We study the existence, multiplicity, and nodal structure of solutions to a superlinear elliptic boundary value problem. Under specific hypotheses on the superlinearity, we show that there exist at least three nontrivial solutions. A pair of solutions are of one sign (positive and negative respectively), and the third solution changes sign exactly once. Our technique is variational, i.e., we study the critical points of the associated action functional to find solutions. First, we define a codimension 1 submanifold of a Sobolev space . This submanifold contains all weak solutions to our problem, and in our case, weak solutions are also classical solutions. We find nontrivial solutions which are local minimizers of our action functional restricted to various subsets of this submanifold. Additionally, if nondegenerate, the one-sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. We also establish that the action level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. Our results extend and complement the findings of Z. Q. Wang ([W]). We include a small sample of earlier works in the general area of superlinear elliptic boundary value problems.

In this paper we propose a hybrid approach for solving vehicle routing problems. The main idea is to combine an Ant System (AS) with a problem specific constructive heuristic, namely the well known Savings algorithm. This differs from previous approaches, where the subordinate heuristic was the Nearest Neighbor algorithm initially proposed for the TSP. We compare our approach with some other classic, powerful meta-heuristics and show that our results are competitive. / Series: Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

A research project submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Masters in Science Education. 2015 / The Curriculum Assessment Policy Statements (CAPS) re-introduced some mathematics topics such as probability. An immediate effect of this re-introduction is that most teachers and learners were not well equipped to deal with this topic. To at least begin addressing this problem, this research explored the errors and misconceptions that learners have when solving probability problems using different representations. The study draws from Nesher’s (1987) theory of errors and misconceptions as well as Sfard’s (2007) theory of commognition in explaining representations and prevalence of errors in learning mathematics. Twenty two Grade 10 learners wrote probability tasks after which their scripts were analysed for errors. Six of those learners were interviewed on the errors they made in solving probability problems with different representations. The findings reveal five main categories of errors and misconceptions. These are: (1) difficulty with construction of visual representations; (2) improper distinction between simple and compound events; (3) application of inappropriate routines; (4) errors associated with familiarity; and, (5) misinterpreted language. The findings also showed that inappropriate choice of representations was caused by misinterpretation of probability terminology. Concurring with Zahner and Corter (2010) the researcher found that learners made a multitude of errors if they constructed and used their own probability representations. Further, learners committed fewer errors where the task provided representations. Results also show that learners were most confident in using tree diagram representations even though they struggled to construct them from scratch. Most learners avoided Venn diagrams, outcome listings and matrix representations even though they would be the most useful in answering the questions. As a result many errors and misconceptions resulted when learners tried to use these representations. The study recommends that teachers take time to discuss probability terminology and the use of different representations with their learners. This promotes both the conceptual and procedural knowledge of probability. Also, to reduce learners’ errors and misconceptions on the topic, teachers need to scaffold the construction of representations by providing partially constructed representations and gradually encourage learners to construct their own probability representations.

The Revised National Curriculum requires that mathematics teachers make shifts in their pedagogical content knowledge about teaching non-routine problems. One of the shifts is to move from a traditional approach of teaching routine tasks to an approach which includes problem solving. Teaching non-routine problems develops flexible forms of knowledge, which allow learners to construct their own strategies and not merely follow steps shown to them by the teacher. In this study I explore what pedagogical content knowledge grade 6 teachers have about problem solving in everyday and mathematical contexts. A qualitative study was used to explore Grade 6 teachers’ pedagogical content knowledge about problem solving in these contexts. Data was collected by means of interviews and lesson observations. The study found variations in the teachers’ content knowledge but very similar pedagogical content knowledge among the teachers. Teachers have different understandings of teaching non-routine problems and they are faced with the challenge of how much to make knowledge accessible to their learners in order to lay a conceptual foundation to solve non-routine problems while not lowering the demands of the task. Teachers taught everyday context tasks differently to mathematical context tasks by focusing on the linguistic aspects of the tasks.

"The purpose of this study was to present descriptively and subjectively analyze the learning problems experienced by a graduate social work student with a ninety-seven per cent loss of vision. This student's second-year block field work placement of four months took place at the Institute of Human Development, Florida State University, Tallahassee, Florida. The focus of this study was to show how this student resolved certain professional learning problems, thus leading to positive learning experiences"--Introduction. / Typescript. / "August, 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Social Work." / Advisor: Dorothy D. Hayes, Professor Directing Study. / Includes bibliographical references (leaves 80-81).

朱英女. / Thesis (M.A.)--香港中文大學, 1988. / Reprint ed. / Includes bibliographical references: leaves 103-107. / Zhu Yingnü. / Thesis (M.A.)--Xianggang Zhong wen da xue, 1988. / 簡語表 --- p.I / 前言 --- p.III / Chapter 第一章 --- 後期維特根斯坦的語言觀 --- p.1 / Chapter 1.1 --- 對早期語言觀的批評 --- p.4 / Chapter 1.2 --- 一個新的進路 --- p.8 / Chapter 1.3 --- 語言遊戲 --- p.10 / Chapter 1.4 --- 對奧古斯殅語言觀的批評 --- p.13 / Chapter 1.5 --- 從用法了解意義 --- p.16 / Chapter 1.6 --- 語言規則 --- p.18 / Chapter 1.7 --- 遵守規則是慣常的活動 --- p.23 / Chapter 1.8 --- 小結 --- p.26 / Chapter 第二章 --- 私有語言論證 --- p.28 / Chapter 2.1 --- 什麼是私有語言？ --- p.28 / Chapter 2.2 --- 怎樣才是一套語言？ --- p.31 / Chapter 2.3 --- 語言的使用判準 --- p.34 / Chapter 2.4 --- 語言的「舞台背景」 --- p.38 / Chapter 2.5 --- 感覺語言與行為 --- p.42 / Chapter 2.6 --- 「私有」的意義 --- p.44 / Chapter 2.7 --- 獨處者的語言 --- p.48 / Chapter 2.8 --- 綜述 --- p.50 / Chapter 第三章 --- 克里普克對私有語言論證的解釋 --- p.55 / Chapter 3.1 --- 私有語言論證的基本結構 --- p.55 / Chapter 3.2 --- 《哲學探究》的中心問題 --- p.57 / Chapter 3.3 --- 懷疑論者的問題 --- p.60 / Chapter 3.4 --- 回答問題的幾個設想 --- p.63 / Chapter ( 1 ) --- 定義 --- p.63 / Chapter ( 2 ) --- 獨一的答案 --- p.64 / Chapter ( 3 ) --- 傾向性分析 --- p.65 / Chapter ( 4 ) --- 最簡單的假設 --- p.67 / Chapter ( 5 ) --- 不可還原的內省經驗 --- p.69 / Chapter ( 6 ) --- 原初狀態的本身 --- p.70 / Chapter 3.5 --- 亦是感覺語言的問題 --- p.71 / Chapter 3.6 --- 懷疑論的解答 --- p.73 / Chapter 3.7 --- 真值條件和斷言條件 --- p.74 / Chapter 3.8 --- 遵守規則的證立條件 --- p.76 / Chapter 3.9 --- 否認遵守規則的私有模式 --- p.78 / Chapter 第四章 --- 對克里普克解釋的回應 --- p.80 / Chapter 4.1 --- 「解釋」在遵守規則的行為中的位置 --- p.80 / Chapter 4.2 --- 意圖：過去的事實 --- p.85 / Chapter 4.3 --- 社團的判斷 --- p.89 / Chapter 4.4 --- 克里普克何以作出這個解釋 --- p.92 / Chapter 4.5 --- 維特根斯坦對懷疑論的態度 --- p.94 / Chapter 4.6 --- 結論 --- p.97 / 注釋 --- p.100 / 參考書目 --- p.103 / 論文撮要 --- p.108

Chan Sik Kin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 38-41). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Preliminary analysis --- p.11 / Chapter 3 --- Liapunov-Schmidt Reduction --- p.19 / Chapter 4 --- The reduced problem: A Minimizing Procedure --- p.32 / Chapter 5 --- Proof of the theorem 1.2 --- p.35 / Bibliography --- p.38

Typescript (photocopy). / Digitized by Kansas Correctional Industries

The effects of order of presentation and amount of experience on errors and solution time were investigated. An interactive narrative puzzle was presented on a computer screen to 60 undergraduate students. Solution of the problem involved the integration of two path segments. Subjects in the Forward Condition were presented the path segments in the order in which they had to be traversed for solution. Subjects in the Backward Condition were exposed to the opposite order of presentation. Amount of experience was varied by permitting one, three, or five readings of the narrative.

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