Student Number : 9804032J - PhD thesis - School of Mathematics - Faculty of Science / The focus of this thesis is the spectral structure of second order self-adjoint differential operators on graphs. Various function spaces on graphs are defined and we define, in terms of both differential systems and the afore noted function spaces, boundary value problems on graphs. A boundary value problem on a graph is shown to be spectrally equivalent to a system with separated boundary conditions. An example is provided to illustrate the fact that, for Sturm-Liouville operators on graphs, self-adjointness does not necessarily imply regularity. We also show that since the differential operators considered are self-adjoint the algebraic and geometric eigenvalue multiplicities are equal. Asymptotic bounds for the eigenvalues are found using matrix Pr¨ufer angle methods. Techniques common in the area of elliptic partial differential equations are used to give a variational formulation for boundary value problems on graphs. This enables us to formulate an analogue of Dirichlet-Neumann bracketing for boundary value problems on graphs as well as to establish a min-max principle. This eigenvalue bracketing gives rise to eigenvalue asymptotics and consequently eigenfunction asymptotics. Asymptotic approximations to the Green’s functions of Sturm-Liouville boundary value problems on graphs are obtained. These approximations are used to study the regularized trace of the differential operators associated with these boundary value problems. Inverse spectral problems for Sturm-Liouville boundary value problems on graphs resembling those considered in Halberg and Kramer, A generalization of the trace concept, Duke Math. J. 27 (1960), 607-617, for Sturm-Liouville problems, and Pielichowski, An inverse spectral problem for linear elliptic differential operators, Universitatis Iagellonicae Acta Mathematica XXVII (1988), 239-246, for elliptic boundary value problems, are solved. Boundary estimates for solutions of non-homogeneous boundary value problems on graphs are given. In particular, bounds for the norms of the boundary values of solutions to the non-homogeneous boundary value problem in terms of the norm of the non-homogeneity are obtained and the eigenparameter dependence of these bounds is studied. Inverse nodal problems on graphs are then considered. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph from which the uniqueness of the potential, for given nodal data, is deduced. An explicit formula for the potential in terms of the nodal points and eigenvalues is given.

IT2018

A research report submitted to the Faculty of Science, University of Witwatersrand, in partial fulfilment for the degree of Masters of Mathematics Education by coursework and research report. Johannesburg, March 2017. / The current mathematics curriculum in South Africa require that learners are provided with opportunities to develop abilities to be methodical, to generalise, to make conjectures and try to justify and prove their conjectures. These objectives call for the use of teaching strategies and tasks that support learners’ participation in the development of mathematical thinking and reasoning. This means that teachers have to be cautious when selecting tasks and deciding on teaching strategies for their classes. Tasks differ in their cognitive and difficulty levels and opportunities they afford for learner to learn mathematics competently. The levels of tasks selected by the teachers; the kinds of questions asked by the teachers during the implementation of the selected tasks and how the questions asked by the teachers and the teachers’ actions at implementations affected the levels of the tasks were the focus of this research report. The study was carried out in one high poverty high school in South Africa. Two teachers were observed teaching and each teacher taught their allocated grades. One teacher was observed teaching Grade 9s while the other taught Grade 11s. Both teacher taught number patterns at the time their lessons were observed. The research was qualitative. Methods of data collection and instruments included lesson observations; collection of tasks used in the observed classes, audio-taping and field notes. Pictures of the teachers’ work and copies of learners’ workbooks also provided some data. The analysis of data shows that the teachers not only selected and used lower-level cognitive demand and ‘easy’ tasks, that did not support mathematical thinking, but also did not lift up the levels and/or maintain the ‘difficulty levels’ of the task at implementation. Teachers were unable to initiate class discussions. Their teaching focused on ‘drill and practice’ learning and teaching practices. / LG2017

M.Sc., Faculty of Science, University of the Witwatersrand, 2011

by Leung Tin Chi. / Bibliography: leaf 32 / Thesis (M.Phil.)--Chinese University of Hong Kong, 1982

by Chi-lok Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Scheduling Problems of Chain-like Task System --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Problem Assumptions and Notations Definition --- p.7 / Chapter 2.3 --- Related Works --- p.9 / Chapter 2.3.1 --- Bokhari's Algorithm --- p.10 / Chapter 2.3.2 --- Sheu and Chiang's Algorithm --- p.12 / Chapter 2.3.3 --- Hsu's Algorithm --- p.12 / Chapter 2.4 --- Decision Algorithms for Un-mergeable Task System --- p.18 / Chapter 2.4.1 --- Feasible Length-K Schedule --- p.18 / Chapter 2.4.2 --- Generalized Decision Test --- p.23 / Chapter 2.5 --- Dominated and Non-dominated Task Systems --- p.26 / Chapter 2.5.1 --- Algorithm for Dominated Task System --- p.26 / Chapter 2.5.2 --- Property of Non-dominated Task System --- p.27 / Chapter 2.6 --- A Searching-Based Algorithm for the Optimization Problem --- p.28 / Chapter 2.6.1 --- Algorithm --- p.29 / Chapter 2.6.2 --- Complexity Analysis --- p.31 / Chapter 2.7 --- A Searching Algorithm Based on a Sorted Matrix --- p.33 / Chapter 2.7.1 --- Sorted Matrix --- p.33 / Chapter 2.7.2 --- Algorithm for the Optimization Problem --- p.35 / Chapter 2.7.3 --- Complexity Analysis --- p.40 / Chapter 2.8 --- A Constructive Algorithm for the Optimization Problem --- p.43 / Chapter 2.9 --- A Modified Constructive Algorithm --- p.46 / Chapter 2.9.1 --- Algorithm --- p.46 / Chapter 2.9.2 --- Worst-Case Analysis --- p.50 / Chapter 2.9.3 --- Sufficient Condition for Efficient Algorithm H --- p.58 / Chapter 2.9.4 --- Average-Case Analysis --- p.62 / Chapter 2.10 --- Performance Evaluation --- p.65 / Chapter 2.10.1 --- Optimal Schedule --- p.65 / Chapter 2.10.2 --- Space Complexity Analysis --- p.67 / Chapter 2.10.3 --- Time Complexity Analysis --- p.68 / Chapter 2.10.4 --- Simulation of Algorithm F and Algorithm H --- p.70 / Chapter 2.11 --- Conclusion --- p.74 / Chapter 3 --- Vehicle Scheduling Problems with Time Window Constraints --- p.77 / Chapter 3.1 --- Introduction --- p.77 / Chapter 3.2 --- Problem Formulation and Notations --- p.79 / Chapter 3.3 --- NP-hardness of VSP-WINDOW-SLP --- p.83 / Chapter 3.3.1 --- A Transformation from PARTITION --- p.83 / Chapter 3.3.2 --- Intuitive Idea of the Reduction --- p.85 / Chapter 3.3.3 --- NP-completeness Proof --- p.87 / Chapter 3.4 --- Polynomial Time Algorithm for the VSP-WINDOW on a Straight Line with Common Ready Time --- p.98 / Chapter 3.5 --- Strong NP-hardness of VSP-WINDOW-TREEP --- p.106 / Chapter 3.5.1 --- A Transformation from 3-PARTITION --- p.107 / Chapter 3.5.2 --- NP-completeness Proof --- p.107 / Chapter 3.6 --- Conclusion --- p.111 / Chapter 4 --- Conclusion --- p.115 / Bibliography --- p.119

Chan Yuet Tai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 107-111). / Chapter 1 --- Introduction to Sturm-Liouville Problem --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Introductory background --- p.2 / Chapter 1.3 --- The Liouville transformation --- p.3 / Chapter 1.4 --- The Sturm-Liouville problem 一 A historical look --- p.4 / Chapter 1.5 --- Where Sturm-Liouville problems come from? --- p.6 / Chapter 1.6 --- Inverse problems of interest --- p.8 / Chapter 2 --- Reconstruction Method I --- p.10 / Chapter 2.1 --- Perturbative inversion --- p.10 / Chapter 2.1.1 --- Inversion problem via Fredholm integral equation --- p.10 / Chapter 2.1.2 --- Output least squares method for ill-posed integral equations --- p.15 / Chapter 2.1.3 --- Numerical experiments --- p.17 / Chapter 2.2 --- Total inversion --- p.38 / Chapter 2.3 --- Summary --- p.45 / Chapter 3 --- Reconstruction Method II --- p.46 / Chapter 3.1 --- Computation of q --- p.47 / Chapter 3.2 --- Computation of the Cauchy data --- p.48 / Chapter 3.2.1 --- Recovery of Cauchy data for K --- p.51 / Chapter 3.2.2 --- Numerical implementation for computation of the Cauchy data . --- p.51 / Chapter 3.3 --- Recovery of q from Cauchy data --- p.52 / Chapter 3.4 --- Iterative procedure --- p.53 / Chapter 3.5 --- Numerical experiments --- p.60 / Chapter 3.5.1 --- Eigenvalues without noised data --- p.64 / Chapter 3.5.2 --- Eigenvalues with noised data --- p.69 / Chapter 4 --- Appendices --- p.79 / Chapter A --- Tikhonov regularization --- p.79 / Chapter B --- Basic properties of the Sturm-Liouville operator --- p.80 / Chapter C --- Asymptotic formulas for the eigenvalues --- p.86 / Chapter C.1 --- Case 1: h ≠ ∞ and H ≠ ∞ --- p.87 / Chapter C.2 --- Case 2: h= ∞ and H ≠∞ --- p.90 / Chapter C.3 --- Case 3: h = ∞ and H = ∞ --- p.91 / Chapter D --- Completeness of the eigenvalues --- p.92 / Chapter E --- d'Alembert solution formula for the wave equation --- p.97 / Chapter E.1 --- "The homogeneous solution uH(x,t)" --- p.98 / Chapter E.2 --- "The particular solution up(x, t)" --- p.99 / Chapter E.3 --- "The standard d'Alembert solution u(x,t)" --- p.101 / Chapter E.4 --- Applications to our problem --- p.101 / Chapter F --- Runge-Kutta method for solving eigenvalue problems --- p.104 / Bibliography --- p.107

Lee Wai Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 129-130). / Text in English; abstract also in Chinese. / Lee Wai Shing. / Contents --- p.i / List of Figures --- p.v / Abstract --- p.vii / Acknowledgement --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- What are inverse problems? --- p.1 / Chapter 1.2 --- Background of this research project --- p.2 / Chapter 1.3 --- Conservative systems and open systems -normal modes (NM's) vs quasi-normal modes (QNM's) --- p.3 / Chapter 1.4 --- Appetizer ´ؤ What our problems are like --- p.6 / Chapter 1.5 --- A brief overview of the following chapters --- p.7 / Chapter Chapter 2. --- Inversion of conservative systems- perturbative inversion --- p.9 / Chapter 2.1 --- Overview --- p.9 / Chapter 2.2 --- Way to introduce the additional information --- p.9 / Chapter 2.3 --- General Formalism --- p.11 / Chapter 2.4 --- Example --- p.15 / Chapter 2.5 --- Further examples --- p.19 / Chapter 2.6 --- Effects of noise --- p.23 / Chapter 2.7 --- Conclusion --- p.25 / Chapter Chapter 3. --- Inversion of conservative systems - total inversion --- p.26 / Chapter 3.1 --- Overview --- p.26 / Chapter 3.2 --- Asymptotic behaviour of the eigenfrequencies --- p.26 / Chapter 3.3 --- General formalism --- p.28 / Chapter 3.3.1 --- Evaluation of V(0) --- p.28 / Chapter 3.3.2 --- Squeezing the interval - evaluation of the potential at other positions --- p.32 / Chapter 3.4 --- Remarks --- p.36 / Chapter 3.5 --- Conclusion --- p.37 / Chapter Chapter 4. --- Theory of Quasi-normal Modes (QNM's) --- p.38 / Chapter 4.1 --- Overview --- p.38 / Chapter 4.2 --- What is a Quasi-normal Mode (QNM) system? --- p.38 / Chapter 4.3 --- Properties of QNM's in expectation --- p.40 / Chapter 4.4 --- General Formalism --- p.41 / Chapter 4.4.1 --- Construction of Green's function and the spectral represen- tation of the delta function --- p.42 / Chapter 4.4.2 --- The generalized norm --- p.45 / Chapter 4.4.3 --- Completeness of QNM's and its justification --- p.46 / Chapter 4.4.4 --- Different senses of completeness --- p.48 / Chapter 4.4.5 --- Eigenfunction expansions with QNM's 一 the two-component formalism --- p.49 / Chapter 4.4.6 --- Properties of the linear space Γ --- p.51 / Chapter 4.4.7 --- Klein-Gordon equation - The delta-potential system --- p.54 / Chapter 4.5 --- Studies of other QNM systems --- p.54 / Chapter 4.5.1 --- Wave equation - dielectric rod --- p.55 / Chapter 4.5.2 --- Wave equation ´ؤ string-mass system --- p.57 / Chapter 4.6 --- Summary --- p.58 / Chapter Chapter 5. --- Inversion of open systems- perturbative inversion --- p.59 / Chapter 5.1 --- Overview --- p.59 / Chapter 5.2 --- General Formalism --- p.59 / Chapter 5.3 --- Example 1. Klein-Gordon equation ´ؤ delta-potential system --- p.66 / Chapter 5.3.1 --- Model perturbations --- p.66 / Chapter 5.4 --- Example 2. Wave equation ´ؤ dielectric rod --- p.72 / Chapter 5.5 --- Example 3. Wave equation ´ؤ string-mass system --- p.76 / Chapter 5.5.1 --- Instability of the matrix [d] = [c]-1 upon truncation --- p.79 / Chapter 5.6 --- Large leakage regime and effects of noise --- p.81 / Chapter 5.7 --- Conclusion . . . --- p.84 / Chapter Chapter 6. --- Transition from open systems to conservative counterparts --- p.85 / Chapter 6.1 --- Overview --- p.85 / Chapter 6.2 --- Anticipations of what is going to happen --- p.86 / Chapter 6.3 --- Some computational experiments --- p.86 / Chapter 6.4 --- Reason of breakdown - An intrinsic error of physical systems --- p.87 / Chapter 6.4.1 --- Mathematical derivation of the breakdown behaviour --- p.90 / Chapter 6.4.2 --- Two verifications --- p.93 / Chapter 6.5 --- Another source of errors - An intrinsic error of practical computations --- p.95 / Chapter 6.5.1 --- Vindications --- p.96 / Chapter 6.5.2 --- Mathematical derivation of the breakdown --- p.98 / Chapter 6.6 --- Further sources of errors --- p.99 / Chapter 6.7 --- Dielectric rod --- p.100 / Chapter 6.8 --- String-mass system --- p.103 / Chapter 6.9 --- Conclusion --- p.105 / Chapter Chapter 7. --- A first step to Total Inversion of QNM systems? --- p.106 / Chapter 7.1 --- Overview --- p.106 / Chapter 7.2 --- Derivation for F(0) --- p.106 / Chapter 7.3 --- Example 一 delta potential system --- p.108 / Chapter Chapter 8. --- Conclusion --- p.111 / Chapter 8.1 --- A summary on what have been achieved --- p.111 / Chapter 8.2 --- Further directions to go --- p.111 / Appendix A. A note on notation --- p.113 / Appendix B. Asymptotic series of NM eigenvalues --- p.114 / Appendix C. Evaluation of functions related to RHS(x) --- p.117 / Appendix D. Asymptotic behaviour of the Green's function --- p.119 / Appendix E. Expansion coefficient an --- p.121 / Appendix F. Asymptotic behaviour of QNM eigenvalues --- p.123 / Appendix G. Properties of the inverse matrix [d] = [c]-1 --- p.125 / Appendix H. Matrix inverse through the LU decomposition method --- p.127 / Bibliography --- p.129

Investigating the psychometric properties of a screening instrument for young children is necessary to ascertain its quality and accuracy. In light of the important role culture plays on human beliefs and parenting styles, a newly translated and adapted test needs to be studied. Evaluating outcomes on a translated version of a test may reveal significant information related to cultural specifications as well as the common nature of child development. The current study examined psychometric properties of the 48-month interval of the Ages and Stages Questionnaires: Social-Emotional Second Edition (ASQ:SE-2) and its Traditional Chinese version (ASQ:SE-TC), using item response theory (IRT). Participants in the U.S. included 3,005 young children/parents dyads; 1,455 dyads were collected to represent a Taiwanese sample. A two-dimensional Rasch Partial Credit Model (2D-RPCM), which was determined to present a better fit than a unidimensional Rasch Partial Credit Model, was used to examine the item fit, item difficulty, reliability, and item information curves to evaluate the psychometric properties on the ASQ:SE and ASQ:SE-TC. Further, differential item functioning was conducted to examine whether items were functioning differently in the two population groups. Lastly, the differences between the distributions of children’s latent traits on the continuum of social and emotional competencies for the U.S. and Taiwanese samples were investigated. Based on findings, the adequacy of psychometric properties is discussed, providing insight into the quality of particular items. Identified differences between the two populations are explored by reviewing literature regarding cultural comparisons of childrearing practices, parenting styles, and cultural beliefs. Future directions for research include examining the cultural equivalence between translated and original versions of other ASQ:SE-2 intervals.

Lam, Hiu Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 58-60). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.