L will denote a completely distributive lattice with an order reversing involution. The concept of an L-uniform space is introduced. An extension theorem concerning L-uniformly continuous functions is proved. A characterisation of L-uniformizability, involving L-complete regularity is given. With respect to L--completely regular spaces it is shown that the topological modification of an L-completely regular space is completely regular. Furthermore it is shown that the topologically generated L-topology of a completely regular space is L-completely regular.

"The purpose of this paper is to record the results of a study of an abstract set upon which a distance function, having certain properties, has been defined. It is assumed that the reader is familiar with the fundamental concepts of set theory"--Introduction. / "June, 1959." / Typescript. / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: H. C. Griffith, Professor Directing Paper. / Includes bibliographical references (leaf 26).

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Prasad Tetali; Committee Member: Dana Randall; Committee Member: Robin Thomas; Committee Member: Vijay Vazirani; Committee Member: William T. Trotter

Thesis (MTech (Education))--Cape Peninsula University of Technology, 2009 / Huidiglik is die uitslae van wiskunde in Suid-Afrika baie swak in vergelyking met ander lande. Selfs die meeste Afrika-Iande presteer beter, Die doel van hierdie studie is om die redes en gevolge vir hierdie swak prestasies vas te stel. Dit is ook die navorser se poging om 'n bydrae te lewer tot beter wiskundige ontwikkeling ten opsigte van getalbegrip in die Grondslagfase deur aanbevelings vir opvoeders daar te stel wat benut kan word om hierdie doel te verwesenlik. In die literatuurstudie is Konstruktivisme, soos gesien deur Piaget en Vygotsky, breedvoerig bespreek. Die Realistiese benadering tot wiskundige ontwikkeling in getalbegrip is ook bestudeer. Verder is gefokus op.verskeie aspekte wat wiskundige ontwikkeling beinvloed, Die rede vir graad een en twee leerders se swak getalbegrip van 1 tot 99 en 'n moontlike oplossing vir hierdie probleem gee aanleiding tot die volgende navorsingsvrae: Dien die Plannemakerprogram as 'n doeltreffende hulpmiddel vir grade een en twee opvoeders om leerders se getalbegrip 1 tot 99 te verbeter? en Verbeter die Realistiese benadering, 5005 gevolg in die Plannemakerprogram, leerders se getalbegrip 1 tot 99? 'n Kwalitatiewe navorsingsontwerp is gebruik om die empiriese studie te voltooi. Vier skole in die Overberg-distrik, twee relatief groot en twee multi-graadskole, is gebruik. Gestruktureerde onderhoude is gevoer met ses graad een- en twee opvoeders en getalbegriptoetse is met hul leerders afgele, Die navorsingsresultate het getoon dat opvoeders wei riglyne benodig vir suksesvol!e ontwikkeling van getalbegrip in die Grondslagfase. Dit bevestig ook dat die grondlegging van goeie getalbegrip in graad een gele word en indien leemtes in hierdie belangrike aanvangsjaar ontstaan, leerders vorentoe probleme ondervind. Leerders by skole een en drie, waar die Plannemakerprogram gevolg is, se uitslae is heelwat hoer as skole twee en vier waar opvoeders ander benaderings gevolg het. Die uitslae van skool een se graad twee leerders, waar die Plannemakerprogram reeds vanaf graad een gevolg is, is ook beduidend hoer as skool drie waar die Plannemakerprogram slegs vanaf graad twee gevolg is. Hierdie navorsingstudie ondersoek, analiseer en bespreek die resultate met aanbevelings.

Please read the abstract in the front section of this document / Thesis (PhD (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

This thesis analyzes the simulation termination problem of implementing global termination conditions and collecting output measures in discrete event simulations. With regard to parallel simulations, this problem is inherently more difficult than the classic termination detection problem for two reasons. The first is that parallel simulation processes are often written as nonterminating; the second is that the decision to terminate can not be independently made by each process contributing to the simulation. A specification of a solution to the termination problem is developed as a sequence of stepwise refinements using UNITY, and proofs are given to demonstrate that each refinement satisfies the preceding specification. Termination conditions are categorized based on stability (if a condition is stable, once it becomes true it will remain true at all future times) and illustrated using space-time diagrams. A discussion is presented of how to implement termination conditions that are a combination of stable and nonstable conditions. This thesis makes two major contributions. The first is an algorithm to implement global termination conditions and to collect the corresponding output measures in discrete event simulation. The specifications and algorithm given in this thesis are architecture-independent and apply to sequential as well as synchronous and asynchronous parallel discrete event simulation algorithms. The second is the development of a generalized, formal framework in which to reason about simulation algorithms. The techniques used in this thesis to solve the simulation termination problem may be applied to solve other problems arising in parallel simulation. / Master of Science

過去二十多年，亞洲國家的學生在一些國際數學能力測試 (如 TIMSS、PISA) 中表現傑出，因此，許多學者嘗試找出這些學生取得優異成績的原因。根據經濟發展與合作組織的報告，在眾多變項中，教師的質素是影響學生學業成果的最重要因素。究竟一個能使學生有效學習的數學課堂，教師應擁有什麼數學知識? 教師應抱持什麼教學信念？ / 很多學者(如Ball, Thames, & Phelps、Shulman等)為鑽研教師的教學知識建立了不少理論，其中Ball和她的團隊利用Shulman有關教師知識的架構而發展出一項針對數學教學所涉及知識的類別，稱為「教學所需的數學知識」(mathematical knowledge for teaching，簡稱MKT)。此外，不少研究顯示除了知識之外，教師信念同樣影響教師的數學教學質素(Mathematical Quality of Instruction, 簡稱 MQI)。 / 本研究旨在了解香港高小經驗數學教師的MKT 和教學信念之現況，同時亦希望找出擁有高MKT及持不同教學信念的教師對其自身MQI之影響。 / 資料蒐集分兩階段進行，第一階段邀請105位擁有五年或以上數學教學經驗的教師參與，透過MKT測試卷和信念問卷分別量度他們的MKT和信念現況。至於第二階段，從第一階段參與的教師中挑選出八位經過測試結果屬於高水平的MKT的教師進行個案研究，研究員先觀察他們四節課堂教學，然後進行課後半結構訪談，測量他們的教學表現及進一步了解他們的教學信念。 / 研究結果顯示：(1) 在職經驗數學教師在圖形空間範疇的MKT成績高於數範疇的MKT成績；(2) 雖然信念問卷結果反映全部教師傾向抱持非傳統的教學信念，但是部分參與個案研究的教師卻抱持傳統教學信念的特徵；(3) 教師的教學質素並非全受著MKT的影響，擁有高MKT水平的教師而又持非傳統信念的教師的教學質素，比持有傳統信念特徵的教師的教學質素好；而(4) 教師在忙碌的教學生活下，大多沒有靈活多變的教學方法。本研究建議政府應推行政策減輕教師的工作量，而師訓機構宜開辦課程讓教師能掌握具體設計(尤其是數範疇) 的學習活動課程，協助教師建立專業交流網絡，創造機會讓教師進行反思，從而提高他們自身的教學能力。 / Over the past two decades, students from Asian countries have outperformed their counterparts in a number of international mathematics achievement studies such as TIMSS and PISA. Many scholars are therefore interested in investigating the reasons for Asian students’ higher performance. According to a research report released by the Organization for Economic Development and Cooperation, among the school variables which affect students’ learning outcomes, the quality of teachers play the most vital role. To provide a classroom environment for students that enables effective learning in mathematics, what kind of knowledge does a teacher need? And what kind of beliefs should a teacher hold? / Many scholars (e.g., Ball, Thames, & Phelps; Shulman, etc.) have proposed theories about the construction of teachers’ knowledge. Ball and her team, based on Shulman’s framework of teachers’ knowledge, developed a framework for "Mathematical Knowledge for Teaching" (MKT). Moreover, research studies have shown that in addition to MKT, a teacher’s beliefs also play an important role in a teacher’s mathematical quality of instruction (MQI). / This study aims to examine the MKT levels and beliefs of teachers who possess 5 or more years’ experience in teaching senior primary level mathematics, and to explore the influence of beliefs about teaching and learning on their MQI for teachers who have a high MKT level. / This study has undergone two stages in collecting data. During the first stage, 105 in-service experienced primary mathematics teachers were invited to complete an MKT instrument and a survey on beliefs about teaching and learning. It aims to explore their MKT levels and types of beliefs. At the second stage, eight teachers from the high MKT score group were selected for lesson observations and semi-structured interviews. Its aims were to explore their teaching performance and further verify their types of beliefs. / Results showed that (1) in-service experienced mathematics teachers generally scored higher MKT scores in the dimension of Shape and Space than in the dimension of Number. (2) They also showed that the scores of all 105 in-service teachers’ beliefs were identified as non-traditional. However, some teachers who were selected to take part in the subsequent case study held the characteristics of traditional transmission-oriented beliefs as revealed in the interview. Moreover, the findings also indicated that (3) the teachers’ instructional ability was not only affected by their MKT, their beliefs also played a part in shaping their pedagogical practices. Among the teachers with high MKT level, those teachers who held non-traditional beliefs outperformed their counterparts in terms of MQI. (4) It was also shown that teachers did not have a rich repertoire of teaching strategies to be used in classroom teaching because they lacked sufficient time to prepare their lessons. / In light of the findings, the Government should revise the current policy to reduce teachers’ workload. In addition, teacher training institutions should offer courses for teachers to design activities facilitating students’ learning in general, and strengthening the learning activities in the dimension of Number in particular. They should help teachers to establish professional exchange networks. By providing more chances for teachers’ to reflect, their teaching proficiency will be improved. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / 陳詠心. / Parallel title from added title page. / Thesis (Ed.D.) Chinese University of Hong Kong, 2015. / Includes bibliographical references (leaves 132-149). / Abstracts also in English. / Chen Yongxin.

The thesis investigates the role a calculator can play in the developing number knowledge of three girls and three boys as part of their mathematics program, during their first two years at primary school. Random sampling was used initially to select six girls and six boys from the twenty-four children entering a 1993 prep class. These twelve children were interviewed on entrance to school and based on the performance of the twelve children on the initial interview, a girl and a boy were chosen from the higher, middle and lower achievers to take part in the full study. The class teachers involved were previously participants in the ‘Calculators in Primary Mathematics’ research program and were committed to the use of calculators in their mathematics program. A case study approach using qualitative methods within the activity theory framework is used to collect relevant data and information, an analysis of five interviews with each child and observations of the children in forty-one classroom lessons provides comprehensive data on the children's developing number knowledge during the two years. The analysis questionnaires establishes each teacher's perceptions of the children's number learning at the beginning and end of each year, compares teacher expectations with children's actual performance for the year and compares curriculum expectations with children's actual performance. A teacher interview established reasons for changes in teaching style; teacher expectations; children's number learning; and was used to confirm my research findings. An activity theory framework provides an appropriate means of co-coordinating perspectives within this research to enable a description of the child's number learning within a social environment. This framework allows for highlighting the mediation offered by the calculator supporting the children's number learning in the classroom. Levels of children's developing number knowledge reached when working with a calculator and as a result of calculator use are mapped against the levels recommended in ‘Mathematics in the National Curriculum’ (National Curriculum Council, December 1988), and the Curriculum and Standards Framework: Mathematics (Board of Studies 2000). Findings from this comparison illustrate that the six children's performance in number was enhanced when using a calculator and indicate that on-going development and understanding of number concepts occurred at levels of performance at least two years in advance of curriculum recommendations for the first two years of school.

The aim of this study was an exploration of the relationship between the use of an Integrated Learning System (ILS), entitled Master Maths, as a supplement to traditional mathematics instruction, and mathematics achievement as measured by the Paper 2 marks of the National Mathematics Examinations for standard grade learners in grade 12. The use of technology in education has increased over the past decade. One way of integrating technology into instructional programmes has been through the use of Integrated Learning Systems (ILSs). The review of the literature traces the history of computer-assisted instruction as conducted on ILSs. The review of recent research studies focuses on the impact of ILSs on learner achievement in mathematics internationally and in the South African context. This study used quantitative and qualitative methods to research the impact of the Master Maths programme on mathematics achievement. Twenty-six learners of the 133 standard grade learners from one high school in New Brighton, Port Elizabeth were selected for each of the experimental and control groups. The experimental group worked on the Master Maths programme for twelve sessions of three hours each. The results of the quantitative analysis show that the intervention did not make a significant difference to the experimental group. The Master Maths programme led to only a 0.56% increase in the marks of the experimental group. The qualitative analysis drew a comparison between the modules of the Master Maths programme and the relevant examination questions in terms of content covered and cognitive levels. The researcher used Bloom’s Cognitive Taxonomy to evaluate the cognitive levels. The data show that it was easier for the learners to obtain higher marks in the module tests than in the examination questions. The data indicate that the module tests were easier than the examination questions in that the cognitive levels of the module tests were lower. The data confirm that there is a gap between the acquisition and evaluation of core skills tested by the modules used in the intervention and the wider knowledge and skills tested in the examination.

The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.