We will explore Crouzeix’s Conjecture, an upper bound on the norm of a matrix after the application of a polynomial involving the numerical range. More formally, Crouzeix’s Conjecture states that for any n × n matrix A and any polynomial p from C → C,∥p(A)∥ ≤ 2 supz∈W (A) |p(z)|.Where W (A) is a set in C related to A, and ∥·∥ is the matrix norm. We first discuss the conjecture, and prove the simple case when the matrix is normal. We then explore a proof for a class of matrices given by Daeshik Choi. We expand upon the proof where details are left out in the original. We also find and fix a small flaw in one section of the original paper.

The results of solution electrochemical and photoelectrochemical studies on a series of well-defined silicon phthalocyanine (SiPc) stacked-ring oligomers are presented. These molecules consist of one (monomer), two (dimer), and three (trimer) SiPc rings which are axially stacked through a O-Si-O backbone with t-butyldimethylsilyl "end cap" groups. The interplanar spacing in the dimer and trimer SiPc is about 3.4 Å which facilitates the through-space molecular orbital overlap that gives them unique spectroscopic and electrochemical properties intermediate between those observed for other systems in which the electroactive centers are non-interacting or have exclusively through-bond interaction. There is a blue-shift in both the Q- and Soret absorbance maxima which is accompanied by an increased oscillator strength as more SiPc subunits are added per molecule. In dichloromethane solution, the cyclic voltammograms of these molecules exhibit multiple, one-electron, chemically reversible oxidations and reductions. The number of oxidations and reductions observed for each molecule increases with the addition of more SiPc rings and the energy difference between successive electron transfers decreases. In addition, there is a large cathodic shift of 0.52 V in the first oxidation potential between the monomer and trimer SiPc indicating a net stabilization of the dimer and trimer towards oxidation with respect to the monomer SiPc. These electrochemical results are shown to correlate well with Ultraviolet Photoelectron Spectroscopic (UPS) and UV-visible absorption spectroscopic data and energy level diagrams for the monomer, dimer, and trimer SiPc as well as higher-order polymeric SiPc are developed. Extensive photoelectrochemical studies on SiPc-modified electrodes are also reported. The effects of the chemical nature, E⁰, and concentration of the solution redox couple, as well as the influence of changing the electrode substrate and incident light intensity and wavelength on the photoresponse characteristics of these electrodes are presented and discussed. A solid-state band model for the dyesensitization process is discussed that treats the SiPc layer as a photoconductor that is capable of causing Fermi level pinning to occur at the SiPc/SnO₂ interface, resulting in an open-circuit photovoltage of about 200 mV which is independent of the solution E⁰. A molecular model is also developed that considers the specific molecular interactions which occur between the SiPc and the substrate, between adjacent SiPc molecules in the dye layer, and between the SiPc molecules and the solution redox species. Photoexcitation of the SiPc layer results in the formation of excitons in which the excited-state is delocalized over an aggregate containing several SiPc molecules.

published_or_final_version / abstract / Chemistry / Master / Master of Philosophy

Faculty of health Sciences, School of Pharmacy, Msc (Med)Research Thesis. / Objective: To assess the relative cost effectiveness of moxifloxacin once-daily empirical monotherapy and ofloxacin/ metronidazole twice daily combination therapy for the treatment of uncomplicated pelvic inflammatory disease in adult female patients. Design: This is a retrospective cost analysis using data from a clinical trial in order to perform the economic anlysis from a funder perspective. The cost analysis is based on the clinical results of the MAIDEN study which is a prospective, randomized, double-blind, multicentre, multinational Phase III study comparing the efficacy and safety of moxifloxacin 400 mg po od for 14 days with ofloxacin 400mg po bid plus metronidazole 400mg po bid for 14 days in patients with uncomplicated pelvic inflammatory disease. Decision analysis is used to characterise the economic outcomes between groups and provide a structure upon which to base the sensitivity analyses. Published 2004 cost values are used throughout. Cost values for moxifloxacin are based on the retail price of Avelon tablets in South Africa as appears on the Orderwise Retail Pharmacy Ordering System (September 2004). Cost values for the comparator, ofloxacin and metronidazole, are based on the cheapest available generics on the South African market i.e. Zanocin 400 and Metazol 400mg respectively. Method: The cost analysis is based on the clinical results obtained from the MAIDEN study. Patients were enrolled in either the moxifloxacin treatment group (Group A) or the ofloxacin / metronidazole comparator group (Group B). Resource utilization included: - cost for study antimicrobials (total number of doses for the study period) - treatment for adverse events occurring up to 7 days after stopping the study medication - treatment for failures (includes patients continued on antimicrobial therapy after the 14 day course of therapy) - cost of additional physician visits to treat adverse events and treatment failures The primary end-point is the overall cost of treatment per patient as determined by: Clinical response 7 to 14 days after the last dose of study medication (Test-of-Cure visit) Since the clinical findings from the MAIDEN study showed that moxifloxacin treatment was at least as efficacious as ofloxacin/metronidazole treatment, a cost-minimization analysis was performed and the results were analysed according to decision analysis. Decision analysis was used to characterise the economic outcomes between the groups and provided a structure upon which to base the sensitivity analyses. The outcomes were depicted on a decision tree which proportionately determined the cost of treatment per patient in the two treatment groups. Results: No significant differences in clinical success rates were detected. Differences were mainly due to the cost of treating adverse events in the two groups. Costs per patient in the monotherapy vs combination therapy comparisons were R10 847.00 for moxifloxacin and R16 630.00 for ofloxacin/metronidazole treatment. Sensitivity analyses revealed that moxifloxacin monotherapy can be cost effective compared with ofloxacin/metronidazole combination therapy in different situations. Conclusion: Per patient, the cost of drug treatment and treatment of adverse events and clinical relapses was R10 847.00 for treatment with moxifloxacin therapy and R16 630.00 for ofloxacin/metronidazole therapy . In comparison to ofloxacin/metronidazole combination therapy, moxifloxacin monotherapy was therefore cost saving.

Since the industrialization of platinum group metals (PGMs), particularly platinum (Pt), palladium (Pd) and rhodium (Rh), the control of trace impurities in these metals has become vital. Reliable analysis of impurities in these metals is, however a difficult task. Nobel metals are difficult to dissolve quickly and quantitatively. Thus, analytical techniques which determine samples in liquid form have become less favourable. They require time consuming digestions and are prone to contamination both from the chemicals and equipment used for the preparation. Direct-analysis techniques are increasingly being used in the platinum group metal (PGM) industry for the determination of impurities in the final products. Spark-optical emission spectroscopy (Spark-OES) for the analysis of metals offers rapid turnaround times. Since the technique is almost non-destructive, little of the product is lost during analysis. The technique is also well established in the PGM industry. It is routinely used by two of the three largest platinum producing companies for the determination of impurities in their products. It is also used for the determination of PGMs after Fire Assay procedure by Anglo American Platinum. The greatest challenge for this technique remains the availability of certified reference materials (CRM) and calibration standards. This study investigates the use of the Spark-OES for the determination of impurities in PGMs (notably gold (Au), silver (Ag), iron (Fe), nickel (Ni), copper (Cu), lead (Pb), magnesium (Mg), manganese (Mn), silicon (Si), aluminium (Al), antimony (Sb), chromium (Cr), tin (Sn), titanium (Ti), zirconium (Zr), calcium (Ca), zinc (Zn), boron (B), cobalt (Co), vanadium (V), molybdenum (Mo), bismuth (Bi), arsenic (As), selenium (Se), tellurium (Te), cadmium (Cd) in refined platinum, palladium and rhodium metals). It is to be used at Anglo American Platinum’s final metal’s laboratory. A method to be used routinely in the laboratory is also developed. The concentration of the impurities determined is used to quantify the overall purity of the PGMs. PGMs, other than the matrix (the metal whose purity is being quantified), are also determined. The use of Spark-OES was evaluated as an alternative to inductively coupled plasma-optical emission spectroscopy (ICP-OES). Due to the lack of CRMs and calibration standards, the study included the preparation of in-house reference material (IRM) for calibration and quality control purposes. The standards were prepared by spiking pure PGM metal sponges (produced by Anglo Platinum) with the metal oxides of the elements of interest. These were melted together using a vacuum induction furnace to produce metal disks. The disks were ground and analysed after dissolution using ICP-OES. The metal disks, and the shavings, were distributed to three other independent laboratories and analyzed by ICP-OES, inductively coupled plasma-mass spectroscopy (ICP-MS) and Spark-OES. The assigned consensus values were used for the calibration of the Spark-OES. The method was validated for linearity, accuracy, precision, robustness, bias and the measurement uncertainty of the method. The metal disks were first tested for homogeneity. It was found that the bottom surface of the rhodium metal was not homogenous. Rapidly cooled moulds, will facilitate almost instantaneous cooling of the metal. This eliminates the migration of elements during cooling. This could assist with homogenizing the metal. Limits of detections (LODs) achieved for the methods ranged from 0.1 mg.kg-1 to 4 mg.kg-1. The highest LOD was for silicon, which was caused by contamination from the crucibles used. The precision for all impurity elements, except ruthenium (Ru), of the three methods (analysis of platinum, palladium and rhodium) was satisfactory. Ru showed poor precision in all the matrices due to the channel installed in the spectrometer. Due to the lack of CRMs, the traceability of the method could not be validated and the accuracy could only be validated by comparing it to in-house reference material. Although the method met the validation criteria, it cannot be used to certify the purity of the product as the traceability could not be validated. It suggests that the method be used for twin stream analysis in conjunction with a primary method. Because of its rapid turnaround time, and its non-destructive nature, the method can be used for plant control purposes, where the level of accuracy required is not as stringent as required on a certificate of analysis.

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Zhang, Xinmiao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 48). / Abstracts in English and Chinese. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Estimation with Known Misclassification Probabilities --- p.3 / Chapter 2.1 --- Model --- p.3 / Chapter 2.2 --- Maximum Likelihood Estimation --- p.5 / Chapter 2.3 --- Statistical Property --- p.6 / Chapter 2.4 --- Mx Estimation --- p.7 / Chapter 2.5 --- Partition Maximum Likelihood (PML) Estimation --- p.9 / Chapter 2.6 --- Starting Value --- p.10 / Chapter 2.7 --- Examples --- p.11 / Chapter 2.7.1 --- Example 1 --- p.11 / Chapter 2.7.2 --- Example 2 --- p.12 / Chapter 2.7.3 --- Example 3 --- p.13 / Chapter 3 --- Estimation by Double Sampling --- p.15 / Chapter 3.1 --- Model and Analysis --- p.16 / Chapter 3.2 --- Statistical Property --- p.17 / Chapter 3.3 --- Mx Estimation and PML Estimation --- p.18 / Chapter 3.4 --- Starting Value --- p.19 / Chapter 3.5 --- Examples --- p.19 / Chapter 3.5.1 --- Example 4 --- p.19 / Chapter 4 --- Simulation --- p.20 / Chapter 4.1 --- Simulation with Known Misclassification Probability --- p.20 / Chapter 4.2 --- Simulation with Double Sampling --- p.22 / Chapter 5 --- Conclusion --- p.24 / Appendix and Tables --- p.26 / References --- p.48

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The main objective of this work is to propose new procedures for the general dynamic factor analysis introduced by Forni et al. (2000). First, we develop an identification method for determining the number of common shocks in the general dynamic factor model. Sufficient conditions for consistency of the criterion are provided for large n (number of series) and T (the series length). We believe that our procedure can shed light on the ongoing debate on the number of factors driving the US or Eurozone economy. Second, we show how the dynamic factor analysis method proposed in Forni et al. (2000), combined with our identification method, allows for identifying and estimating joint and block-specific common factors. This leads to a more sophisticated analysis of the structures of dynamic interrelations within and between the blocks in suchdatasets. Besides the framework of the general dynamic factor model we also propose a consistent lag window spectral density estimator based on multivariate M-estimators by Maronna (1976) when the underlying data are coming from the alpha mixing stationary Gaussian process.