Diss. (sammanfattning) Stockholm : Karol. inst., 2002. / Härtill 4 uppsatser.

Thesis (Ph. D.)--University of Hawaii at Manoa, 2003. / On title page "[eta]" appears as Greek symbol and "c̳" is a subscript. Includes bibliographical references (leaves 124-128).

Η συγκεκριμένη μελέτη περίπτωσης πραγματοποιήθηκε στην Γ΄τάξη ενός επαρχιακού Δημοτικού σχολείου. Ο στόχος ήταν να ερευνηθεί ο βαθμός κατανόησης από τους μαθητές, της έννοιας του κλάσματος ως αριθμού,του συμβολισμού του κλάσματος και των ισοδύναμων κλασμάτων, μετά από διδασκαλία με βάση το σχολικό βιβλίο και λαμβάνοντας υπόψιν τις υποδείξεις και προτάσεις του βιβλίου του δασκάλου. Τα αποτελέσματα της έρευνας έδειξαν ότι σημειώθηκαν οι "αναμενόμενες" και, από άλλες έρευνες, δυσκολίες, στην έννοια των ισοδύναμων κλασμάτων, στην τοποθέτηση κλασμάτων στην αριθμογραμμή και στη χρήση, κατά τη διδασκαλία, σκίτσων - νομισμάτων. Αξιοσημείωτη είναι η δυσκολία που αντιμετώπισαν οι μικροί μαθητές στις μετατροπές κλασμάτων σε δεκαδικούς αριθμούς ή άλλα κλάσματα. Τέλος, η χρήση χειροπιαστών αντικειμένων κατά τη διδασκαλία, έπαιξε καταλυτικό ρόλο στην κατανόηση των εννοιών. / The sample case study was assessed in the C class of a Provincial Community Primary school. The aim of the study was to examine the degree of the pupils' comprehension of the concept of a fraction as a numerical figure, the symbolism of the fraction and the equivalent fractions after having been taught based on the school book and taking into account the directions and suggestions stated in the teachers' book. The results of the study showed similar expected outcomes as with other studies, that is, the difficulties to perceive the meaning of equivalent fractions, to set fractions up the number-line and the use of pictures of coins during tuition. It is worth noting the difficulty the young pupils faced with the conversion of fractions into decimal numbers or other fraction numbers. Finally, the use of manipulative objects during tuition played a significant role in the understanding of the concepts.

Acompanha: Caderno pedagógico: as oficinas na formação continuada de professores: uma estratégia a partir do pró-letramento matemática para a construção do conceito de frações. / O presente estudo teve como objetivo contribuir na formação de professores dos Anos Iniciais do Ensino Fundamental a partir de oficinas pedagógicas baseadas no fascículo de frações do Programa de Formação Continuada Pró-Letramento Matemática. A pesquisa foi realizada no ano de 2011, em um curso de 30 horas para 16 professores do 2º ano do 2º ciclo (antiga 4ª série) do Ensino Fundamental de 9 anos da Rede Municipal de Ensino de Ponta Grossa. Do ponto de vista metodológico, o estudo inseriu-se em uma pesquisa interpretativa de natureza qualitativa e finalidade aplicada. Para coleta de dados foram utilizados os seguintes instrumentos: o questionário, o Pré-teste, o Pós-teste, o Diário Coletivo e o Diário de Bordo. A partir do questionário foi possível traçar o perfil dos professores e com os resultados do Pré-teste foram detectadas as dificuldades e os obstáculos que os professores encontravam no ensino de frações. Após análise e avaliação dos dados obtidos com estes instrumentos foram organizadas sete oficinas pedagógicas visando o aprofundamento do conteúdo e estratégias de ensino. Os registros do Diário de bordo foram utilizados pela pesquisadora/aplicadora para a confirmação e/ou adaptações nas oficinas de modo atender as necessidades dos professores participantes do curso. Ao término das oficinas foi aplicado o Pós-teste, que juntamente com os comentários registrados pelos professores no Diário Coletivo, serviu de parâmetro para avaliar o aprendizado durante as oficinas. Ao final do curso percebeu-se que as atividades realizadas nas oficinas pedagógicas contribuíram para ampliar o conhecimento dos professores tanto no aprofundamento conceitual quanto nas estratégias de ensino, e ainda evidenciou-se a relevância do acompanhamento sistematizado das equipes centrais das instituições mantenedoras (Secretarias Municipais e Estaduais de Educação), para que o professor não se sinta isolado diante do desafio de ensinar e possa propiciar aos seus alunos uma aprendizagem cada vez melhor. Como produto final desse trabalho, apresenta-se o Caderno Pedagógico: “As oficinas na formação continuada de professores - uma estratégia do Pró-Letramento Matemática para a construção do conceito de frações”, cuja finalidade é a de orientar ações pedagógicas no ensino de frações. / The present study aimed to contribute to the training of teachers of the first years of elementary school from teaching workshops based on issue fractions of the Program of Continuing Education Pro-Literacy Mathematics. The survey was conducted in 2011, on a course of 30 hours for 16 teachers in the 2nd year of the 2nd cycle (former 4th grade) elementary school for nine years the Municipal School of Ponta Grossa city. From the methodological point of view, the study entered into a qualitative interpretative research and applied purposes. For data collection we used the following instruments: the questionnaire, the Pre-test, the Post-test, the Collective Diary and Diary. From the questionnaire it was possible to trace the profile of teachers and the results of the pre-test were found difficulties and obstacles that teachers were in teaching fractions. After analysis and evaluation of data obtained with these instruments were organized seven workshops aimed at deepening the educational content and teaching strategies. The logbook records were used by the Researcher/Applicator for confirmation and adaptations in the workshops so teachers meet the needs of course participants. At the end of the workshops was applied post-test, which along with the comments recorded by teachers in the Journal Collective, served as parameter for assessing learning during the workshops. At the end of the course it was noted that activities in educational workshops contributed to the knowledge of teachers both in conceptual and deepening the teaching strategies, and also revealed the importance of systematic monitoring of the core teams of institutions offering (Municipal Education and State), so that the teacher does not feel isolated from the challenge of teaching and can provide its students with a learning better. As a final product of this work, we present the Educational Booklet: "The workshops on continuing education of teachers - a strategy of Pro-Literacy Mathematics for the construction of the concept of fractions”, whose purpose is to guide pedagogical practices in teaching fractions.

This thesis is focused on the problem of growth and integrability in multi-valued dynamics generated by $SL_2 (\mathbb{Z})$ actions. An important example is given by Markov dynamics on the cubic surface $$x^2+ y^2 +z^2 = 3xyz,$$ generating all the integer solutions of this celebrated Diophantine equation, known as Markov triples. To study the growth problem of Markov numbers we use the binary tree representation. This allows us to define the Lyapunov exponents $\Lambda (x)$ as the function of the paths on this tree, labelled by $x \in \mathbb{R}P^1$. We prove that $\Lambda (x)$ is a $PGL_2 (\mathbb{Z})$-invariant function, which is zero almost everywhere but takes all values in $\left[ 0, \ln \varphi \right]$ (where $\varphi$ denotes the golden ratio). We also show that this function is monotonic, and that its restriction to the Markov-Hurwitz set of most irrational numbers is convex in the Farey parametrisation. We also study the growth problem for integer binary quadratic forms using Conway's topograph representation. It is proven that the corresponding Lyapunov exponent $\Lambda_Q(x) = 2 \Lambda(x)$ except for the paths along the Conway river. Finally, we study the tropical version of the Markov dynamics on the tropical version of the Cayley cubic proposed by Adler and Veselov, and show that it is semi-conjugated to the standard action of $SL_2(\mathbb{Z})$ on a torus. This implies the dynamics is ergodic, with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.

La digestion anaérobie représente un élément clé dans la dynamique de développement durable contribuant à la valorisation verte des déchets organiques sous forme de biogaz et d'engrais. Dans le cadre de ce travail de thèse franco-libanais, nous cherchons à valoriser le marc de raisin, déchet majeur et principal sous-produit issu de la viniculture, par la digestion anaérobie, afin de générer de l’énergie dite verte sous forme de méthane. Dans un premier temps, le potentiel méthane du marc de raisin est démontré, validant notre biomasse végétale comme source potentielle d’énergie. Des informations détaillées sur les productions maximales de méthane à partir du marc entier, des pulpes et des pépins, séparément, sont obtenues en mode batch à 37 °C. La faisabilité technique du procédé est alors validée suite à une extrapolation au mode continu. De plus, des essais d’acclimatation du digesteur continu à la température moyenne de la vallée de la Bekaa (25 °C) permettent de simuler et d’adapter le système au milieu libanais. D’autre part, nous avons mené une caractérisation bio-physico-chimique de différents cépages de marcs de raisins en provenance de différentes régions viticoles. La diversité du contenu lignocellulosique et du potentiel méthanogène des substrats choisis a été mise en évidence. Une corrélation négative existe, en particulier, entre le potentiel méthane et les teneurs en lignine et en cellulose. Afin d’intensifier la production de méthane, le dimensionnement des digesteurs anaérobies en mode continu est optimisé en déterminant un optimum de fonctionnement pour une charge appliquée de 3,7 kg DCO m-3 j-1 et un temps de séjour de 20 jours. Enfin, nous évaluons les effets d’une variété de prétraitements (congélation, traitement alcalin, traitement acide, ultrasons et champs électriques pulsés) sur la production de méthane et sur la biodégradabilité des fractions. Le couplage du traitement alcalin à 10% NaOH avec la congélation à -20 °C s’avère être le meilleur procédé d’intensification. / Anaerobic digestion is considered to be a crucial part of a sustainable development strategy, contributing to the green valorization of organic waste as biogas and fertilizers. As part of this doctoral thesis, we explored the valorization of grape pomace, the major waste and main by-product of winemaking, by anaerobic digestion, to generate green energy in the form of methane. Firstly, the methane potential of grape pomace is demonstrated, validating our vegetal biomass as a potential source of energy. Detailed information on the maximum production of methane from whole pomace, pulps and seeds are obtained in batch mode at 37 °C. The technical feasibility of the process is then validated following an extrapolation to the continuous mode. Moreover, the acclimation of the continuous digester at the average temperature of the Beqaa valley (25 °C) allowed to simulate and adapt the current system to the Lebanese environment. On another note, we carried out a bio- physico-chemical characterization of different grape varieties from different wine-growing areas. The diversity of the lignocellulosic content and the methane potential of the selected substrates was highlighted. A negative correlation exists, in particular, between the methane potential and the lignin and cellulose fractions. In order to intensify methane production, we conducted an optimization of anaerobic digesters dimensioning in continuous mode by determining an optimum of operation for an applied load of 3.7 kg COD m-3 d-1 and a residence time of 20 days. Finally, we evaluated the effects of a variety of pretreatments (freezing, alkaline treatment, acid treatment, ultrasounds and pulsed electric fields) on the methane production and the biodegradability of lignocellulosic fractions. The coupling of the alkaline treatment using 10% NaOH with freezing at -20 °C seems to be the best intensification process.

On réfléchit à une façon de déterminer une fraction rationnelle postcritiquement finie à conjugaison par une transformation de Möbius près, à l'aide de graphes planaires finis munis de la dynamique de l'application. Étant donné une telle fraction rationnelle f, on définit un ensemble G(f) de classes d'équivalences de graphes admissibles (des graphes planaires finis invariants, connexes, qui contiennent l'ensemble postcritique de f). On montre que si G(f) est non vide, alors f et g sont conjuguées par une transformation de Möbius si et seulement si l'intersection entre G(f) et G(g) est non vide. Cela nous amène à réfléchir à la construction de graphes admissibles pour une fraction rationnelle postcritiquement finie. On montre qu'une intersection non vide entre les bords de deux composantes de Fatou périodiques contient au moins un point périodique. On construit des graphes admissibles pour certains éléments de la famille des fractions rationnelles quadratiques dont l'un des deux points critiques est l'image de l'autre, avec des techniques utilisables dans de nombreuses autres familles. / We think of a way to determine a postcritically finite rational map up to Möbius conjugacy, using planar finite graphs fitted with the dynamics of themap. Givensuch a rational map f, we define a set G(f)of equivalence classes of admissible graphs (invariant, connected planar finite graphs containing the postcritical set). We show that if G(f) is non-empty, then f and g are Möbius conjugate if and only if G(f)nG(g) unequal Ø. This leads us to think about the construction of admissible graphs for a postcritically finite rational map. We show that a non-empty intersection between the boundaries of two periodic Fatou components contains at least one periodic point. We construct admissible graphs for some elements of the family of quadratic rational maps whose one of the two critical points is the image of the other, with technics usable in a lot of other families.

M.Ed. (Mathematics Education) / The purpose of this study was to explore teachers’ pedagogical content knowledge (PCK) in the teaching of equivalent fractions in grade 3 learners. The study used a qualitative paradigm, where a case study design was adopted. Five teachers were sampled through a purposive sampling strategy. The data were collected through three methods: observations, interviews and document analysis. Four teachers were observed in their classrooms and were also interviewed, the fifth teacher was only interviewed through an interview schedule. Interview and observation data were analyzed and interpreted using the ladder of inference, which follows a three-rung approach. The documents in the form of learners’ class works, home works, tests and teachers’ lesson plans were analyzed. Learners’ works correlated with their teachers’ preparations and presentations. The findings indicate that teachers lacked PCK in the teaching of fractions in particular, equivalent fractions. Teachers were not competent, fluent and not independent in the teaching of fractions. It is also clear from data collected that fractions are notoriously difficult for teachers to teach and for learners to learn. On the whole, teachers seem to lack requisite skills to contextualize content to the appropriate cognitive levels of their learners. Teachers demonstrated poor content and conceptual knowledge ...

Cette thèse en approximation diophantienne se divise en deux parties. La première concerne la géométrie paramétrique des nombres introduite récemment par W. M. Schmidt et L. Summerer. Cette théorie décrit, à des fonctions bornées près, le comportement des minimas successifs d'une famille de corps convexes à un paramètre, à savoir celle qui intervient naturellement dans les problèmes d'approximation rationnelle simultanée de nombres réels linéairement indépendant sur Q. Nous démontrons dans le cas d'un seul nombre que le comportement qualitatif des deux minima est équivalent à la donnée du développement en fraction continue de la distance de ce nombre à l'entier le plus proche. Nous nous intéressons aussi à une conjecture de Schmidt de 1983 démontrée par N. Moshchevitin en 2012, et nous l'améliorons. De plus, nous démontrons que notre amélioration est meilleure possible. La seconde partie de la thèse concerne les nombres extrémaux introduits par D. Roy en 2003. Ces nombres sont transcendants réels et se comporte similairement aux nombres quadratiques réels quant à certaines propriétés d'approximation diophantienne. On peut leur associer une suite canonique de matrices symétriques à coefficients entiers. Dans leur papier [23], D. Roy et E. Villani considèrent une classe particulière de nombres extrémaux et étudient les suites de matrices correspondantes à la fois d'un point de vue analytique et d'un point de vue algébrique. Nous considérons ici une classe plus restreinte de nombres extrémaux, dits de type Markoff. Nous commençons par établir une conjecture qui généralise les résultats de Roy et Villani, sur la base de résultats numériques. Cette conjecture dépend d'une quantité entière appelé degré, et nous la démontrons en degré au plus 6. La preuve dépend d'une construction, elle-même valide en tous degrés.

Abstract Pressure screening, nowadays the most widely used method for cleaning pulp, has been traditionally investigated as a debris removal process. The aim of this thesis, however, was to study it with a view to the fractionation of pulps, examining systematically and extensively the effects of screening parameters on fractionation under actual working conditions in order to provide an insight into its possibilities and limitations as a fractionation method. The experimental work was performed with a full-scale two-stage pressure screen connected to an industrial TMP process. Fractionation of the pulp was analysed in terms of consistency, freeness, optical fibre length distribution, coarseness index and Bauer-McNett fractions. Two sampling systems were used during the screening experiments, manual and semiautomatic. The latter was assessed to be more reliable, as reflected in lower stochastic variation and the absence of a systematic bias in the mass balance errors over the screen. The poorer reliability of the manual sampling system was offset by the large number of screening tests, however. The results of the screening experiments showed that with a given design of the screen plate, the separation of each fraction was dependent almost exclusively on the mass and volumetric reject rates. The mass flow of fines, defined as the Bauer-McNett P200 fraction, was dependent mostly on the volumetric reject rate, while the mass flow of fibrous fractions (R200, R50, R30, R16) depended mostly on the mass reject rate. The mass reject rate obtained in pressure screening was a result of the choice of operating parameters, but fractionation efficiency could not be affected by using different combinations of these parameters (feed consistency, rotor tip speed and slot velocity) if the mass and volumetric reject rates were kept constant. The slot width together with the contouring of the screen plate affected the fractionation efficiency as compared with the situation at constant mass and volumetric reject rate. Increased fractionation was obtained by reducing the slot width and contouring. The pulp passage ratio, which combines the mass and volumetric reject rates into one parameter, was found to be a expedient way of expressing the fractionation of pulp, as it was possible to present fractionation uniformly as a function of this ratio. The change in freeness was found to correlate quite well with that in Bauer-McNett fractions, and it was a good indicator of fractionation efficiency in screening. Apart from fractionation according to length (or Bauer-McNett fractions), the slotted pressure screen was also found to classify the fibres according to their coarseness. The coarseness difference was partially dependent on the fibre length, but additionally the coarseness in the accept pulp for any given fibre length class was always lower than that in the reject pulp. The difference obtained seemed to depend on the passage ratio of the pulp. This thesis provides new information for the modelling of pulp quality and the design of fractionation experiments, fractionation processes and screen room control strategies.