Dissertations / Theses on the topic 'Partitive Division'

Thesis (M.A.)--Central Connecticut State University, 1997.
Thesis advisor: Kathy A. Martin-Troy, Ph. D. "... in partial fulfillment of the requirements for the degree of Master of Arts in Biology." Includes bibliographical references (leaves 23-24).

La ségrégation, ou partition, des chromosomes et des plasmides bactériens est l'étape fondamentale du cycle cellulaire qui assure la transmission de l'ensemble du génome aux cellules filles. C'est l'équivalent procaryote de la mitose. Des systèmes de ségrégation, appelés les loci par, ont été identifiés sur les plasmides à bas nombre de copies, et des homologues de ces systèmes de partition sont présents sur la majorité des chromosomes bactériens. Le système code deux protéines, une ATPase et une protéine qui se fixe spécifiquement sur une région centromérique. Ces deux protéines interagissent entre elles, permettent la localisation subcellulaire des réplicons et assurent ainsi leur maintien dans les générations futures. Au laboratoire, nous étudions l'un des systèmes modèles majeurs, le système de partition du plasmide F d'Escherichia coli, afin de déterminer le mécanisme moléculaire assurant le processus de ségrégation et son contrôle pendant le cycle cellulaire. La stabilité du plasmide F est assurée par le système de partition sopABC. Après la réplication du plasmide, la protéine SopB s'assemble sur le centromère sopC pour former un complexe de partition qui permet aux copies du plasmide d'être positionnés au centre de la cellule. Avant la division cellulaire les plasmides migrent aux positions 1/4 et 3/4 de la cellule et assurent ainsi l'héritage des réplicons dans les futures cellules filles. L'ATPase SopA est essentielle dans le processus de partition, mais son rôle n'est pas bien défini. SopA pourrait être impliquée dans les étapes de positionnement et/ou de déplacement des plasmides de part et d'autre de la cellule. SopA possède plusieurs activités. In vivo, SopA agit comme autorépresseur de l'opéron sopAB en se fixant sur la région promotrice. De plus elle interagit avec le complexe de partition et forme des polymères en présence d'ATP. Nous avons montré que cette activité est régulée par SopB et par l'ADN. L'activité ATPase de SopA est essentielle pour la partition. Elle est légèrement stimulée par SopB et par l'ADN, mais lorsque ces deux facteurs sont présents, elle est fortement stimulée. Nous avons entrepris de caractériser les interactions existantes entre ces trois protagonistes. Ainsi, nous avons démontré que cette stimulation nécessite une interaction de SopA avec SopB d'une part et avec l'ADN d'autre part. Nous avons également montré que le site centromérique sopC potentialise la stimulation de l'activité ATPase par l'intermédiaire de SopB. Nous nous sommes intéressés ensuite à l'interaction SopA-SopB, et nous avons mis en évidence que SopB stimule l'activité ATPase de SopA via un motif arginine finger. Pour finir, nous avons montré que in vivo, la stimulation de l'activité ATPase de SopA joue un rôle dans la régulation de l'opéron sopAB mais aussi dans la partition du plasmide F
Mitotic segregation of chromosomes and plasmids, termed partition in bacteria, is a fundamental step of the cell cycle that ensures the transmission of the whole genome to daughter cells. It is governed by specific genetic loci named par, first identified in low copy number plasmids and later found to be present as homologues in most bacterial chromosomes. Par loci encode two proteins, an ATPase and a DNA binding protein, and include a cis-acting centromeric site. These components interact with each other to direct the subcellular localization that ensures stability of their replicons. To determine the molecular mechanisms of the partition process and its control during the cell cycle, we study the Sop partition system of the Escherichia coli plasmid, F. Sop is one of the best-known partition systems. After F plasmid replication, SopB protein binds to the sopC centromeric site to form a partition complex. The complex on each plasmid copy interacts with SopA, an ATPase, and activates it to move the plasmid molecules towards the two cell poles. SopA ATPase is essential to the segregation process but its role is not defined. SopA has many activities. In vivo it represses its own operon by binding to the sopAB promoter. Moreover, in addition to its interaction with the partition complex it polymerizes in the presence of ATP. We have shown that SopB and DNA regulate this activity. Although the ATP-binding site on SopA is essential for partition, ATP hydrolysis by SopA is very weak. It is stimulated modestly by DNA and by SopB and strongly in the presence of both. We have characterized the interactions necessary for stimulation of ATP hydrolysis. First we found that the SopB-sopC partition complex is required for maximal stimulation. Then we showed that SopB and DNA contact SopA by two distinct interactions to fully activate ATPase activity. We also found that SopB activates SopA ATPase through an arginine finger motif. Finally, we have shown that in vivo, stimulation of the ATPase activity is necessary for both regulation of the sopAB operon and partition of plasmid F to be efficient

Le maintien des réplicons bactériens (partition) présente des similitudes avec la mitose eukaryote. Dans l'équipe du Dr Lane nous étudions le plasmide F d'Escherichia coli qui possède un système actif de partition. Ce dernier est très répandu parmi les réplicons bactériens. Il est composé de deux protéines, SopA et SopB produites à partir d'un opéron dont la transcription est réprimée par SopA. En aval se trouve le site sopC sur lequel SopB se fixe pour former le complexe de partition. Nous avons centrés nos recherches sur l'activité ATPase de SopA au moyen d'une mutagenèse du site actif de fixation de l'ATP. J'ai purifié ces protéines mutées et les ai caractérisées, tout en menant en parallèle une étude de leur fonction in vivo. Nous avons établi l'importance de l'activité ATPase dans la partition et dans la répression. La localisation de SopB a été déterminée au moyen d'une protéine SopB-GFP. Nous avons montré qu'elle était dépendante de SopA, sopC et du site actif ATPase de SopA.

La ségrégation du matériel génétique est une étape fondamentale du cycle cellulaire permettant la transmission du patrimoine génétique au cours des générations. Dans les cellules eucaryotes, la mitose est l'étape qui permet la répartition des chromosomes dupliqués dans chaque cellule fille. Des systèmes actifs, dédiés à la ségrégation de l'ADN sont retrouvés sur la majorité des plasmides et chromosomes bactériens. Ces systèmes ParABS, dits "de partition", sont constitués de deux protéines, ParA et ParB, et d'une séquence centromérique, parS. La protéine ParA est une ATPase capable de positionner les plasmides dans le cytoplasme tout au long du cycle cellulaire. Son comportement dynamique fait d'elle le moteur de la partition. La protéine ParB, l'autre acteur de la partition est une protéine adaptatrice entre la molécule d'ADN et la protéine motrice. ParB se fixe sur le centromère parS pour former une complexe nucléoprotéique appelée "complexe de partition". En utilisant différents modes de fixation à l'ADN et en établissant des interactions protéine-protéine multiples, ParB est aussi capable de s'organiser en complexe de partition dit "étendu" qui est le substrat de la réaction de ségrégation. La formation du complexe de partition "étendu" est la première étape du mécanisme de partition et est essentielle au processus de ségrégation. L'architecture de ce complexe n'est connue pour aucun des systèmes de partition parABS. L'un des systèmes modèles majeurs du mécanisme de ségrégation de l'ADN chez les bactéries est le système sopABC du plasmide F d'E. Coli. Ainsi, au cours de ma thèse, j'ai initié plusieurs projets en parallèle, visant à caractériser à la fois in vivo et in vitro, les différentes interactions impliquées dans l'organisation du complexe de partition et du complexe de partition "étendu" de ce plasmide. En collaboration avec l'équipe du Dr Véronique Le Berre au Laboratoire d'Ingénierie des Systèmes Biologiques et des Procédés (LISBP-INSA) de Toulouse, j'ai contribué à la validation des déterminants nucléiques impliqués dans l'interaction de notre séquence centromérique modèle, parS, avec le motif hélice-tour-hélice (HTH) de ParB. Ensuite, nous avons identifié un nouveau motif en dehors du motif HTH et montré qu'une arginine de ce motif est essentielle à l'interaction spécifique avec le centromère. Ces résultats ont révélé une caractéristique conservée dans le règne bactérien : les protéines ParB contiennent un domaine de liaison au centromère, composé de deux motifs séparés et essentiels. Le cœur de mes travaux de thèse a été de comprendre l'organisation du complexe de partition "étendu" et son rôle dans la partition. Par une combinaison d'approche moléculaire in vivo et in vitro, j'ai montré que l'architecture de ce complexe étendu en dehors des sites parS, nécessitait deux types d'interaction, des interactions protéine/protéine mais également ADN/protéine. Afin d'étudier le mode d'interaction de ParB avec des séquences nucléiques non spécifiques avoisinant le complexe de partition, j'ai mis en œuvre une technique d'immunoprécipitation de chromatine, couplée à des techniques de hautes détections (qPCR ou ChIPseq). Cette technique nous a permis de montrer que ParB est capable de s'étendre sur une distance d'environ vingt kilobases de part et d'autre du centromère. Nos résultats semblent incompatibles avec le précédent modèle dans lequel ParB serait capable de polymériser sur l'ADN, à la manière d'un protéo-nucléofilament qui s'initierait au niveau du centromère. Ainsi ces travaux, nous ont permis de proposer un nouveau modèle dans lequel le complexe de partition "étendu" serait une structure concentrée, dynamique et résultant d'interactions stochastiques entre ParB et l'ADN avoisinant ParS
Segregation of genetic material over generations is an essential process ensuring that every daughter cell receives a copy of each DNA molecule. Similarly to Eukaryotes, Prokaryotes possess cytoskeletal machineries, named Par, responsible for DNA segregation. Bacterial Par systems, found on chromosomes as well as on various low copy number plasmids, are composed of three elements: a ParA protein, a ParB protein and a centromere site, parS. ParA ATPase is able to position plasmids in the cytoplasm during the cell cycle. Its dynamic pattern make it the motor of the partition. The centromere binding protein (CBP) ParB, binds the centromere to form a nucleoprotein assembly called the "partition complex". Using different mode of DNA binding and multiple protein-protein interactions, ParB is also capable of organizing into higher order complexes called the "extended" partition complex. This complex is the substrate for the partitioning process. Formation of the extended partition represents the first step in partition and is essential to segregation. The architecture of this complex is not known for any partition system parABS. Here, we focus on the assembly of the F partition complex. During my PhD, I initiated several projects in parallel to characterize the different interactions involved in the organization of the partition complex and the extended partition complex of this plasmid with in vivo and in vitro approaches. In collaboration with the laboratory of Dr. Veronique Le Berre in Toulouse (LISBP -INSA), we determined sopC basis involved in specific SopB-sopC interactions. Then, we identified a new ParB determinant, outside of the helix-turn-helix DNA binding motif, responsible for specific DNA binding to the centromere. These findings reveal that ParB have an extended DNA binding domain, composed of two separate DNA binding motifs. We extended our analysis to chromosomal ParB and show that this second centromere binding motif is highly conserved in a wide range of bacteria. Using in vivo and in vitro approaches, we show that the extended partition complex architecture requires both protein-protein and protein-DNA interactions. To investigate the overall organization of the SopB-sopC extended partition complex, we use chromatin immunoprecipitation (ChIP) coupled with high throughput sequencing. This technique allowed us to visualize that SopB is able to extend around sopC over ~20 Kb. Our results are thus inconsistent with previous models suggesting that SopB polymerize side by side in a proteo-nucleofilament emanating from the centromere. So, we propose a new model in which the extended partition complex of F plasmid assembles in a nucleoprotein complex from stochastic binding of SopB on neighboring sopC DNA

La ségrégation de l'ADN, appelée partition chez les procaryotes, permet à tout organisme de transmettre son patrimoine génétique au cours des générations. Il existe des systèmes actifs de partition présents sur la majorité des plasmides et des chromosomes bactériens. Ces systèmes sont essentiels pour la ségrégation active des plasmides à bas nombre de copies tel que le plasmide F d'Escherichia coli, étudié au sein de notre équipe. Son système de partition, appelé sop, est composé de deux gènes, sopA et sopB et d'une séquence centromérique sopC. SopB se fixe à sopC pour former le complexe de partition, reconnu par l'ATPase SopA. SopA polymérise en présence d'ATP. Ce comportement pourrait être le " moteur " de la ségrégation des réplicons. Les travaux de notre équipe ont démontré que SopA se fixe de manière ATP-dépendante à l'ADN non spécifique. Cette fixation inhibe la polymérisation de SopA. SopB, de par sa capacité à se fixer à l'ADN non spécifique, contrebalance ainsi cette inhibition. Nous proposons un modèle dans lequel la polymérisation de SopA serait régulée dans la cellule par l'ADN du nucléoïde. En présence du plasmide, SopB, présent à forte concentration autour du complexe de partition, masquerait l'ADN, créant un environnement dans lequel SopA initierait sa polymérisation. Cette régulation de la dynamique de SopA serait nécessaire au processus de partition. Afin d'étayer notre modèle, nous avons recherché un domaine d'interaction à l'ADN dans SopA. Nous avons identifié un mutant ayant perdu sa fixation ATP-dépendante à l'ADN. Seules les activités de SopA dépendante de cette reconnaissance de l'ADN ont été affectées : l'inhibition de la polymérisation, la stimulation de l'activité ATPase basale et la localisation intracellulaire. Cette mutation entraîne aussi une perte majeure de stabilité du plasmide F correspondant. Ceci confirme l'implication de l'ADN du nucléoïde dans la régulation du comportement dynamique de SopA nécessaire à la partition du plasmide F
The segregation of the DNA, also called partition for procaryotes, is the process allowing any organisms to transmit its genetic heritage to next generation. In bacteria, mitotic stability of plasmids and many chromosomes depends on replicon-specific systems which comprise a centromere, a centromere-binding protein and an ATPase. We have taken as a model, the low-copy number plasmid F of Escherichia coli. Centromere-binding protein SopB binds to sopC centromere and forms the partition complex. This nucleoproteic complex is recognized by the SopA "Walker-box" ATPase. SopA shares with other partition ATPase the capacity of self assembly in presence of ATP. This dynamic self-assembly would allow active partition during bacterial division. Previous work in our team showed SopA is also able to bind to non specific DNA in an ATP-dependant manner whereby polymerization is inhibited. Indeed, DNA inhibited this polymerization and cause breakdown of pre-formed polymers. SopB counteracted this DNA effect by binding itself to and masking DNA. We had proposed a model in which the polymerization is spacially regulated. Nucleoid DNA prevent inappropriate SopA polymerization but when SopB is present in high concentration, it create a DNA-depleted zone within SopA can initiate polymerization. The regulation of the dynamic behaviour of the "driving" protein of the system would be necessary for the process of partition. To support our model, we looked for a DNA binding domain in SopA. We have found a SopA mutant, defective for ATP dependent DNA binding. Only the activities of SopA dependent on this binding were affected: the inhibition of the polymerisation is abolished, as the stimulation of the ATPase activity and the intracellular localization. Moreover, this mutant is defective for plasmid stabilization. This last result confirms the implication of the nucleoïd DNA in regulation of the dynamic behavior of SopA, which is necessary for the partition of the plasmide F

The interplay between spectrum and structure of graphs is the recurring theme of the three more or less independent chapters of this thesis. The first chapter provides a method to relate the eigensolutions of two matrices, one being the principal submatrix of the other, via an arbitrary annihilating polynomial. This is extended to lambda-matrices and to matrices the entries of which are rational functions in one variable. The extension may be interpreted as a possible generalization of other known techniques which aim at reducing the size of a matrix while preserving the spectral information. Several aspects of an application in order to reduce the computational costs of ordinary eigenvalue problems are discussed. The second chapter considers the straightforward extension of the well known concept of equitable partitions to weighted graphs, i.e. complex matrices. It provides a method to divide the eigenproblem into smaller parts corresponding to the front divisor and its complementary factor in an easy and stable way with complexity which is only quadratic in matrix size. The exploitation of several equitable partitions ordered by refinement is discussed and a suggestion is made that preserves hermiticity if present. Some generalizations of equitable partitions are considered and a basic procedure for finding an equitable partition of complex matrices is given. The third chapter deals with isospectral and unitary equivalent graphs. It introduces a construction for unitary equivalent graphs which contains the well known GM-switching as a special case. It also considers an algebra of graph matrices generated by the adjacency matrix that corresponds to the 1-dimensional Weisfeiler-Lehman stabilizer in a way that mimics the correspondence of the coherent closure and the 2-dimensional Weisfeiler-Lehman stabilizer. The algebra contains the degree matrix, the (combinatorial, signless and normalized) Laplacian and the Seidel matrix. An easy construction produces graph pairs that are simultaneously unitary equivalent w.r.t. that algebra.

Le droit au partage est généralement présenté comme étant consubstantiel du droit de propriété. Ce dernier est garanti par des textes de valeur constitutionnelle. Or, sinon le caractère prétendument absolu du droit de propriété, du moins son rang dans la hiérarchie des droits réels, s'opposent à ce que ses titulaires se trouvent contraints de l’exercer selon des modalités non voulues. Il importe dès lors que l'opération de partage soit impérativement réalisée dans le respect du droit dont elle est le prolongement, c'est-à-dire conformément aux intérêts patrimoniaux des copartageants. C'est « l’égalité » du partage, entendue alors comme synonyme de neutralité économique de l’opération. Mais les relations entre égalité et partage excèdent largement la notion d'égalité face à l'opération de « conversion » que réalise le partage. L'égalité n'est pas seulement le critère et l'impératif du partage ; elle en est aussi la « cause-impulsive », et parfois le motif. En certaines occasions, c'est parce qu'une certaine forme d'égalité aura été voulue qu'un partage s'établira. A cet égard, l’égalité ne se borne alors plus à assurer la bonne fin du partage ; elle le suscite. Le partage ne saurait se réduire en effet à la simple réalisation d’une opération commutative. Il assure la réception, en droit privé et, plus particulièrement, au sein du droit patrimonial de la famille, d’une aspiration égalitaire transcendant la technique juridique. La convention de partage n’est pas seulement un contrat nécessairement « égalitaire » ; elle emporte « contractualisation de l’égalité ». L’égalité en constitue le principal objet
The right to an equal share in the partition / to equal partition is undisociable from the right to property which is garanteed by Constitutional law. So, If not because of the allegedly absolute right to property, at least because of its rank in the hierarchy of real rights, the right to division is contrary to any acceptance by the entitled (of the proceedings) on undesired modalities. This is equality in partition in the full sense of the word – a synonym, here, of economic neutrality of the operation. The relations between equality and partition go beyond the notion of equality as understood in partition proper. Equality, indeed, is not solely the criterium and imperative for the partition, it is also the cause and, at times, even, the mobile. It may happen that partitioning is made because a certain kind of equality is desired. In these cases, equality has not its goal limited in bringing about a partition; but partition finds its cause in equality. It can, thus, be said that partitioning sums up both the efficacy of the right to property and the enforcement of a principle of equality that far exceeds its legal technique. It is clear, indeed, that the partition can’t be reduced to the sole effectiveness of a necessarily accumulative operation. It helps acknowledge, within, the private law and the patrimonial family law, in particular, that aspiration for equality that far transcends legal procedures. In that sense, the partition agreement must not be regarded solely as a necessarily egalitarian contract; it entails that equality itself be a contract. Indeed, equality is not a characteristic of the operation but its main object

L'objectif de cette thèse est de définir la chorégraphie de ségrégation des deux chromosomes circulaires de Vibrio cholerae, c'est à dire le positionnement de l'information génétique au cours de la croissance de la cellule, ainsi que les mécanismes dirigeant ces ségrégations. Il a longtemps été supposé que les bactéries étaient trop petites pour avoir une organisation intra-cellulaire, et le manque de techniques appropriées ne permettait pas d'infirmer cette hypothèse. Or la taille des chromosomes comparée à celle de la bactérie impose une compaction et aujourd'hui, de nouvelles techniques de microscopie et d'analyse génétique permettent d'affirmer que les chromosomes bactériens étudiés jusqu'à maintenant ont tous une organisation et une chorégraphie de ségrégation précises et différentes selon les espèces. Toutes les espèces étudiées à ce jour ont un chromosome circulaire unique : la réplication du chromosome commence à une origine unique bidirectionnelle, les deux fourches de réplication se déplacent le long des deux bras de réplication (ou réplichores) et finissent la réplication au terminus, diamétralement à l'opposée de l'origine de réplication sur la carte du chromosome. Peu d'espèces ont été étudiées, et Vibrio cholerae émerge progressivement comme un nouveau modèle : son génome est réparti sur deux chromosomes, et la chorégraphie de plusieurs chromosomes dans une cellule n'a jamais été décrite. De plus, cette espèce semble être au croisement évolutif entre Caulobacter crescentus et Escherichia coli : Vibrio cholerae a d'une part une morphologie en croissant, des systèmes de partition aux origines et un positionnement de l'origine du chromosome I, semblables à C. crescentus, et d'autre part un système de compaction du terminus et un set de gènes impliqués dans la maintenance du chromosome ayant co-évolué, qu'on ne retrouve que dans peu d'espèces proches d'E. coli. Une autre caractéristique intéressante de V. cholerae est que le chromosome II semble avoir été acquis récemment et n'est donc peut être pas gouverné par les mêmes mécanismes que le chromosome I, comme en témoignent le positionnement de son origine et son terminus, inédits pour des chromosomes bactériens. Parmi les Vibrios (environ 60 espèces principalement retrouvées dans les environnements aquatiques), certaines espèces sont des pathogènes dévastateurs pour les poissons, le corail, les crustacés ou les fruits de mer. Mais la plus documentée est Vibrio cholerae, car elle provoque chez l'Humain une maladie provoquée par l'ingestion d'eau contaminée qui peut être mortelle si le patient n'est pas réhydraté à temps. Bien que facilement traitable, le choléra fait encore de nombreuses victimes dans les pays en développement où les structures de santé et les règles d'hygiène font parfois défaut. Ainsi l'étude de Vibrio cholerae présente un intérêt médical, mais également par extension aux autres Vibrios, un intérêt environnemental non négligeable.

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Secretaria da Educação do Estado de São Paulo
The intention of this dissertation is the elaboration and analysis of a didactic sequence focused to the teaching and learning of examinations for students of the 9th school year in Brazil, using a computer tool. The whole experiment was developed in a Project of Argumentations and test in the Mathematics school (AProvaME), that he/she has as objective to prepare a map on the conceptions of arguments and the students tests in the state of São Paulo in Brazil. The elaboration of the didactic sequence was inspired in the questionaire of Álgebra of the project AProvaME, based and analyzed under the perspective from the levels of Balacheff (1988) classification of experiments and ideas related to the roles and functions of the experiments of Villiers (2001) and in the ideas suggested in PCN. We used as computer resource EXCEL, which allowed to the the students the electronic construction of spreadsheets enlarging the data for better analyze and elaborate their conjectures, arguments, justifications and validations. It was analyzed three protocols of three couples with ages between 13 and 14 years old of a private school in the city of Lorena in São Paulo which contributed voluntarialy with the experimentation. Analysing the results we can establish that predominated the formulations of the arguments and conjectures limked to conceptual pragmatic experiments described in the natural languages that in the whole process were empirically justified. A relevant and decisive fact in the successful research was the role of the mediator teacher during the moments of interventions that it provided the interaction student-teacher, necessary in the development of didactic sequences that aim the teaching and learning of arguments and mathematical experiments
O propósito desta dissertação é a elaboração e análise de uma seqüência didática para o ensino e aprendizagem de provas e argumentações, destinada a alunos do 9º ano do Ensino Fundamental, utilizando uma ferramenta computacional. Todo o experimento foi desenvolvido no âmbito do Projeto Argumentação e prova na Matemática Escolar (AProvaME), que tem como objetivo construir um mapa sobre as concepções de argumentação e prova de alunos do Estado de São Paulo. A elaboração da seqüência didática foi inspirada no questionário de Álgebra do projeto AProvaME e fundamentada e analisada sob a perspectiva da classificação de provas de Balacheff (1988), das idéias relacionadas aos papéis e funções das provas de De Villiers (2001) e nas sugestões apresentadas nos PCN. Utilizamos como recurso computacional o Excel, que permitiu aos alunos a construção de planilhas eletrônicas ampliando os dados para melhor analisar e elaborar suas conjecturas, argumentos, justificativas e validações. Foram analisados os protocolos de três duplas, com idades de 13 e 14 anos, de uma escola particular da cidade de Lorena-SP, que contribuíram voluntariamente com a experimentação. Analisando os resultados, constatamos que predominaram as formulações de argumentos e conjecturas vinculadas a provas pragmáticas descritas na língua natural, que, em todo o processo, foram justificadas empiricamente. Um fato relevante e determinante no sucesso da pesquisa foi o papel do professor mediador, durante os momentos de intervenções, favorecendo a interação aluno-professor, necessária no desenvolvimento de seqüências didáticas que visam o ensino e aprendizagem de argumentações e provas matemáticas

abstract: Researchers have described two fundamental conceptualizations for division, known as partitive and quotitive division. Partitive division is the conceptualization of a÷b as the amount of something per copy such that b copies of this amount yield the amount a. Quotitive division is the conceptualization of a÷b as the number of copies of the amount b that yield the amount a. Researchers have identified many cognitive obstacles that have inhibited the development of robust meanings for division involving non-whole values, while other researchers have commented on the challenges related to such development. Regarding division with fractions, much research has been devoted to quotitive conceptualizations of division, or on symbolic manipulation of variables. Research and curricular activities have largely avoided the study and development of partitive conceptualizations involving fractions, as well as their connection to the invert-and-multiply algorithm. In this dissertation study, I investigated six middle school mathematics teachers’ meanings related to partitive conceptualizations of division over the positive rational numbers. I also investigated the impact of an intervention that I designed with the intent of advancing one of these teachers’ meanings. My findings suggested that the primary cognitive obstacles were difficulties with maintaining multiple levels of units, weak quantitative meanings for fractional multipliers, and an unawareness of (and confusion due to) the two quantitative conceptualizations of division. As a product of this study, I developed a framework for characterizing robust meanings for division, indicated directions for future research, and shared implications for curriculum and instruction.
Dissertation/Thesis
Doctoral Dissertation Mathematics Education 2019

Dissertação de mestrado em Estudos da Criança (área de especialização em Ensino e Aprendizagem da Matemática)
Esta dissertação pretende perceber como as crianças de 4 e 5 anos entendem as relações lógicas da divisão partitiva de quantidades discretas. Procurou-se saber como as crianças pequenas compreendem a relação inversa entre divisor e quociente, quando o dividendo se mantém constante; e ainda, como entendem a divisão de quantidades discretas em partes iguais. Este estudo incidiu num grupo de crianças de 4 e 5 anos de Jardins de Infância do concelho de Esposende, Braga. As crianças realizaram tarefas de divisão partitiva de quantidades discretas envolvendo conjuntos de 12 e 24 unidades. Os dados foram recolhidos a partir de entrevistas individuais, estruturadas, tendo-se recorrido à gravação áudio e vídeo. Utilizou-se uma metodologia quantitativa na análise dos dados. Esta análise centrou-se nas estimativas das crianças para o quociente nas divisões, nos procedimentos por elas utilizados, nas suas justificações e nos desempenhos por elas apresentados. Os resultados indicam que as crianças de 4 e 5 anos conseguem efectuar estimativas para o quociente quando varia o divisor e o dividendo se mantém constante. Nestas condições, o grupo de 5 anos obteve melhor desempenho e justificou as suas respostas dando a entender que compreendem a relação inversa entre o divisor e quociente. O procedimento mais utilizado pelas crianças consistiu na distribuição um-a-um, tendo o procedimento de agrupamentos baseados na percepção sido também frequentemente usado por muitas crianças. Os procedimentos que envolveram contagens foram pouco utilizados pelas crianças, apesar de terem sido mais frequentes no grupo etário de 5 anos. A maioria das crianças de 4 anos e a quase totalidade das crianças de 5 anos revelaram entender intuitivamente a necessidade de obtenção de partilhas equitativas. Uma grande maioria das crianças de 5 anos, contrastando com uma minoria das crianças de 4 anos, justificou as suas respostas baseando-se em factos numéricos. Não existiram diferenças significativas nos desempenhos das crianças nas tarefas de divisão devido à idade, tendo-se observado melhor desempenho nas tarefas que envolveram um dividendo menor. O facto de algumas crianças de 4 anos não saberem contar correctamente os objectos ao seu dispor, não impediu a realização correcta das tarefas. Assim, este estudo faz-nos pensar na possibilidade de se estudar mais aprofundadamente a inclusão de problemas de divisão partitiva de quantidades discretas, mantendo-se constante o dividendo e variando o divisor, nas práticas do pré – escolar.
The present dissertation tries to understand how 4 and years-old children comprehend the logical relations of partitive division of discrete quantities. It was intended to know how small children comprehend the inverse relation between divisor and quotient, when the dividend stays constant, and even more, how they understand the division on discrete quantities in equal parts. This study focused on a group of 4 and 5 years-old children of Kindergartens from Esposende, Braga. The children performed tasks of partitive division of discrete quantities involving sets of 12 and 24 units. The data was collected from structured, individual interviews, with the use of audio and video recorders. A quantitive methology was used in the data analysis. This analysis focused on the children’s estimates on divisions, on procedures they used, on justifications and performances presented. The results point out that 4 and 5 years-old children can make estimates for the quotient when the divisor varies and the dividend stays constant. In these circumstances, the 5 years-old group had better performances and justified their answers, implying they understand the inverse relation between division and quotient. The procedure most used by children consisted on one-by-one distribution, having been also frequently used by many the perception based procedures of grouping. The procedures that involved counting were barely used, even though it was more frequently in the 5 years-old. Most of the 4 years-old and almost the total of the 5 years-old children revealed an intuitive understanding of the need of getting equitative partitions. A great majority of 5 years-old children opposing to a minority of 4 year-old children, justified their answers based on numeric facts. There were no significative diferences on children’s performances on the division tasks due to age, having been observed better accomplishments on tasks involving a minor dividend. The fact that some 4 years-old children don’t know how to count the avaiable objects properly, didn’t restrain the correct development of the tasks. Thus, this study makes us think about the possibility of deepening a study on the inclusion of partitive division problems with discrete quantitites, maintaining constant the dividend and ranging the divisor, on Pre-School pratice.

In most cases in a deceased person’s estate, there are problems with co-ownership where more than one family member inherits the deceased family estate assets. To escape the perils of co-ownership the beneficiaries consensually agree to divide the inherited communallyshared asset/s. This agreement can take place immediately after the death of the family estate owner or some time later regarding some or all of the said assets. On the conclusion of the division agreement, the contractual party who receives the awarded assets enjoys sole ownership and the other contractual parties by agreement retract their ownership. In a jurisprudential content analysis of forty-six recorded family deceased division agreements from Old Babylonian Larsa and Nippur, essential elements are identified which are the framework and qualification requirements for a family deceased division agreement. Within this framework the concepts, terms and elements of the agreement are categorised as natural and incidental elements, which reflect the specific law traditions and choices of contractual parties and show the unique scribal traditions in the different Old Babylonian city-states of Larsa, Nippur and Sippar. The aim of the study is to shed a more focused light on the interpretation of recorded Old Babylonian division agreements and to show that the division agreement was a successful, timeless, estate administration mechanism and tool to obviate any undesirable consequences of co-ownership of the bequeathed property.
Old Testament & Ancient Near Eastern Studies
D. Litt. et Phil. (Ancient Near Eastern Studies)

The Mycobacterium genus contains major human pathogens, like Mycobacterium tuberculosis and Mycobacterium leprae, which are the causative agents of Tuberculosis and Leprosy, respectively. They have evolved as successful human pathogens by adapting to the adverse conditions prevailing inside the host, which include host immune activation, nutrient depletion, hypoxia, and so on. During such adaptation for the survival and establishment of persistent infection inside the host, the pathogen, like M. tuberculosis, regulates its cell division. It is known that M. tuberculosis enters a state of non-replicating persistence (NRP) inside the host, to establish latent infection, which helps the survival of the pathogen under adverse host conditions such as hypoxia and nutrient depletion. The pathogen can reactivate itself, to come out of the NRP state, and establish active infection at a later stage, when conditions are suitable for its proliferation. The altered physiological state of the latent bacterium makes it tolerant to drugs, which are only effective against proliferating tubercle bacilli. In view of this unique behavioural physiology of tubercle bacilli, it is important to study the process of cell division and how it is regulated in the NRP and actively growing states. The work reported in the thesis is an attempt to understand these aspects of mycobacterial cell division. iii Chapter 1. Introduction: This chapter gives a detailed introduction to bacterial cell division and its regulation in various organisms, like Escherichia coli, Bacillus subtilis, Caulobacter crescentus, and others. In the background of this information, the major studies on mycobacterial cell division and its regulation are presented. Chapter 2. Materials and Methods: This chapter describes in detail all the materials and methods used in the experiments, which are presented in the four data chapters, 3-6. Chapter 3. Ultrastructural Study of the Formation of Septal Partition and Constriction in Mycobacteria and Delineation of its Unique Features: Mycobacteria have triple-layered complex cell wall, playing an important role in its survival under adverse conditions in the host. It is not known how these layers in the mother cell participate during cell division. Therefore, the ultrastructural changes in the different envelope layers of Mycobacterium tuberculosis, Mycobacterium smegmatis, and Mycobacterium xenopi, during the process of septation and septal constriction, were studied, using Transmission and Scanning Electron Microscopy. The unique aspects of mycobacterial septation and constriction were identified and were compared with those of E. coli and Bacillus subtilis septation. Further, based on all these observations, models were proposed for septation in M. tuberculosis and M. smegmatis. Chapter 4. Identification of Asymmetric Septation and Division in Mycobacteria and Its Role in Generating Cell Size Heterogeneity: Bacterial populations are known to harbour phenotypic heterogeneity that helps survival under stress conditions, as this heterogeneity comprises subpopulations that have differential susceptibility to stress conditions. The iv heterogeneity has been known to lead to the requirement for prolonged drug treatment for the elimination of the tolerant subpopulation. Hence, it is important to study the different mechanisms, which operate to generate population heterogeneity. Therefore, in this chapter, studies were carried out to find out whether asymmetric septation and division occur in mycobacteria to generate cell size heterogeneity. Subpopulations of mycobacterial mid-log phase cells of M. tuberculosis, M. smegmatis, and M. xenopi were found to undergo asymmetric division to generate cell size heterogeneity. The asymmetric division and the ultrastructure and growth features of the products of the division were studied. Chapter 5. Study of Mycobacterial Cell Division Using Growth-Synchronised Cells: In this chapter, different stages of cell septation and constriction were studied using growth-synchronised M. smegmatis cells. Phenethyl alcohol (PEA), which has been found to reversibly arrest mycobacterial cells, was used for growth synchronisation. The growth-synchronised mycobacterial cells, which were released from PEA block, were studied at different stages of septation and septal constriction, at the ultrastructural and molecular levels. Chapter 6. Identification of the Stage of Cell Division Arrest in NRP Mycobacteria: The exact stage at which the NRP tubercle bacilli are arrested in cell division is currently unknown. In Wayne’s in vitro model for hypoxia-responsive tubercle bacilli, gradual depletion of oxygen leads to hypoxic stress, inducing the bacilli to enter non-replicating persistence (NRP) state. Using this model, the stage of cell division arrest in M. tuberculosis was characterised at the ultrastructural and molecular levels. Hypoxia-stressed M. smegmatis was used as an experimental system for contrast. The thesis concludes with salient findings, a bibliography, and the list of publications.

The interplay between spectrum and structure of graphs is the recurring theme of the three more or less independent chapters of this thesis. The first chapter provides a method to relate the eigensolutions of two matrices, one being the principal submatrix of the other, via an arbitrary annihilating polynomial. This is extended to lambda-matrices and to matrices the entries of which are rational functions in one variable. The extension may be interpreted as a possible generalization of other known techniques which aim at reducing the size of a matrix while preserving the spectral information. Several aspects of an application in order to reduce the computational costs of ordinary eigenvalue problems are discussed. The second chapter considers the straightforward extension of the well known concept of equitable partitions to weighted graphs, i.e. complex matrices. It provides a method to divide the eigenproblem into smaller parts corresponding to the front divisor and its complementary factor in an easy and stable way with complexity which is only quadratic in matrix size. The exploitation of several equitable partitions ordered by refinement is discussed and a suggestion is made that preserves hermiticity if present. Some generalizations of equitable partitions are considered and a basic procedure for finding an equitable partition of complex matrices is given. The third chapter deals with isospectral and unitary equivalent graphs. It introduces a construction for unitary equivalent graphs which contains the well known GM-switching as a special case. It also considers an algebra of graph matrices generated by the adjacency matrix that corresponds to the 1-dimensional Weisfeiler-Lehman stabilizer in a way that mimics the correspondence of the coherent closure and the 2-dimensional Weisfeiler-Lehman stabilizer. The algebra contains the degree matrix, the (combinatorial, signless and normalized) Laplacian and the Seidel matrix. An easy construction produces graph pairs that are simultaneously unitary equivalent w.r.t. that algebra.