and student learning. The analysis was based on variation theory (Marton &amp; Tsui, 2004) and focused on what participants considered to be critical aspects of the objects of learning and on the components embedded in that knowledge. The result shows the identified critical aspects of the two objects of learning and, furthermore, how the teachers’ knowledge about those critical aspects gradually changed and became more refined and specified in relation to their students’ understanding. The thesis provides an insight into the value of the teachers’ enhanced knowledge of the object of learning, in relation to how PCK can be understood.

Nos ultimos anos o uso de tecnicas de aprendizado computacional tornou se uma das tarefas comumente realizadas, pois tem inumeras aplicacoes de reconhecimento de padroes, tais como: reco- nhecimento de voz, classificacao de texto, reconhecimento facial, diagnostico por imagens medicas, entre outras. Dessa forma, um grande numero de tecnicas que lidam com este tipo de problema tem sido desenvolvido ate o momento. Neste trabalho apresentamos uma alternativa para melhorar a taxa acerto de classificacao do classificador binario SLS, que apresentou resultados comparaveis com as SVMs. Nesse metodo, o Gradiente Descendente e utilizado para otimizar a posicao final dos conjuntos de segmentos de reta que representarao cada classe. Embora convirja rapidamente a um valor otimo, muitas vezes e possivel o algoritmo parar em uma regiao de otimos locais, que nao representa o minimo global. Dado esse problema, foram utilizados diferentes algoritmos evolutivos em combinacao com o Gradiente Descendente a fim de melhorar a acuracia do classificador SLS. Adicionalmente a aplicacao de algoritmos evolutivos na fase de treinamento do classificador SLS, foram exploradas duas propostas: (i) explorar o uso de diferente numero de segmentos de reta para representar a distribuicao de dados de cada classe. Dado que no algoritmo original do metodo SLS o numero de segmentos de reta e igual para cada classe, o qual pode significar alguma perda de acuracia ou sobreposicao dos segmentos de reta; (ii) estimar a melhor combinacao de segmentos de reta a serem usados para cada classe. O uso de diferentes quantidades de segmentos de reta por classe pode ser de ajuda na obtencao de melhores porcentagens de acerto, mas determinar uma quantidade otima que permita representar cada classe, e um trabalho dificil. Assim, usamos o algoritmo X-Means, que e um algoritmo de agrupamento, para estimar o numero de segmentos de reta. As propostas exibiram bons resultados que possibilitam a aplicacao do classificador SLS, com um algoritmo de treinamento hibrido, em problemas reais. / During the past years, the use of machine learning techniques have become into one of the most frequently performed tasks, due to the large amount of pattern recognition applications such as: voice recognition, text classification, face recognition, medical image diagnosis, among others. Thus, a great number of techniques dealing with this kind of problem have been developed until now. In this work, we propose an alternative training algorithm to improve the accuracy of the SLS binary Classifier, which produces good results that can be compared to Support Vector Machines. In that classifier, the Gradient Descent method has been used to optimize the final positions of two sets of straight line segments that represent each class. Although, this method quickly converges to an optimum, it is possible that the algorithm stops at a local optimum region, which does not guarantee a global minimum. Given that problem, we combine evolutive optimization algorithms with the gradient descent method to improve the accuracy of the SLS Classifier. In addition to our proposal of using evolutive algorithms, we also developed two proposals: (i) we explore the use of different number of straight line segments to represent the data distribution. Since the original SLS classifier algorithm uses the same number of segments for each class, which could lead to a loss of accuracy or straight line segments overlapping. So, using different number of segments could be the way to improve the accuracy; (ii) estimate the best combination of straight line segments to represent each class. Finding an optimal combination, can be a very difficult problem, so we propose the X-Means algorithm to determine the number of segments. The proposed methodology showed good results which can be used to solve some other real problems with the SLS classifier using the proposed hybrid training algorithm.

[pt] O desenvolvimento tecnológico, cada vez mais acelerado, possibilitou o aperfeiçoamento das mídias disponíveis em um nível nunca antes visto, contribuindo significativamente para a transmissão de diversos conteúdos aos discentes no âmbito escolar como, por exemplo, os pontos notáveis de um triângulo e a Reta de Euler trazendo vários conceitos como: mediana, baricentro, bissetriz, incentro, ortocentro, mediatriz e o circuncentro, sendo este último o foco deste estudo que busca apresentar a proposta da utilização do vídeo A Comunidade: Circuncentro do triângulo, como forma de aperfeiçoar o conhecimento deste conceito de uma maneira dinâmica, através da solução de um problema prático sobre o local mais ideal para a construção de uma horta, sem que a mesma cause problemas às fazendas vizinhas. O tema se justifica devido à necessidade de desenvolvimento de competências necessárias para que os alunos estejam preparados para superar desafios sobre o conteúdo, sendo este possível através da análise de problemas práticos, como o apresentado no vídeo sugerido, que agrega valor ao processo de ensino aprendizagem desenvolvido em sala de aula. / [en] The increasingly accelerated technological development has enabled the improvement of available media at a level never seen before, significantly contributing to the transmission of various contents to students in the school environment, such as the remarkable points of a triangle and the Euler Line bringing several concepts such as: median, barycenter, bisector, incenter, orthocenter, bisector and circumcenter, the latter being the focus of this study that seeks to present the proposal to use the video The Community: Circumcenter of the triangle as a way to improve the knowledge of this concept in a dynamic way, through the solution of a practical problem about the most ideal place to build a vegetable garden, without causing any problems to neighboring farms. The theme is justified due to the need to develop the skills necessary for students to be prepared to overcome challenges regarding the content, which is possible through the analysis of practical problems, such as the one presented in the suggested video, which adds value to the teaching-learning process developed in the classroom.

Wir beschäftigen uns mit Problemen der algorithmischen Gruppentheorie und untersuchen dabei die Komplexität von komprimierten Versionen des Wortproblems und des Konjugationsproblems für endlich erzeugte Gruppen. Das Wortproblem fragt für eine feste, endlich erzeugte Gruppe ob ein gegebenes Wort über der Erzeugermenge das neutrale Element der Gruppe repräsentiert. Wir betrachten das gegebene Wort jedoch in einer komprimierten Form, als Straight-line Program (SLP) und untersuchen die Komplexität dieses Problems, das wir \'komprimiertes Wortproblem\' nennen. SLPs sind kontextfreie Grammatiken, die genau einen String erzeugen. Die Eingabegröße ist dabei stets die Größe des gegebenen SLPs. Eine Hauptmotivation ist dabei, dass für eine feste endlich erzeugte Gruppe das Wortproblem ihrer Automorphismengruppe durch eine Turingmaschine in Polynomialzeit auf das komprimierte Wortproblem der Gruppe selbst reduzierbar ist. Wir untersuchen das komprimierte Wortproblem für die verbreiteten Gruppenerweiterungen HNN-Erweiterungen (amalgamierte Produkte und Graphprodukte) und können zeigen, dass sich Instanzen des komprimierten Wortproblems von einer Turingmaschine in Polynomialzeit auf Instanzen des komprimierten Wortproblems der Basisgruppe (respektive Basisgruppen und Knotengruppen) reduzieren lassen. Weiterhin zeigen wir, dass das komprimierte Wortproblem für endlich erzeugte nilpotente Gruppen von einer Turingmaschine in Polynomialzeit entscheidbar ist. Wir betrachten außerdem eine komprimierte Variante des Konjugationsproblems. Das unkomprimierte Konjugationsproblem fragt für zwei gegebene Wörter über den Erzeugern einer festen endlich erzeugten Gruppe, ob sie in dieser Gruppe konjugiert sind. Beim komprimierten Konjugationsproblem besteht die Eingabe aus zwei SLPs und es wird gefragt, ob die beiden Wörter die von den SLPs erzeugt werden in der Gruppe konjugierte Elemente präsentieren. Wir konnten zeigen, dass sich das komprimierte Konjugationsproblem für Graphgruppen in Polynomialzeit entscheiden lässt. Weiterhin haben wir das Wortproblem der äußeren Automorphismengruppen von Graphprodukten endlich erzeugter Gruppen untersucht. Durch den engen Zusammenhang des komprimierten Konjugationsproblems einer Gruppe mit dem Wortproblem der äußeren Automorphismengruppe konnten wir zeigen, dass sich das Wortproblem der äußeren Automorphismengruppe eines Graphprodukts von endlich erzeugten Gruppen durch eine Turingmaschine in Polynomialzeit auf Instanzen von simultanen komprimierten Konjugationsproblemen der Knotengruppen und Instanzen von komprimierten Wortproblemen der Knotengruppen reduzieren lässt. Als Anwendung gelten obige Resultate auch für right-angled Coxetergruppen und Graphgruppen, da beide spezielle Graphprodukte sind. So folgt beispielsweise, dass das komprimierte Wortproblem einer right-angled Coxetergruppe in Polynomialzeit entscheidbar ist.

Wir beschäftigen uns mit Problemen der algorithmischen Gruppentheorie und untersuchen dabei die Komplexität von komprimierten Versionen des Wortproblems und des Konjugationsproblems für endlich erzeugte Gruppen. Das Wortproblem fragt für eine feste, endlich erzeugte Gruppe ob ein gegebenes Wort über der Erzeugermenge das neutrale Element der Gruppe repräsentiert. Wir betrachten das gegebene Wort jedoch in einer komprimierten Form, als Straight-line Program (SLP) und untersuchen die Komplexität dieses Problems, das wir \''komprimiertes Wortproblem\'' nennen. SLPs sind kontextfreie Grammatiken, die genau einen String erzeugen. Die Eingabegröße ist dabei stets die Größe des gegebenen SLPs. Eine Hauptmotivation ist dabei, dass für eine feste endlich erzeugte Gruppe das Wortproblem ihrer Automorphismengruppe durch eine Turingmaschine in Polynomialzeit auf das komprimierte Wortproblem der Gruppe selbst reduzierbar ist. Wir untersuchen das komprimierte Wortproblem für die verbreiteten Gruppenerweiterungen HNN-Erweiterungen (amalgamierte Produkte und Graphprodukte) und können zeigen, dass sich Instanzen des komprimierten Wortproblems von einer Turingmaschine in Polynomialzeit auf Instanzen des komprimierten Wortproblems der Basisgruppe (respektive Basisgruppen und Knotengruppen) reduzieren lassen. Weiterhin zeigen wir, dass das komprimierte Wortproblem für endlich erzeugte nilpotente Gruppen von einer Turingmaschine in Polynomialzeit entscheidbar ist. Wir betrachten außerdem eine komprimierte Variante des Konjugationsproblems. Das unkomprimierte Konjugationsproblem fragt für zwei gegebene Wörter über den Erzeugern einer festen endlich erzeugten Gruppe, ob sie in dieser Gruppe konjugiert sind. Beim komprimierten Konjugationsproblem besteht die Eingabe aus zwei SLPs und es wird gefragt, ob die beiden Wörter die von den SLPs erzeugt werden in der Gruppe konjugierte Elemente präsentieren. Wir konnten zeigen, dass sich das komprimierte Konjugationsproblem für Graphgruppen in Polynomialzeit entscheiden lässt. Weiterhin haben wir das Wortproblem der äußeren Automorphismengruppen von Graphprodukten endlich erzeugter Gruppen untersucht. Durch den engen Zusammenhang des komprimierten Konjugationsproblems einer Gruppe mit dem Wortproblem der äußeren Automorphismengruppe konnten wir zeigen, dass sich das Wortproblem der äußeren Automorphismengruppe eines Graphprodukts von endlich erzeugten Gruppen durch eine Turingmaschine in Polynomialzeit auf Instanzen von simultanen komprimierten Konjugationsproblemen der Knotengruppen und Instanzen von komprimierten Wortproblemen der Knotengruppen reduzieren lässt. Als Anwendung gelten obige Resultate auch für right-angled Coxetergruppen und Graphgruppen, da beide spezielle Graphprodukte sind. So folgt beispielsweise, dass das komprimierte Wortproblem einer right-angled Coxetergruppe in Polynomialzeit entscheidbar ist.

Title: Dependence of the position of teams in the FIFA rankings on the achievements in top tournaments Objectives: The aim of this work is determining dependency team standings in the FIFA rankings on the achievements in top tournaments. This is the World Cup in 1998, 2002, 2006, 2010 and 2014. Methods: In my thesis I used a method of regression and correlation analysis, correlation coefficient and regression straight line. Results: All results are presented in the analytical part of the work. It was found dependency between the observed data and an increasing trend. The biggest dependence was at the World Cup 2014, which reached high levels. The smallest was at the World Cup 1998 and 2002, where the results reached below average. Keywords: Football, FIFA, FIFA World rankings, FIFA World Cup, correlation analysis, regression analysis, correlation coefficient, regression straight line