在這篇論文裡，我們主要有兩部分。首先，我們給了一類新的素有理函數，接下來，我們整理了一個有理函數分解的反例，並討論了Weierstrass p-函數的素性。 / In this thesis, our primary objective here is two parts. First, we give a new class of prime rational functions. Next, we study an example in the factorization of the rational functions and the primeness of the Weierstrass p-function.

The purpose of this study was to determine whether fourth-, sixth-, and eighth-grade students used rational number reasoning to solve comparison of area tasks, and whether the tendency to use such reasoning increased with grade level. The areas to be compared were not similar and therefore, could not directly be compared in a straightforward manner. The most viable solution involved comparing the part-whole relationships inherent in the tasks. Rational numbers in the form of fractional terms could be used to express the part-whole relationships. The use of fractional terms provided a means for students to express the areas to be compared in an abstract manner and thus free themselves from the perceptual aspects of the tasks. The study examined how students solve unique problems in a familiar context where rational number knowledge could be applied. It also noted the effect of introducing fraction symbols into the tasks after students had indicated how they would solve the problems without any reference to fractions. Data were gathered through individual task-based interviews which consisted of 21 tasks, conducted with 36 elementary and junior high school students (12 students each in the fourth, sixth, and eighth grades). Each interview was video and audio taped to provide a record of the students' behavioral and verbal responses. The student responses were analyzed to determine the strategies the students used to solve the comparison of area tasks. The student responses were classified into 11 categories of strategies. There were four Part-Whole Categories, one Part-Whole/Direct Comparison Combination category and six Direct Comparison categories. The results of the study indicate that the development of rational number instruction should include: learning sequences which take students beyond the learning of a set of fraction concepts and skills, attention to the interaction of learning and the visual aspects of instructional models, and the careful inclusion of different types of fractions and other rational number task variables. This study supports the current national developments in curriculum and evaluation standards for mathematics instruction which stress the ability of students to problem solve, communicate, and reason.

This thesis introduces a new quantity called loop number, and shows the conditions in which loop numbers become knot invariants. For a given knot diagram D, one can traverse the knot diagram and count the number of loops created by the traversal. The number of loops recorded depends on the starting point in the diagram D and on the traversal direction. Looking at the minimum or maximum number of loops over all starting points and directions, one can define two positive integers as loop numbers of the diagram D. In this thesis, the conditions under which these loop numbers become knot invariants are identified. In particular, the thesis answers the question when these numbers are invariant under flypes in the diagram D.

The thesis considers the career of the Student Christian Movement (SCM) which was founded in 1892 to promote missions and to recruit students for missionary work. As it grew, the SCM extended its operations to the founding and servicing of Christian Unions in colleges and progressively abandoned its evangelical roots and come to play a major part in the development of liberalism and ecumenism. In the nineteen sixties it became more radical than liberal and developed an interest in Marxism and alternative life styles. The career of the conservative evangelical Inter-Varsity Fellowship (IVF), formed as a result of a number of schisms from SCM, is also charted. These two movement organisations are considered in the light of ideas derived from the sociology of social movements. In the Introduction a brief critical account of various dominant theories of social movement origination is presented and elements of an alternative, voluntaristic, and essentially processual account are advanced. The careers of SCM and IVF are used to suggest correctives to a number of theoretical insights that have been developed on the basis of an exaggeration of the division between stable society and social movement. Particular topics dealt with include the growth and spread of social movements, goal transformation, schism and decline. It is argued that the rapid rise of SCM can be understood as resulting from (a) the existence of a wealthy milieu which accepted the movement as legitimate and (b) the SCM's attitude towards its own purpose and ideology which was open and inclusive. This denominationalism allowed the SCM to utilise the resources of the milieu and to recruit rapidly. It also laid the foundation for an erosion of purpose and identity. Many of the problems that promoted the decline of the SCM were caused by the particular nature of its constituency, recruiting as it did among students and experiencing therefore a high membership turnover, but a full understanding of the contrast between the decline of SCM and the stability of lVF requires consideration of the ideologies that informed the two organisations. For this reason the final chapter is concerned with the reasons for the precariousness of liberal protestantism and the strength of conservative evangelicalism.

In recent times the emergence of the property cycle and the effects that it has on the property market has caused the relevant parties involved in the market to start placing more emphasis on how these cycle works. The overall objective of this study is to try establish if the interpretation of market related data affects at what position these parties are relative to one another on the property curve. The study concentrates on the use of market indicators, indices and variables in trying to determine an individual’s position on the property market curve. It also concentrates on how this market data is retrieved and what effect it has on how they interpret the data. The methodology adopted for this study involves the collecting and interpretation of market related indices and indicators relevant to the property market over a ten year period from 1996 through to 2006. This data was then used to establish the key indicators used. A questionnaire was sent out to the relevant parties involved in the property market to ascertain the extent of what the main sources of market information are and how this data is collected and interpreted. This was limited to individuals in the Gauteng region. The data was examined and collected in the form of line graphs, histograms and pie charts. The data was then examined and presented in four areas: the major sources of information used by parties for market related data, to try and establish where these parties are relative to one another on the property curve, the effect that the different types of sources of information has on each party and finally to try determine by how much these parties lag or lead one another on the curve.

A thesis submitted to the Wits School of Education, Faculty of Humanities, University of the Witwatersrand in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / This study explores the potential for Web-based diagnostic assessments in the classroom, with specific focus on certain common challenges experienced by learners in the development of their rational number knowledge. Two schools were used in this study, both having adequate facilities for this study, comprising a well-equipped computer room with one-computer-per-learner and a fast, reliable broadband connection. Prior research on misconceptions in the rational numbers has been surveyed to identify a small set of problem types with proven effectiveness in eliciting evidence of misconceptions in learners. In addition to the problem types found from prior studies, other problem types have been included to examine how the approach can be extended. For each problem type a small item bank was created and these items were presented to the learners in test batteries of between four and ten questions. A multiple-choice format was used, with distractor choices included to elicit misconceptions, including those previously reported in prior research. The test batteries were presented in dedicated lessons to learners over four consecutive weeks to Grade 7 (school one) and Grade 8 (school two) classes from the participating schools. A number of test batteries were presented in each weekly session and, following the learners’ completion of each battery, feedback was provided to the learner with notes to help them reflect on their performance. The focus of this study has been on diagnosis alone, rather than remediation, with the intention of building a base for producing valid evidence of the fine-grained thinking of learners. This evidence can serve a variety of purposes, most significantly to inform the teacher on each learners’ stage of development in the specific micro-domains. Each micro-domain is a fine-grained area of knowledge that is the basis for lesson-sized teaching and learning, and which is highly suited to diagnostic assessment. A fine-grained theory of constructivist learning is introduced for positioning learners at a development stage in each micro-domain. This theory of development stages is the foundation I have used to explore the role of diagnostic assessment as it may be used in future classroom activity. To achieve successful implementation into time-constrained mathematics classrooms requires that diagnostic assessments are conducted as effectively and efficiently as possible. To meet this requirement, the following elements of diagnostic assessments were investigated: (1) Why are some questions better than others for diagnostic purposes? (2) How many questions need to be asked to produce valid conclusions? (3) To what extent is learner self-knowledge of item difficulty useful to identify learner thinking? A Rasch modeling approach was used for analyzing the data, and this was applied in a novel way by measuring the construct of the learners’ propensity to select a distractor for a misconception, as distinct from the common application of Rasch to measure learner ability. To accommodate multiple possible misconceptions used by a learner, parallel Rasch analyses were performed to determine the likely causes of learner mistakes. These analyses were used to then identify which questions appeared to be better for diagnosis. The results produced clear evidence that some questions are far better diagnostic discriminators than others for specific misconceptions, but failed to identify the detailed rules which govern this behavior, with the conclusion that to determine these would require a far larger research population. The results also determined that the number of such good diagnostic questions needed is often surprisingly low, and in some cases a single question and response is sufficient to infer learner thinking. The results show promise for a future in which Web-based diagnostic assessments are a daily part of classroom practice. However, there appears to be no additional benefit in gathering subjective self-knowledge from the learners, over using the objective test item results alone. Keywords: diagnostic assessment; rational numbers; common fractions; decimal numbers; decimal fractions; misconceptions; Rasch models; World-Wide Web; Web-based assessment; computer-based assessments; formative assessment; development stages; learning trajectories.

Prescribing medicines is the primary intervention that most doctors offer to influence their patients’ health; however concerns have been expressed about the extent to which graduates are prepared by medical schools to assume prescribing responsibility. Both students and clinical teachers have identified a gap between workplace prescribing demands placed on newly qualified doctors and their preparation for this complex activity during undergraduate training. This study explored the exit-level prescribing performance of final-year students in the Graduate Entry Medical Programme at the University of the Witwatersrand compared with students’ perceptions of their prescribing competence. The results indicated a disparity between students’ competence and confidence. Examination marks showed that 83.6% of students were competent to prescribe according to the graduating standards of the University; however, questionnaire data revealed that 66% of students did not feel that their training had enabled them to prescribe rationally. This inconsistency was explored by analysis of the examination papers according to Bloom’s Revised and the SOLO Taxonomies. It was concluded that students score well on questions which test recall and application of knowledge, but some do not manage questions involving evaluation. Since prescribing is a complex skill that requires evaluative competence, this may explain why, despite high examination scores, students remain insecure. Exploration of the structure of knowledge through a Bernsteinian lens revealed that curricular components including problem-based learning and horizontal integration constrain epistemic access to the structure of rational prescribing knowledge for some students. It is recommended that rational prescribing skills should be taught as a synchronous strand within the curriculum, rather than in the current integrated mode. Learning could also be improved by innovative pedagogies associated with active learning and improved feedback.

Syftet med denna studie är att ta reda på hur lärare i grundskolans tidigare år introducerar och arbetar med området bråkräkning. Utifrån detta syfte så formulerades tre stycken frågeställningar: Hur beskriver lärare att de introducerar området för sina elever? Hur beskriver lärare i grundskolans tidigare år att de arbetar med området? Samt är lärare medvetna om någon svårighet med området bråk? För att kunna besvara dessa tre frågeställningar genomfördes kvalitativa intervjuer med sju stycken lärare som arbetar runt om i Sverige. Studiens resultat visar att bråkräkning är någonting som upplevs som svårt av många elever samt att grunden till förståelse för området ligger vid en tydlig introduktion av både området i sig, men även av väsentliga begrepp. De intervjuade lärarna har även beskrivit hur de introducerar och arbetar med området bråkräkning och detta diskuteras sedan i enighet med tidigare forskning.

Para os ingredientes farmacêuticos ativos (APIs) que exibem baixa solubilidade e/ou estabilidade e/ou perfis de dissolução, dentre outros problemas capazes de afetar a sua eficácia terapêutica, as abordagens da Engenharia de Cristais têm se destacado nas últimas décadas como uma solução satisfatória. No estado sólido, os APIs podem apresentar polimorfos, sais, solvatos, co-cristais, amorfos e combinações dos mesmos. Associadas a cada estado sólido estão propriedades físicas e químicas, que podem ou não variar em relação ao API de referência. Neste contexto, este projeto de pesquisa visou o planejamento racional, síntese e caracterização de novas formas sólidas do pró-fármaco 5-fluorocitosina (5-FC) e do fármaco antineoplásico 5-fluorouracila (5-FU), ambos exibindo problemas fisicoquimicos que dificultam sua aplicação em formas de dosagem sólidas. Apesar de ser usado como fungicida, recentemente o 5-FC se tornou um dos pró-fármacos mais utilizados na terapia antineoplásica por meio de terapia dirigida por gene-enzima-pró-fármaco, uma vez que na presença da enzima citosina-desaminase o 5-FC é convertido em 5-FU dentro das células cancerosas. Para esta finalidade, coformadores adequados foram pre-selecionados a partir de análises estatísticas, realizadas utilizando o banco de dados estruturais da Cambridge, com base na competição entre sintons. Na sequencia, protocolos de cristalização foram desenvolvidos de acordo com duas técnicas: evaporação a partir de solvente e mecanoquímica, em particular a moagem com gota-solvente (SDG). Nove estruturas foram obtidas com 5-FC. Todas foram analisadas por microscopia óptica de luz polarizada e por difração de raios X por monocristal. Seis foram caracterizadas como co-cristais e três como sais. Sais farmacêuticos são preferidos nas formulações porque melhoram consideravelmente a solubilidade dos APIs. Co-cristais farmacêuticos têm sido cada vez mais explorados porque promovem igualmente melhorias nas propriedades físico-químicas dos APIs sem alterar a natureza dos mesmos. Os sais de 5-FC foram obtidos com os ácidos fumárico, oxálico e maleico e os cocristais com os ácidos adípico, tereftálico, málico, succínico e benzóico. Além disso, obteve-se um co-cristal multi-API, de 5-FC e 5-FU. Estes resultados, obtidos por meio do desenho racional de novas formas sólidas, concordaram bem com a regra de três, implementada para variações nos valores pKa (pKabase - pKaácido). Esta regra permite estimar a formação sais/co-cristais durante o processo de cristalização. Os sais de 5-FC foram analisados estrutural e termicamente. Os cocristais de 5-FC foram estruturalmente avaliados. O co-cristal multi-API também foi sintetizado por mecanoquímica, como parte dos esforços direcionados à aplicação dos princípios da Química Verde para a produção em larga escala de fármacos. Os resultados obtidos nesta tese oferecem um API como forte candidato a coformador, a saber a 5-FC, e introduz um co-cristal multi-API como potencial candidato para a terapia antineoplásica. / For active pharmaceutical ingredients (APIs) exhibiting low solubility and/or stability and/or dissolution profiles, among other problems capable of affecting their therapeutic efficacy, Crystal Engineering approaches have been highlighted in recent decades as a satisfactory solution. In the solid state, APIs may exhibit polymorphs, salts, solvates, hydrates, cocrystals, amorphous and combinations of them. Associated to each solid form are physical and chemical properties which may or not vary in relation to the reference API. In this context, this research project aimed to rationally design, synthesize and characterize new solid forms of the prodrug 5-fluorocytosine (5-FC) and the antineoplastic drug 5-fluorouracil (5-FU), by considering that both exhibit physicochemical issues that difficult their application in solid dosage forms. Although used as a fungicide, 5-FC has become one of the most used prodrugs for cancer treatment by gene-directed enzyme prodrug therapy, as in the presence of the enzyme cytosine deaminase, 5-FC is converted into 5-FU inside cancer cells. For this purpose, suitable coformers were pre-selected from statistical analyses performed on the Cambridge Structural Database, based on the synthon competition approach. Briefly, crystallization protocols were designed following two techniques: slow evaporation from solution and mechanochemistry, in particular the solvent-drop grinding (SDG). Nine structures were obtained with 5-FC. All were analyzed by polarized light optical microscopy and by single-crystal X-ray crystallography. Six were characterized as cocrystals and three as salts. Pharmaceutical salts are preferred in formulations because they considerably improve the solubility/stability of APIs. Cocrystals have been increasingly explored because they promote improvements in the physicochemical properties of the API while not altering the API´s nature. 5-FC salts were obtained with fumaric, oxalic, and maleic acids, and 5-FC cocrystals with adipic, terephthalic, malic, succinic, and benzoic acids. In addition, a multi-API cocrystal of 5-FC and 5-FU was obtained. These results, derived from the rational design of new solid forms, agreed well with the rule of three, implemented to variations in pKa values (pKabase - pKaácido). This rule allows for estimation of the salts/cocrystals formation during the crystallization process. The 5-FC salts were structurally and thermally analyzed. The 5-FC´s cocrystals were structurally evaluated. The multi-API co-crystal was also synthesized by SDG as part of the ongoing efforts toward Green Chemistry application for drug scale-up production. The results obtained in this thesis offer a strong API candidate to be used as a coformer, namely the 5-FC, and introduce a multi-API co-crystal as a potential candidate for anticancer therapy.

O objetivo da presente pesquisa consiste numa tradução e comentários dos três primeiros capítulos do primeiro tratado do Liber de Philosophia Prima sive Scientia Divina Livro referente à Primeira Filosofia ou Ciência Divina -, de Avicena, que trata do estabelecimento da Metafísica como ciência filosófica a partir da delimitação de seu objeto formal e de sua conveniente denominação. / The objective of this research entails translation and commentaries of three first chapters of first treaty of Liber de Philosophia Prima sive Scientia Divina Book on First Philosophy or Divine Science -, of Avicenne, which is about the establishment of Metaphysic as philosophic science from delimitation of its subject matter and its convenient denomination.