Au croisement de la sociologie des classes populaires et de la sociologie urbaine et du logement, cette thèse porte sur l’une des formes de logements non ordinaires qui a connu un développement massif en France dans les territoires ruraux et périurbains au cours de ces dernières années : le camping résidentiel. Combinant immersion ethnographique dans plusieurs campings de la région parisienne, entretiens, archives et statistiques, elle montre l’existence d’une véritable stratification interne à cet habitat, eu égard à l’hétérogénéité des situations résidentielles, des trajectoires, des ressources, des expériences et des styles de vie des habitants. Le camping résidentiel est de fait apparu comme remplissant trois grandes fonctions sociales segmentant la population qui le pratique : il peut représenter une alternative à la maison individuelle inaccessible, figurer un déclassement subjectif et objectif ou encore s’apparenter à une solution qui pallie la pénurie de logement abordable et évite le dénuement extrême de la rue. Inscrite dans une approche à la fois contextualiste et dispositionnaliste des manières d’habiter, la thèse rapporte ces trois fonctions du camping – qui constituent également trois styles de vie distincts – aux caractéristiques particulières de cette forme d’habitat non ordinaire mais aussi à des ressources, des trajectoires et des socialisations résidentielles différentes articulées à des dispositions populaires relativement homogènes. Elle identifie enfin les causes, les dynamiques et les conséquences des pratiques de délogement en documentant « de l’intérieur » un cas de fermeture de terrain de camping, exemple de concrétisation du risque associé au statut juridique de cet habitat. / This thesis finds itself at the intersection of the sociology of the working classes and urban sociology, with a special interest in housing. It focuses on one of the forms of non-ordinary housing that has seen massive development in France in recent years: residential camping. Combining an ethnographic immersion in several campsites in the Paris region, interviews, archives and statistics, it shows the existence of a real internal stratification within this habitat, taking into account the heterogeneity of residential situations, trajectories, resources, experiences and lifestyles of the inhabitants. Residential camping has in fact emerged as fulfilling three major social functions which segment the population that practices it: it may represent an alternative to the inaccessible single-family home, or stand as a subjective and objective downgrading or even be a solution that makes up for the shortage of affordable housing, thus preventing the extreme destitution of living in the streets. Illustrating a dispositionalist-contextualist approach to ways of living, the thesis connects these three functions of camping - which also constitute three distinct lifestyles - with the particular characteristics of this unusual form of housing. It also links it to different resources, trajectories and forms of residential socialisation corresponding to relatively homogeneous popular dispositions. Finally, it identifies the causes, dynamics and consequences of eviction practices by documenting "from the inside" the case of a campground closure, an example of the risk involved in the legal status of this habitat.

Ce mémoire s'emploie à étudier les corps quadratiques réels ainsi qu'un élément particulier de tels corps quadratiques réels : l'unité fondamentale. Pour ce faire, le mémoire commence par présenter le plus clairement possible les connaissances sur différents sujets qui sont essentiels à la compréhension des calculs et des résultats de ma recherche. On introduit d'abord les corps quadratiques ainsi que l'anneau de ses entiers algébriques et on décrit ses unités. On parle ensuite des fractions continues puisqu'elles se retrouvent dans un algorithme de calcul de l'unité fondamentale. On traite ensuite des formes binaires quadratiques et de la formule du nombre de classes de Dirichlet, laquelle fait intervenir l'unité fondamentale en fonction d'autres variables. Une fois cette tâche accomplie, on présente nos calculs et nos résultats. Notre recherche concerne la répartition des unités fondamentales des corps quadratiques réels, la répartition des unités des corps quadratiques réels et les moments du logarithme de l'unité fondamentale. (Le logarithme de l'unité fondamentale est appelé le régulateur.) / This memoir aims to study real quadratic fields and a particular element of such real quadratic fields : the fundamental unit. To achieve this, the memoir begins by presenting as clearly as possible the state of knowledge on different subjects that are essential to understand the computations and results of my research. We first introduce quadratic fields and their rings of algebraic integers, and we describe their units. We then talk about continued fractions because they are present in an algorithm to compute the fundamental unit. Afterwards, we proceed with binary quadratic forms and Dirichlet's class number formula, which involves the fundamental unit as a function of other variables. Once the above tasks are done, we present our calculations and results. Our research concerns the distribution of fundamental units in real quadratic fields, the disbribution of units in real quadratic fields and the moments of the logarithm of the fundamental unit. (The logarithm of the fundamental unit is called the regulator.)

Un problème de géométrie algébrique réelle s'exprime sous forme d’un système d’équations et d’inéquations polynomiales, dont l’ensemble des solutions est un ensemble semi-algébrique. L'objectif de cette thèse est de montrer comment les algorithmes de ce domaine peuvent être décrits formellement dans le langage du système de preuve Coq.Un premier résultat est la définition formelle et la certification de l’algorithme de transformation de Newton présentée dans la thèse d'A. Bostan. Ce travail fait intervenir non seulement des polynômes, mais également des séries formelles tronquées. Un deuxième résultat est la description d'un type de donnée représentant les ensembles semi-algébriques. Un ensemble semialgébrique est représenté par une formule logique du premier ordre basée sur des comparaisons entre expressions polynomiales multivariées. Pour ce type de données, nous montrons comment obtenir les différentes opérations ensemblistes et allons jusqu'à décrire les fonctions semi-algébriques. Pour toutes ces étapes, nous fournissons des preuves formelles vérifiées à l'aide de Coq. Enfin, nous montrons également comment la continuité des fonctions semi-algébrique peut être décrite, mais sans en fournir une preuve formelle complète. / A real algebraic geometry problem is expressed as a system of polynomial equations and inequalities, and the set of solutions are semi-algebraic sets. The objective of this thesis is to show how the algorithms of this domain can be formally described in the language of the Coq proof system. A first result is the formal definition and certification of the Newton transformation algorithm presented in A. Bostan's thesis. This work involves not only polynomials, but also truncated formal series. A second result is the description of a data type representing semi-algebraic sets. A semi-algebraic set is represented by a first-order logical formula based on comparisons between multivariate polynomial expressions. For this type of data, we show how to obtain the different set operations all the way to describing semialgebraic functions. For all these steps, we provide formal proofs verified with Coq. Finally, we also show how the continuity of semi-algebraic functions can be described, but without providing a fully formalized proof.

The study aimed to explore misconceptions that Grade 9 learners at a school in Soweto had concerning the topic of fractions. The study was based on the ideas of constructivism in a bid to understand how learners build on existing knowledge as they venture deeper into the development of advanced constructions in the concept of fractions. A case study approach (qualitative) was employed to explore how Grade 9 learners describe the concept of fractions. The approach offered a platform to investigate how Grade 9 learners solve problems involving fractions, thereby enabling the researcher to discover the misconceptions that learners have/display when dealing with fractions. The research allowed the researcher to explore the root causes of the misconceptions held by learners concerning the concept of fractions. Forty Grade 9 participants from a township school were subjected to a written test from which eight were purposefully selected for an interview. The selection was based on learners’ responses to the written test. The researcher was looking for a learner script that showed application of similar but incorrect procedures under specific sections of operations of fractions, for example, multiplication of fractions. Both performance extremes were also considered, the good and the worst performers overall. The written test and the interviews were the primary sources of data in this study. The study revealed that learners have misconceptions about fractions. The learners’ definitions of what a fraction is were neither complete nor precise. For example, the equality of parts was not emphasised in their definitions. The gaps brought about by the learner conception of fractions were evident in the way problems on fractions were manipulated. The learners did not treat a fraction as signifying a specific point on the number system. Due to this, learners could not place fractions correctly on the number line. Components of the fraction were separated and manipulated as stand-alone whole numbers. Consequently, whole number knowledge was applied to work with fractions. A lack of conceptual understanding of equivalent fractions was evident as the common denominator principle was not applied. In the multiplication of fractions, procedural manipulations were evident. In mixed number operations, whole numbers were multiplied separately from the fractional parts of the mixed number. Fractional parts were also multiplied separately, and the two answers combined to yield the final solution. In the division of fractions, the learners displayed a lack of conceptual knowledge of division of fractions. Operations were made across the division sign numerators separate from the denominators. This reveals that a fraction was not taken as an outright number on its own by learners but viewed as one number put on top of the other which can be separated. Dividing across, learners rendered division commutative. A procedural attempt to apply the invert and multiply procedure was also evident in this study. Learners made procedural errors as they showed a lack of conceptual understanding of the keep-change-flip division algorithm. The study revealed that misconceptions in the concept of fraction were due to prior knowledge, over-generalisation and presentation of fractions during instruction. Constructivism values prior knowledge as the basis for the development of new knowledge. In this study, learners revealed that informal knowledge they possess may impact negatively on the development of the concept of fractions. For example, division by one-half was interpreted as dividing in half by learners. The prior elaboration on the part of a whole sub-construct also proved a barrier to finding solutions to problems that sought knowledge of fractions as other sub-constructs, namely, quotient, measure, ratio and fraction as an operator. Over generalisation by learners in this study led to misconceptions in which a procedure valid in a particular concept is used in another concept where it does not apply. Knowledge on whole numbers was used in manipulating fractions. For example, for whole numbers generally, multiplication makes bigger and division makes smaller. The presentation of fractions during instruction played a role in some misconceptions revealed by this study. Bias towards the part of a whole sub-construct might have limited conceptualisation in other sub-constructs. Preference for the procedural approach above the conceptual one by educators may limit the proper development of the fraction concept as it promotes the use of algorithms without understanding. The researcher recommends the use of manipulatives to promote the understanding of the fraction concept before inductively guiding learners to come up with the algorithm. Imposing the algorithm promotes the procedural approach, thereby depriving learners of an opportunity for conceptual understanding. Not all correct answers result from the correct line of thinking. Educators, therefore, should have a closer look at learners’ work, including those with correct solutions, as there may be concealed misconceptions. Educators should not take for granted what was covered before learners conceptualised fractions as it might be a source of misconceptions. It is therefore recommended to check prior knowledge before proceeding with new instruction. / Mathematics Education / M. Ed. (Mathematics Education)

Au Burkina Faso, des études menées par Kiélem et Barro (2007) et Traoré (2012) ont montré des insuffisances dans la formation initiale des enseignants. Par exemple, Traoré (2012) révèle que les programmes de formation initiale à l’enseignement des mathématiques sont inadaptés aux besoins des finissants. Notre étude vise la compréhension des pratiques des stagiaires du primaire et du post-primaire sur l’enseignement de la fraction, un concept difficile pour les élèves. Une combinaison de plusieurs théories et concepts dont les théories des situations didactiques (Brousseau, 1986a) et des champs conceptuels (Vergnaud, 1981), les concepts d’incidents didactiques (Roditi, 2003; Rogalski, 2003), de postures épistémologiques (DeBlois & Squalli, 2002) et d’adaptations (DeBlois & Maheux, 2005) et des conceptions des mathématiques (Noël & Mura, 1999) a permis de répondre à la question suivante : « Quelles sont les pratiques des stagiaires du primaire et du post-primaire dans leur enseignement de la fraction en classe du cours moyen deuxième année (CM2) du primaire et en classe de sixième du post-primaire? » Notre méthodologie de recherche est basée sur une étude de cas multiples. Huit projets d’enseignement, dont quatre pour chaque niveau de scolarité (6e et 7e années) ont été analysés. Des entrevues semi-dirigées qui ont suivi les réalisations de leçons ont été également analysées. Nos analyses montrent que lors des réalisations de cours, les stagiaires rencontrés se préoccupent davantage de l’achèvement des contenus planifiés, ce qui pourrait expliquer qu’ils pratiquent un enseignement transmissif. Certaines conceptions, comme les mathématiques sont transparentes, pourraient soutenir ces pratiques et expliquer que les stagiaires rencontrés adoptent tous une posture d’ancien élève. En outre, nos analyses montrent que les stagiaires du primaire font des adaptations normatives ou d’évitement alors que quatre formes d’adaptation sont manifestées par les stagiaires du post-primaire. Cette compréhension des pratiques des stagiaires pourrait favoriser une formation initiale à l’enseignement des mathématiques amenant les stagiaires à se décentrer de leur posture de l’ancien élève par des transformations dans leurs conceptions des mathématiques et dans leurs préoccupations d’enseignement dans un contexte de large effectif de classe. / In Burkina Faso, studies carried out by Kiélem and Barro (2007) and Traoré (2012) showed deficiencies in the initial training of teachers. For example, Traoré (2012) reveals that initial training to the teaching of mathematics programs are inadequate to the needs of graduates. Our study aims at the understanding of teaching practice of students of primary and post-primary on the fraction in CM2 and first form. The fraction that is taught in the two levels of education is hardly learned by students. A combination of several theories and concepts including theories of didactic situations (Brousseau, 1986a) and conceptual fields (Vergnaud, 1981), the concepts of didactic incidents (Roditi, 2003; Rogalski, 2003), epistemological postures (DeBlois & Squalli, 2002) and adaptations (DeBlois & Maheux, 2005) and mathematical concepts (Noël & Mura, 1999) allowed to answer the following question "what are the practices of primary and post primary trainees in teaching the fraction in the medium second-year (CM2) of the primary level and the first form of the post-primary? "Our research methodology is based on a multiple case study. Eight teaching projects, including four for the Grade 6 and four for the Grade 7 were analyzed. Semi-structured interviews that followed the lessons performance were also analyzed. Our analyzes show that during the performance of course, the trainees encountered are concerned about the completion of the planned contents, which may explain that they practice a transmissive teaching. Some designs, such as mathematics are transparent, could support these practices and explain that trainees encountered all adopt a posture of former student. In addition, our analyzes show that primary school trainees manifest normative adaptations or avoidance while four forms of adaptation are manifested by the post-primary trainees. This understanding of the practical trainees could promote initial training in the teaching of mathematics bringing trainees to decenter from the posture of former student by changes in their conceptions of mathematics and their concerns of teaching in a context of a large class.

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Underachievement in Mathematics hangs over South African Mathematics learners like a dark cloud. TIMSS studies over the past decade have confirmed that South African learners‟ results (Grades 8 and 9 in 2011) remained at a low ebb, denying them the opportunity to compete and excel globally in the field of Mathematics. It is against this backdrop that the researcher investigated the meaningful understanding of the important yet challenging algebraic concept of Proportion. The theoretical as well as the empirical underpinnings of the fundamental idea of Proportion are highlighted. The meaningful learning of Algebra was explored and physical, effective and cognitive factors affecting meaningful learning of Algebra, views on Mathematics and learning theories were examined. The research narrowed down to the meaningful understanding of Proportion, misconceptions, and facilitation in developing Proportional reasoning. This study was embedded in an interpretive paradigm and the research design was qualitative in nature. The qualitative data was collected via task sheets and interviews. The sample informing the central phenomenon in the study consisted of a heterogeneous group of learners and comprised a kaleidoscope of nationalities, both genders, a variety of home languages, differing socio-economic statuses and varying cognitive abilities. The findings cannot be generalised. Triangulation of the literature review, the analysis of task sheets and interviews revealed that overall the participants have a meaningful understanding of the Proportion concept. However, a variety of misconceptions were observed in certain cases. Finally, recommendations are made to address the meaningful learning of Proportion and its associated misconceptions. It is hoped that teachers read and act on the recommendations as it is the powerful mind and purposeful teaching of the teacher that can make a difference in uplifting the standard of Mathematics in South African classrooms! / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2014

Thesis (PhD (Electical and Electronic Engineering))--University of Stellenbosch, 2006. / Resonant frequencies and Q-factors of microwave ring resonators are predicted using interpolation- based modelling. A robust and efficient multivariate adaptive rational-multinomial combination interpolant is presented. The algorithm models multiple resonance frequencies of a microwave ring resonator simultaneously by solving an eigenmode problem. To ensure a feasible solution when using the Method of Moments, a frequency dependent scaling constant is applied to the output model. This, however, also induces a discontinuous solution space across the specific geometry and requires that the frequency dependence be addressed separately from other physical parameters. One-dimensional adaptive rational Vector Fitting is used to identify and classify resonance frequencies into modes. The geometrical parameter space then models the different mode frequencies using multivariate adaptive multinomial interpolation. The technique is illustrated and evaluated on both two- and three-dimensional input models. Statistical analysis results suggest that models are of a high accuracy even when some resonance frequencies are lost during the frequency identification procedure. A three-point rational interpolant function in the region of resonance is presented for the calculation of loaded quality factors. The technique utilises the already known interpolant coefficients of a Thiele-type continued fraction interpolant, modelling the S-parameter response of a resonator. By using only three of the interpolant coefficients at a time, the technique provides a direct fit and solution to the Q-factors without any additional computational electromagnetic effort. The modelling algorithm is tested and verified for both high- and low-Q resonators. The model is experimentally verified and comparative results to measurement predictions are shown. A disadvantage of the method is that the technique cannot be applied to noisy measurement data and that results become unreliable under low coupling conditions.

Thesis (MEd)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Many learners, even at high school level, have difficulty with fractions and computations involving fractions. A report from the Department of Basic Education (DBE, 2012c: 15) has highlighted that the lack in basic fraction sense was one of the areas of concern that contributed to the low achievement in matriculation mathematics examinations in 2012. Fractions play an important role in our ever-advancing technological society. Many occupations today rely heavily on the ability to compute accurately, proficiently, and insightfully with fractions. High school learners’ understanding or the lack thereof is carried over to their tertiary studies and workplaces. It is for that reason that in this dissertation, the learning and understanding of fractions and their role in the high school curriculum are studied through a critical literature review. Fractions are compound constructs and can therefore be interpreted in many different ways, depending on the area of study within mathematics. The concept of fractions consists of five sub-constructs, namely, part-whole, ratio, operator, quotient, and measure (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). This thesis starts with discussion of the background of the study and its importance. Thereafter the elements that assist in the understanding of the fraction concept is discussed. Then, the five different sub-constructs are elaborated on, and how these different sub-constructs are used in the high school curriculum is demonstrated. The conclusion offers some implications for classroom teaching and mathematics teachers’ professional development. / AFRIKAANSE OPSOMMING: Talle leerders, tot op hoërskool vlak, ervaar probleme met breuke en berekeninge met breuke nie. ‘n Verslag van die Departement van Basiese Onderwys (DBE, 2012c: 15) het beklemtoon dat die gebrek aan basiese breuk vaardighede een van die oorsake was wat daartoe gely het dat die prestasie in die 2012 matriek wiskunde eksamen so laag was. Breuke speel ‘n belangrike rol in ons voortdurende tegnologiese voor uitgaande samelewing. Talle beroepe vandag is grootliks afhanklik van die akkurate, bekwame en insiggewende berekeninge van breuke. Hoërskool leerders se begrip, of die gebrek daaraan word oorgedra na hul tersiêre studies en werksplekke. Dit is vir dié rede dat hierdie tesis die leer en begrip van breuke en hul rol in die hoërskool kurrikulum bestudeer deur middel van ‘n kritiese literatuur studie. Breuke is ‘n saamgestelde konsep en kan vir hierdie rede op verskillende wyses geïnterpreteer word, afhangende van die area van studie in wiskunde. Die konsep van ‘n breuk bestaan uit vyf sub-konstrukte, naamlik deel-van-‘n-geheel, ‘n verhouding, operateur, kwosiënt en meting (Behr, Lesh, Post, & Silver, 1983; Kieren, 1980). Hierdie tesis begin met ‘n bespreking oor die agtergrond van hierdie studie en die belangrikheid daarvan. Daarna word die faktore wat bydra tot die verstaan van die breuk konsep. Dit word gevolg deur ‘n uitbreiding op die vyf verskillende sub-konstrukte en waar hierdie verskillende sub-konstrukte in die hoërskool kurrikulum voorkom. Die bevinding bied ‘n paar implikasies vir onderrig. Hierdie studie fokus nie op die ontwerp van enige take of ander leermateriaal vir ‘n intervensie program nie, maar konsentreer op die belangrike kwessies rondom breuke. My hoop is dat die bevindinge van hierdie studie implikasies inhou vir wiskunde onderwysers se professionele ontwikkeling deur hul te motiveer om nuwe leerondersteuningsmateriaal te ontwikkel en die aanbieding van breuke in klaskamers aan te pas sodat die begrip van breuke by leerders ten volle ontwikkel kan word.

Šiame darbe yra aprašomi lengvai biologiškai skaidomų organinių teršalų koncentracijos Palangos miesto nuotekose atskirais metų laikotarpiais bei Visagino miesto nuotekose tyrimai. Tyrimams nuotekos buvo imamos prieš biologinį valymą, o veiklusis dumblas – iš veikliojo dumblo reaktoriaus pabaigos. Tyrimai atlikti VGTU Vandentvarkos katedros ir UAB „Palangos vandenys“ laboratorijose. Siekiant įvertinti teršalų bei technologinių parametrų kitimą bei jų įtaką nuotekų išvalymo rodikliams, buvo surinkti ir statistiškai apdoroti Palangos ir Visagino nuotekų tyrimų rodikliai. Analizuota nuotekų parametrų priklausomybė, teršalų koncentracijų kaita, nuotekų valymo efektyvumas ir jų įtaka biologiniam fosforo šalinimui. Ištirtos lengvai biologiškai skaidomų organinių teršalų koncentracijos. Gauti rezultatai palyginti su anksčiau VGTU Vandentvarkos katedroje atliktų magistrantų nuotekų tyrimų rezultatais. / It is the investigations for determination of concentration of easily biodegradable organic matter in wastewater of Palanga and Visaginas towns. Wastewater for investigations have been taken before biological treatment, and active sludge - from the end of active sludge reactor. The investigations were carried out at Water Management Department of Vilnius Gediminas Technical University and UAB "Palangos vandenys" laboratories. The indicators of investigations of Palanga and Visaginas wastewater were collected and statistically processed in order to evaluate alteration of technological parameters and their influence to indicators of wastewater cleaning. It was analysed the relationship between various parameters in order to find out their influence for the efficiency of biological phosphorus removal. It was studied easily biodegradable organic pollutant concentrations too. The results are compared with previously results of investigations, made by postgraduates of Water Management Department of Vilnius Gediminas Technical University.