The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

Recent investigations into the permeability of the kidney capillaries have produced conflicting reports. This study was an attempt to better describe the permeability of the kidney capillaries by using narrow-range macromolecular dextran fractions in four molecular sizes: MW 61,400, MW 77,000, MW 118,000, and MW 147,000. Permeability was measured by dextran concentration differences in plasma and kidney lymph. Permeability decreased as the dextran molecular weight increased. Molecular weights 61,400 and 77,000 penetrated into the kidney lymph. Molecular weight 118,000 exhibited greater difficulty in penetrating to the lymph. The largest fraction penetrated into the kidney lymph with greatest difficulty. Plasma expansion by saline infusion increased the permeability of all dextran fractions.

TITLE: Developing activity and creativity in teaching fractions in grades 6 to 7 AUTHOR: Bc. Lenka Svobodová DEPARTMENT: Department of Mathematics and Mathematical Education SUPERVISOR: Mgr. Marie Tichá, CSc. ABSTRACT: The thesis consists of a theoretical and an empirical part. The theoretical part looks at the cognitive process of pupils and possible ways of their motivation during fractions teaching. Furthermore, it outlines causes of lack of understanding of fractions and re-education interventions. The empirical part is based on a qualitative research whose main objective was to determine how the pupils are able to solve fractions exercises independently and creatively and what mistakes they make. Various methods of data collection were used, such as interview with teachers on issues of fractions teaching in elementary schools, non- standardised test with fractions exercises and additional interview with the tested pupils. This part also describes the preparation of the research, the research itself and its results KEYWORDS: fractions, interpretations, representations, models, misunderstandings, mistakes, teaching approaches

(anglicky) Fractions are one of the critical areas of mathematics at the primary school. The aim of this diploma thesis is to describe didactic practices used when teaching fractions (focusing in particular on additive operations with fractions) and find out what strategies are used by pupils at selected tasks focusing on fractions, what difficulties are an obstacle in their successful solutions and what images of fractions the pupils have. The theoretical part contains considerations about the teaching and learning of mathematics, the overview of the historical development of the concept of fraction and operations with fractions and also selected research related to fractions. The practical part begins with the analysis of the textbooks where their semantic and structural models used for fractions are identified as well as ways of teaching operations with fractions. My own research consists of semi-structured interviews with five experienced teachers (one of whom is not qualified) on the topic of teaching fractions, the difficulties of pupils with fractions and of clinical interviews with three pupils from different primary schools. Interviews with the pupils were focused on the solution of the diagnostic tasks in the field of fractions. Transcripts of both interviews were analyzed by techniques...

L’étude de l’optimisation des paramètres de production et de conservation de la farine de taro (Colocasia esculenta) a permis de mieux comprendre le processus cuisson des tubercules et de conservation des farines. Les études de cuisson des tubercules de six variétés ont montré que la variété Ibo coco est celle qui cuit vite eu égard à ses bonnes propriétés thermiques. La cuisson des tubercules dans l’eau du robinet est la meilleure technique de cuisson comparativement aux cuissons, à la vapeur, dans les solutions acides de citron et de tamarin. Les études de détermination des conditions optimales de séchage et conservation de la farine de taro, en utilisant un vieillissement accéléré à 10 et 20 jours de stockage à 45°C à des activité en eau de 0,07 ; 0,11 ; 0,23 ; 0,33 ; 0,53 ont permis de montrer que le vieillissement après 10 jours de stockage était suffisant pour les analyses. De plus l’application du modèle BET lors des mesures des isothermes de sorption, a permis de déterminer les teneurs en eau mono moléculaires théorique et expérimentale. Les farines produites à la teneur en eau mono moléculaire, donnent une pâte ayant des caractéristiques texturales proches de celles de la pâte traditionnelle. Toutefois, la conservation de ces farines à différentes fractions particulaires (75, 150 et 250 µm) montre que les petites fractions particulaires (75 µm) absorbent plus vite de l’eau que les autres fractions particulaires et sont de ce fait plus susceptibles à la détérioration / Study on optimization of production and storage parameters of the taro (Colocasia esculenta) flour permitted to better understanding the cooking process of taro tubers and storage of taro flours. Studies of cooking of six varieties of tubers showed that the variety Ibo coco had high rate of cooking, due to his good thermal properties. The cooking of tubers in the faucet water is the best technique of cooking compared to, steam cooking, lemon and tammarin cooking. Studies of determination of the optimal conditions of drying and storage of the taro flour, using an accelerated ageing to 10 and 20 days of storage at 45°C to the activity in water of 0,07; 0,11; 0,23; 0,33; 0,53, showed that the ageing after 10 day of storage was sufficient for analyses. Besides the use of the BET model on sorption isotherms, permitted to obtain values of theorical and experimental monolayer water. Flours produced at this monolayer water, give dough with the textural characteristics near of those of traditional dough. However, the storage of these flours with different particle sizes (75, 150 and 250 µm) shown that the small particle size (75 µm) absorb water quickly than other flours, and they are more susceptible to the deterioration

Doctor of Philosophy / Curriculum and Instruction Programs / David S. Allen / This study examined pedagogical content knowledge (PCK) and Mathematics knowledge for teaching (MKT) from a perspective that blends existing definitions, questions, and methodologies into a unique method of collecting and analyzing data. Many studies on MKT have been done using qualitative methodology, usually with the researcher interviewing or testing individual teachers and analyzing that data using qualitative methods. A smaller number of studies have attempted to measure MKT using a quantitative approach, often times involving paper pencil tests with multiple choice and some open ended questions. Current research is also heavily weighted towards pre-service elementary teachers in the area of MKT for division by fractions (Depaepe et al, 2013). Although it may be true many pre-service elementary teachers have difficulties with division by fractions (Li & Kulm, 2008; Ma, 1999; Tirosh, 2000), we do not know if these problems persist for in-service elementary teachers or with secondary teachers at any stage. This study used a survey created by the researcher as a synthesis of existing questions from research on MKT with regards to division by fractions. This survey was delivered through an online format. The qualitative data in this research was then coded into quantitative data using a rubric developed by the researcher producing MKT scores that could be analyzed using statistical methods and generalized to a larger population. This study examined the Mathematical Knowledge for Teaching held by elementary, middle, and secondary in-service teachers in the domain of division by fractions. In particular this research asked if there was a relationship between the educational background, training, and experience of teachers and their MKT. Multiple analysis including independent t-tests, independent one-way ANOVAs, and Multiple regression analyses revealed that Middle school teachers and teaching at a middle school were significant predictors of increased MKT scores when compared with other groups of teachers. The type of teaching license, the type of degree held, total experience, and grade level experience were all found to have no significant relationship to MKT scores.

ABSTRACT This study investigates what and how mathematics (for teaching) is constituted in classroom practice. Specifically mathematical knowledge for teaching fractions in Grade 7. One teacher was studied to gain insight into the mathematical problemsolving the teacher does and the dilemmas he faces as he goes about his work. The analysis of the data show that the mathematical problem-solving that this particular teacher engaged in can be classified as demonstrating, encouraging and working with learner ideas. He appealed to mathematics (rules & empirical), experience (everyday) and the curriculum (tests and exams) to fix meaning. The mathematical problem solving and appeals he made threw up dilemmas of representing the content, competing goals and student thinking. This aided in providing a description of what mathematics for teaching is in this practice. The report concludes with a discussion of what teachers need to know or study in order to become better mathematics teachers and where do they find these courses to accommodate their need to improve as mathematics teachers.

L’objet de cette thèse concerne les propriétés du groupe symétrique à travers deux problèmes. Le premier consiste à étudier l’action du groupe symétrique sur la fraction (...). En appliquant certaines opérations sur les graphes et les cartes, nous donnons des algorithmes et des formules combinatoires pour déterminer complètement la fraction réduite suivante : (...). L’auteur C. Greene a introduit cette fraction rationnelle pour généraliser des identités liées a la règle de Murnaghan-Nakayama. Nous utilisons (...) pour établir un nouvel algorithme de décomposition en éléments simples à l’aide des graphes. Dans la seconde partie, nous cherchons a développer les puissances paires du Vandermonde au moyen de fonctions symétriques. En particulier, nous proposons une écriture hyperdéterminantale des coefficients du développement des puissances paires du Vandermonde dans la base des fonctions de Schur. Nous obtenons plusieurs identités reliant les puissances paires du Vandermonde et les polynômes de Jack. Puis nous introduisons une q-déformation des puissances paires du Vandermonde que nous exprimons grâce aux polynômes de Macdonald / The main purpose of this document is the symmetric group. In particular, we study the two following problems. First, the symmetric group acts naturally on the rational function (…), by permuting the variables. With the help of some operations on the graphs, we give algorithms and combinatorial formulas allowing us to compute the reduced fraction (…). The author C. Greene has introduced these rational functions in the aim to generalize some identities related to the Murnaghan-Nakayama rules. We use these properties to give an original algorithm to perform partial decompositions of fractions with the help of graphs. In the second problem, we study the expansion of the even powers of the Vandermonde in several basis of symmetric functions. In this part, we give identities between symmetric functions and hyperdeterminants and we use them to obtain an hyperdeterminental expression of the coefficients in Schur’s basis. We investigate also the relation between the even powers of the Vandermonde and Jack’s functions. Finally, we introduce a q-deformation of the even powers of the Vandermonde and we relate it to some specialisations of Macdonald’s polynomials

Este estudo se insere na problemática do ensino e aprendizagem de frações no Ensino Fundamental e, mais particularmente, no que se refere à apreensão das frações como números pelos alunos. Essa concepção da fração como número é descrita na literatura da área de Educação Matemática como problemática para os alunos, pois, em geral, concebem a representação a/b (com a e b naturais e b não nulo) apenas como um duplo processo de contagem no modelo parte-todo. Decidiu-se, portanto, investigar o impacto de algumas abordagens que ampliassem a referida concepção de fração como parte de inteiro. Para tanto, tomou-se como base algumas pesquisas que destacam diferentes ideias e situações para conceituar frações, bem como a Teoria dos Registros de Representação Semiótica de Raymond Duval, devido à importância de se abordar as frações em seus vários significados, por meio de diferentes representações. Realizou-se um estudo experimental de caráter qualitativo, inspirado na metodologia de Design Experiment, envolvendo 24 alunos do 6º ano do Ensino Fundamental de uma escola em Goiânia. A elaboração das atividades fundamentou-se nos constructos teóricos do levantamento bibliográfico, bem como em um questionário inicial que permitiu identificar as principais dificuldades do grupo de alunos. As hipóteses consideradas no design foram: ênfase em atividades de conversão de representações entre os registros numérico ou figural e o gráfico (reta numérica), em ambos os sentidos; e foco na ideia da fração como representação do resultado de uma divisão de dois números naturais. Foram realizadas quatro atividades, com diversas tarefas em cada uma delas, ao longo de nove encontros no âmbito das aulas regulares de Matemática. As propostas transitaram entre trabalhos em grupo e individuais, envolvendo recursos tradicionais e materiais concretos, sendo que a coleta de dados deu-se essencialmente a partir das observações da pesquisadora e dos registros orais e escritos das produções dos alunos. Na atividade principal, foi introduzido um recurso para realizar a divisão de segmentos em partes congruentes, visando dar condição para os alunos representarem frações não decimais em retas numéricas, sem realizar a conversão para a representação decimal e/ou efetuar aproximações imprecisas. As análises mostraram que houve, em geral, um amadurecimento dos estudantes em relação às ideias apresentadas, aproximando-os da concepção de fração como número uma vez que explicitaram compreensão de aspectos de equivalência e ordem ao posicionar frações em retas numéricas e perceberam que tais frações correspondiam a resultados de divisões entre dois números naturais, isto é, a quocientes vistos como quantidades. / This study concerns the issue of teaching and learning fractions in Basic Education and, more particularly, regards the students apprehension of fractions as numbers. The notion of fractions as numbers is described in Mathematics Education literature as problematic for students, since, generally, they understand the representation a/b (where a and b are natural numbers and b is different than zero) only as a double counting process in the part-whole model. Therefore, we decided to investigate the impact of some approaches that broadened the notion of fraction as part of a whole. In order to achieve that, we used as a basis research that highlights different ideas and situations to conceptualize fractions, as well as Raymond Duvals Theory of Registers of Semiotic Representation, due to the importance of approaching fractions in their diverse meanings, through different representations. We carried out an experimental study of qualitative character, inspired by the Design Experiment methodology, with 24 students in the 6th grade from a school in Goiânia. The activities were written based on the theoretical constructs analyzed in the bibliographic search, as well as based on an initial questionnaire that allowed us to identify the main difficulties that the student group had. The hypotheses considered in the design were: emphasis on activities concerning representation conversion between numerical or figural registers and graphical (number line) in both directions; and focus on the idea of fraction as the representation of a division of two natural numbers. Four activities were carried out, with several tasks in each one, along nine meetings in the context of regular Math classes. The activities varied between group and individual tasks, involving traditional resources and concrete materials, with the data collection taking place essentially through the researchers observations and oral and written records of the students productions. In the main activity, we introduced a resource to facilitate the division of segments into congruent parts, aiming to help the students depict non-decimal fractions in number lines without converting them into the decimal register and/or using inaccurate approximations. The analysis shows that, generally, there was an improvement in the students concerning the ideas presented in the activities, bringing them closer to the concept of fractions as numbers, as they demonstrated understanding aspects of equivalence and order by placing fractions in number lines and realized that these fractions corresponded to the results of divisions of two natural numbers, that is, quotients perceived as quantities.

O nitrogênio é um dos nutrientes mais importante para o desenvolvimento vegetal e o mais aplicado em diversos cultivos agrícolas. Dentre as fontes nitrogenadas disponíveis, a ureia se destaca pela alta concentração de nitrogênio que remete a diversas vantagens econômicas e operacionais no sistema de produção agrícola. Porém, além da alta propensão a perdas, principalmente por volatilização do gás amônia, tem-se a necessidade do aumento da produtividade de modo ao suprimento da demanda crescente de alimentos, fibras e energia. Considerando estes fatores, uma nova vertente de pesquisa está voltada a aplicação de tecnologias à ureia, os fertilizantes com tecnologia agregada, estes vêm se destacando e disponibilizando alternativas de novos produtos no mercado. Com base no exposto, buscou-se elucidar os seguintes questionamentos: As novas tecnologias aplicadas a ureia possibilitam a redução de perdas por volatilização? As novas tecnologias aplicadas a ureia modificam suas transformações inorgânicas e orgânicas no solo e, em caso de modificação, estas são positivas? O uso de tecnologias aplicadas à ureia reflete em maior longevidade na liberação do nutriente ao solo e, em caso positivo, esta longevidade é vantajosa? A combinação entre produtos de ureia convencional e com tecnologia agregada, pode trazer mais benefícios à cultura do milho do que a aplicação destes isoladamente? A substituição do uso/aplicação da ureia em sua forma tradicional pela ureia com tecnologia agregada, na cultura do milho, é viável economicamente ao produtor? Para responder a estes questionamentos, foram conduzidos três experimentos: análise das frações do nitrogênio provenientes da ureia com inibidores de urease; caracterização da dinâmica de liberação e longevidade de fertilizantes recobertos com polímero e enxofre e; análise da produtividade da cultura do milho submetida a adubação com ureia recoberta com polímero e enxofre. O primeiro experimento permitiu concluir que o uso dos inibidores de urease tem potencial para a redução das perdas por volatilização da amônia, foi possível verificar as transformações sofridas pelo nitrogênio desde a aplicação do grânulo até a disponibilidade das formas minerais, passando a incorporação do nutriente na matéria orgânica. No segundo experimento concluiu-se que os fertilizantes recobertos com polímero e enxofre possuem dinâmica exponencial de liberação, o recobrimento é dependente da temperatura, umidade do solo, tipo de material de recobrimento e processo de recobrimento (espessura), que afetam diretamente a liberação e longevidade destes fertilizantes. O último experimento mostrou que uso dos fertilizantes recobertos com polímero e enxofre aumenta a produtividade, os parâmetros biométricos, os parâmetros nutricionais e a qualidade do grão da cultura do milho. As combinações realizadas entre os produtos recobertos com a ureia convencional são alternativas para o aumento da produtividade resultando em valores superiores a aplicação dos insumos de forma isolada. A combinações de produtos recobertos com a ureia convencional é opção para a redução do investimento e, consequentemente, custo de produção resultante da adoção destas novas tecnologias. / Nitrogen is one of the most important nutrient for plant development and the most applied nutrient in several agricultural crops. Among the available nitrogen sources, urea differ due to the high concentration of nitrogen that result to several economic and operational advantages in the agricultural production system. However, in addition to the high propensity for losses, mainly due to the ammonia volatilization, the need to increase yield in order to supply the growing demand for food, fiber and energy emerge. Considering these factors, a new research area is focused to aggregate technologies to the urea, they call fertilizers with aggregate technology, this type of fertilizers emerges and brings new alternatives about new products available in the market. Based on the above, it was tried to answer the following questions: Does the new technologies applied to urea fertilizer enable the reduction of losses by volatilization? The new technologies applied to urea modify their inorganic and organic transformations in the soil and, if modified, are these positive? Does the use of technologies applied to urea reflect in a longer longevity in the release of the nutrient to the soil and, if so, is this longevity advantageous? Can the combination of conventional urea and aggregate technology products brings more benefits to cons than the application of these alone? Is the substitution of urea in its traditional form by urea with aggregate technology in the maize crop economically viable to the producer? In order to answer these questions, three experiments were conducted: analysis of nitrogen fractions from urea and urea mixed with inhibitors; Dynamics of release and longevity of polymer sulfurcoated fertilizers; Analysis of maize yield submitted to fertilization with polymer and sulfur coated urea. The first experiment allowed concluding that the use of the urea inhibitors has potential to reduce losses by ammonia volatilization, it was possible to verify all the changes undergone by the nitrogen fertilization since the granule until the availability of the mineral forms, including the immobilization of the nutrient in organic matter. In the second experiment, it was concluded that the polymer and sulfur coated fertilizers have exponential dynamics of release, the coating depends on the temperature, soil moisture, type of coating material and coating process (thickness), which directly affects the release and longevity of these fertilizers. The last experiment showed that the use of the coated fertilizers with polymer and sulfur increases the yield, the biometric parameters, the leaf concentration and the quality of grain to the maize crop. The combinations performed between the coated products with conventional urea are alternatives for increasing the yield resulting in higher application values of the inputs in an isolated way. Combinations of coated products with conventional urea is an alternative to reduce investment and consequently the production costs, it will result the adoption of these new technologies.