The Schrödinger equation ... is considered. The solution of this equation is reduced to the problem of finding the eigenvectors of an infinite matrix. The infinite matrix is truncated to a finite matrix. The approximation due to the truncation is carefully studied. The band structure of the eigenvalues is shown. The eigenvectors of the multiwells potential are presented. The solutions of Schrödinger equation are calculated. The results are very sensitive to the value of the parameter y. Localized solutions, in the case that the energy is slightly greater than the maximum value of the potential, are presented. Wigner and Weyl functions, corresponding to the solutions of Schrödinger equation, are also studied. It is also shown that they are very sensitive to the value of the parameter y.

在這篇論文中，我們將探討整函數與半純函數之分解。並且著重於質函數與擬質函數之研究。同時，我們證明某類多項式其次數為合成數且為質函數。 / In this thesis, we study some factorizations of entire and meromorphic functions, in particular, prime and pseudo-prime functions. Also, we obtain a new class of prime polynomials of composite degree.

We present a robust measurement of the rest-frame UV luminosity function (LF) and its evolution during the peak epoch of cosmic star formation at 1 < z < 3. We use our deep near-ultraviolet imaging from WFC3/UVIS on the Hubble Space Telescope and existing Advanced Camera for Surveys (ACS)/WFC and WFC3/IR imaging of three lensing galaxy clusters, Abell 2744 and MACS J0717 from the Hubble Frontier Field survey and Abell 1689. Combining deep UV imaging and high magnification from strong gravitational lensing, we use photometric redshifts to identify 780 ultra-faint galaxies with M-UV < -12.5 AB mag at 1 < z < 3. From these samples, we identified five new, faint, multiply imaged systems in A1689. We run a Monte Carlo simulation to estimate the completeness correction and effective volume for each cluster using the latest published lensing models. We compute the rest-frame UV LF and find the best-fit faint-end slopes of alpha = -1.56 +/- 0.04, alpha = -1.72 +/- 0.04, and alpha = -1.94 +/- 0.06 at 1.0 < z < 1.6, 1.6 < z < 2.2, and 2.2 < z < 3.0, respectively. Our results demonstrate that the UV LF becomes steeper from z similar to 1.3 to z similar to 2.6 with no sign of a turnover down to MUV = -14 AB mag. We further derive the UV LFs using the Lyman break "dropout" selection and confirm the robustness of our conclusions against different selection methodologies. Because the sample sizes are so large and extend to such faint luminosities, the statistical uncertainties are quite small, and systematic uncertainties (due to the assumed size distribution, for example) likely dominate. If we restrict our analysis to galaxies and volumes above >50% completeness in order to minimize these systematics, we still find that the faint-end slope is steep and getting steeper with redshift, though with slightly shallower (less negative) values (alpha = -1.55 +/- 0.06, -1.69 +/- 0.07, and -1.79 +/- 0.08 for z similar to 1.3, 1.9, and 2.6, respectively). Finally, we conclude that the faint star-forming galaxies with UV magnitudes of -18.5 < M-UV < -12.5 covered in this study produce the majority (55%-60%) of the unobscured UV luminosity density at 1 < z < 3.

The work in this thesis is focussed on obtaining fast, e cient solutions to the Schroedinger-Poisson model of electron states in microelectronic devices. The self-consistent solution of the coupled system of Schroedinger-Poisson equations poses many challenges. In particular, the three-dimensional solution is computationally intensive resulting in long simulation time, prohibitive memory requirements and considerable computer resources such as parallel processing and multi-core machines. Consequently, an approximate analytical solution for the coupled system of Schroedinger-Poisson equations is investigated. Details of the analytical techniques for the approximate solution are developed and the original approach is outlined. By introducing the hyperbolic secant and tangent functions with complex arguments, the coupled system of equations is transformed into one for which an approximate solution is much simpler to obtain. The method solves Schroedinger's equation rst by approximating the electrostatic potential in Poisson's equation and subsequently uses this solution to solve Poisson's equation. The complete iterative solution for the coupled system is obtained through implementation into Matlab. The semi-analytical method is robust and is applicable to one, two and three dimensional device architectures. It has been validated against alternative methods and experimental results reported in the literature and it shows improved simulation times for the class of coupled partial di erential equations and devices for which it was developed.

Liu, Tak Chi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 112-125). / Abstracts and appendixes in English and Chinese. / Abstract --- p.i / Acknowledgements --- p.viii / List of tables --- p.ix / List of figures --- p.xiii / List of Abbreviations --- p.xvi / List of Unit Abbreviations --- p.xvii / Table of Contents --- p.xviii / Chapter Chapter 1: --- Background and introduction --- p.P.1 / Chapter Chapter 2: --- Subjects and methods --- p.P.7 / Chapter 2.1 --- Study design and sampling frame --- p.P.7 / Chapter 2.2 --- Sample size calculation --- p.P.8 / Chapter 2.3 --- Study population --- p.P.11 / Chapter 2.4 --- Subject assessment in participating kindergartens --- p.P.13 / Chapter 2.5 --- Quality control for spirometric measurements --- p.P.18 / Chapter 2.6 --- Statistical analysis --- p.P.20 / Chapter 2.7 --- Outcome measures --- p.P.23 / Chapter 2.8 --- Participation and role in this study --- p.P.24 / Chapter Chapter 3: --- Results --- p.P.26 / Chapter 3.1 --- Comparison between the study populations in the training and research phases --- p.P.26 / Chapter 3.1.1 --- "Response rate, participation rate and success rate" --- p.P.27 / Chapter 3.1.2 --- Other factors --- p.P.31 / Chapter 3.2 --- Comparison between participants and non-participants in the research phase --- p.P.33 / Chapter 3.3 --- Comparison of factors between the subjects who succeed and failed to provide acceptable spirometric maneuvers which meet ATS/ERS standards in the research phase --- p.P.36 / Chapter 3.4 --- Comparison of lung function parameters between subjects who met and failed to meet the health criteria in the research phase --- p.P.39 / Chapter 3.4.1 --- Gestational birth age (< 37 weeks vs > 37 weeks) --- p.P.40 / Chapter 3.4.2 --- Birth weight (< 2.5kg vs > 2.5kg) --- p.P.41 / Chapter 3.4.3 --- Children with vs without current wheeze --- p.P.43 / Chapter 3.4.4 --- Children with vs without history of asthma ever --- p.P.44 / Chapter 3.4.5 --- Children with vs without recent respiratory tract infections (RTIs) --- p.P.45 / Chapter 3.5 --- The test-retest reliability --- p.P.47 / Chapter 3.6 --- "Relationship between lung function parameters and demographic, early-life, anthropometric and environmental factors in subjects who satisfied both health and ATS/ERS criteria in our research phase" --- p.P.50 / Chapter 3.6.1 --- Demographic factors --- p.P.51 / Chapter 3.6.2 --- Early-life factors --- p.P.53 / Chapter 3.6.3 --- Anthropometric factors --- p.P.56 / Chapter 3.6.4 --- Environmental factors --- p.P.57 / Chapter 3.7 --- Reference standards for incentive spirometry: Reference equations and normograms --- p.P.59 / Chapter Chapter 4: --- Discussions --- p.P.76 / Chapter 4.1 --- Pioneering incentive spirometry in Hong Kong preschoolers: Training and research phases --- p.P.77 / Chapter 4.2 --- Participants and non-participants in the research phase --- p.P.79 / Chapter 4.3 --- Subjects who succeed and failed to give acceptable maneuvers which meet ATS/ERS standards in the research phase --- p.P.81 / Chapter 4.4 --- "The relationship between demographic, anthropometric and environmental factors and spirometric parameters in local young children" --- p.P.84 / Chapter 4.41 --- Environmental tobacco smoke exposure and maternal smoking --- p.P.85 / Chapter 4.42 --- Place of birth --- p.P.87 / Chapter 4.43 --- Obesity and underweight --- p.P.89 / Chapter 4.44 --- Breastfeeding practice --- p.P.91 / Chapter 4.45 --- "Birth factors: mode of delivery, birth weight and gestation birth age" --- p.P.92 / Chapter 4.46 --- "Indoor environment: pets, moulds and others" --- p.P.94 / Chapter 4.5 --- Evaluation of the test-retest reliability --- p.P.95 / Chapter 4.6 --- The relationship between the health criteria and spirometric parameters in local young children --- p.P.96 / Chapter 4.7 --- The spirometric reference standards in Chinese preschool children in Hong Kong: Comparisons with published findings in different ethnic groups --- p.P.98 / Chapter 4.8 --- Drawbacks and limitations of this study --- p.P.105 / Chapter 4.9 --- Future research directions --- p.P.110 / Chapter Chapter 5: --- Conclusions --- p.P.111 / References --- p.P.112 / Appendices --- p.P.126 / Chapter Appendix I: --- Invitation letter --- p.P.127 / Chapter Appendix II: --- Reply form --- p.P.129 / Chapter Appendix III: --- Consent --- p.P.130 / Chapter Appendix IV: --- ISAAC questionnaire --- p.P.134 / Chapter Appendix V: --- Subject report --- p.P.163

Typescript (photocopy). / Digitized by Kansas Correctional Industries

Multi-Polynomial Higher Order Neural Network Group Models (MPHONNG) program developed by the author will be studied in this thesis. The thesis also investigates the use of MPHONNG for financial data and rainfall data simulation and prediction. The MPHONNG is combined with characteristics of Polynomial function, Trigonometric polynomial function and Sigmoid polynomial function. The models are constructed with three layers Multi-Polynomial Higher Order Neural Network and the weights of the models are derived directly from the coefficents of the Polynomial form, Trignometric polynomial form and Sigmoid polynomial form. To the best of the authors knowledge, it is the first attempt to use MPHONNG for financial data and rainfall data simulation and prediction. Results proved satisfactory, and confirmed that MPHONNG is capable of handling high frequency, high order nonlinear and discontinuous data. / Master of Science (Hons)

Persons with chronic fatigue syndrome (CFS) often complain of an inability to maintain activity levels and experience a variety of orthostatic symptoms such as dizziness, trembling, nausea, postural hypotension with bradycardia or tachycardia, sweating, palpitations, paleness, and syncope. Orthostatic intolerance (OI) may be defined as an inability to maintain systolic blood pressure (SBP) within 20 mmHg of resting level upon moving from a supine to upright posture. The primary objective of this study is to determine whether men and women with CFS are more susceptible to OI during a 3-stage head-up tilt (HUT) than non CFS, sedentary subjects matched by age, sex, and ethnicity. The secondary objective is to examine whether possible underlying mechanisms may be predictively associated with OI susceptibility in CFS. Possible causes of OI include autonomic nervous system (ANS) dysfunction and altered hematological profile. Thus, specific aims included within this objective are: 1) to determine whether there are differences in resting cardiovascular function {i.e., blood pressure [BP], heart rate [HR], stroke volume [SV], cardiac output [CO], total peripheral resistance [TPR], and contractility [i.e., ejection fraction (EF), fractional shortening (FS), and the velocity of circumferential shortening corrected by HR (VCFc)]}, ANS function {i.e., beta1-, beta2-, and alpha-receptor sensitivities, baroreceptor sensitivity [BRS], and vagal function [i.e., respiratory sinus arrhythmia (RSA), RSA envelope (RSAE), high frequency (HF) spectral component, and HR range]}, and hematological profile [i.e., red blood cell volume (RBCV), plasma volume (PBV), and total blood volume (TBV)] between CFS and non-CFS groups; and 2) to determine whether cardiovascular, ANS, and hematological measures differentially predicted OI during HUT. The results indicate that OI susceptibility does not occur with greater prevalence in persons with CFS than non-CFS sedentary persons. However, power analyses revealed that with a much larger sample size group differences in OI susceptibility would be found. The CFS group was distinguished from the control group only by differences in blood volume measures. There appears to be no substantive group differences in a range of cardiovascular and ANS measures; moreover, none of these measures, including the blood volume measures, accounted for differences in OI susceptibility. Compensatory mechanisms may be present in CFS for the diminished blood volume that could explain the lack of group differences in OI susceptibility. In addition, future research may find some clues relevant to CFS pathophysiology in the assessment of hemodynamic responses during orthostatic challenge in the present subjects.

The human immunodeficiency virus (HIV)-1 can compromise pulmonary function at all stages of the disease. The present study examined whether there were differences in resting and exercising pulmonary function among sedentary, resistance-trained and aerobically-trained, early symptomatic, HIV-1+ men. Forty five subjects, 15 per group, were enrolled. An analysis of variance (ANOVA) showed differences in demographics for age [F (2, 42) = 5.14, p<0.01)], weight [F (2, 42) = 4.84, p<0.01)], body mass index [F (2, 42) = 9.50, p<0.01)] and average years HIV-1+ [F (2, 42) = 4.78, p<0.01)]. A multiple analysis of covariance (MANCOVA) showed differences in resting pulmonary function [F (8, 72) = 7.164, P = 0.01]. Univariate ANOVA's and Bonferroni post-hoc comparisons showed the aerobically-trained group had higher forced expiratory volume in one second (FEV1) than the resistance-trained and sedentary groups (p<0.05 and p<0.01, respectively), higher forced vital capacity (FVC) (p<0.01, for both), higher maximum voluntary ventilation (p<0.01, for both) and higher FEV1/FVC ratios than the sedentary group only (p<0.01). The resistance-trained group also showed higher FEV1 (p<0.01) and FEV1/FVC (p<0.01) than the sedentary group. For exercising pulmonary function, significant differences in our MANCOVA were found [F (12, 68) = 12.73, P = 0.001]. Univariate ANOVA's and Bonferroni post-hoc comparisons showed that the aerobically-trained group had higher dyspnea index than the resistance-trained and sedentary groups (p<0.01 and p<0.05, respectively), higher ventilatory efficiency (RR/VE max) than the resistance-trained and sedentary groups (p<0.05 and p<0.01, respectively), higher maximum minute ventilation (VE max) (p<0.01, for both), higher peak oxygen consumption (peak VO2) (p<0.01, for both) and lower dead space (VD/VT) (p<0.01, for both). The resistance-trained group also showed higher peak VO2 (p<0.01), lower VD/VT (p<0.01) and lower RR/VE max (p<0.01) than the sedentary group. Results suggest that aerobically-trained, and to a lesser extent, resistance-trained seropositives possessed superior resting and exercising pulmonary function compared to sedentary seropositive males.

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