Background: In 2007 the population density for the Gauteng Province was 614 persons per km2 and in the Northern Cape Province it was 2.9 persons per km2 . High population density leads to an increase in crime. This was evident in the percentage distribution of total crime reported from 2000 to 2003 of 27.4% in Gauteng Province, while the percentage distribution of total crime reported in the Northern Cape for the same period of time was 2,8%. Stress and insomnia can be caused by crime which is influenced by population density. Crime and high population density, may cause stress and fear, which may lead to insomnia and anxiety, which in turn may lead to an increase in benzodiazepine usage. Objective: The general objective of this study was to investigate the benzodiazepine usage in the private health care sector in South Africa based on age, sex, geographical areas, prescriber type and days between refills. Methods: The data were obtained from a medicine claims database of a pharmacy benefit management company covering the periods from 1 January 2006 to 31 December 2006 and 1 January 2008 to 31 December 2008. The statistical analysis was performed by making use of the Statistical Analysis System®. A drug utilisation review was performed. Results: Patients claiming benzodiazepines represented about 7.25% of all patients in total database in 2006 and 7.97% in 2008. Female patients claimed more benzodiazepines than male patients in both Gauteng (67.24% in 2006 & 67.36% in 2008 respectively) and Northern Cape Province (67.77% in 2006 & 67.70% in 2008 respectively). Patients aged 40 years to 65 years claimed the highest number of benzodiazepine items, while patients younger than 12 years claimed the lowest number of benzodiazepine items. The number of patients that claimed benzodiazepines in the Northern Cape was lower than those in Gauteng. The percentage of patients that claimed benzodiazepines in 2006 was 7.91% in Gauteng versus 8.96% in Northern Cape. In 2008 the percentage of patients that claimed benzodiazepines was 8.47% in Gauteng versus 9.51% in Northern Cape. The percentage of benzodiazepine prescriptions claimed in Gauteng was 4.79% in 2006 and 5.10% in 2008. In the Northern Cape the percentages of benzodiazepine prescriptions claimed in 2006 and 2008 were 4.62% and 4.30% respectively. General medical practitioners prescribed most of the benzodiazepine prescriptions in both Northern Cape and Gauteng Province. Trade name products that were mostly prescribed in the Gauteng was Adco–Alzam® 0.5 mg and in the Northern Cape it was Brazepam® 3 mg for both 2006 and 2008. Conclusion: The difference in the prescribing patterns of benzodiazepines in Gauteng and the Northern Cape was not statistically significant. Recommendations for future research were made. / Thesis (M.Pharm (Pharmacy Practice))--North-West University, Potchefstroom Campus, 2011.

This thesis is concerned with the efficient numerical simulation of finite element based flexible multibody systems. Scaling operations are systematically applied to the governing index-3 differential algebraic equations in order to solve the problem of ill conditioning for small time step sizes. The importance of augmented Lagrangian terms is demonstrated. The use of fast sparse solvers is justified for the solution of the linearized equations of motion resulting in significant savings of computational costs. Three time stepping schemes for the integration of the governing equations of flexible multibody systems are discussed in detail. These schemes are the two-stage Radau IIA scheme, the energy decaying scheme, and the generalized-α method. Their formulations are adapted to the specific structure of the governing equations of flexible multibody systems. The efficiency of the time integration schemes is comprehensively evaluated on a series of test problems. Formulations for structural and constraint elements are reviewed and the problem of interpolation of finite rotations in geometrically exact structural elements is revisited. This results in the development of a new improved interpolation algorithm, which preserves the objectivity of the strain field and guarantees stable simulations in the presence of arbitrarily large rotations. Finally, strategies for the spatial discretization of beams in the presence of steep variations in cross-sectional properties are developed. These strategies reduce the number of degrees of freedom needed to accurately analyze beams with discontinuous properties, resulting in improved computational efficiency.

Die Hauptthemen dieser Arbeit sind einerseits eine tiefgehende Analyse von nichtlinearen differential-algebraischen Gleichungen (DAEs) vom Index 2, die aus der modifizierten Knotenanalyse (MNA) von elektrischen Schaltkreisen hervorgehen, und andererseits die Entwicklung von Konvergenzkriterien für Waveform Relaxationsmethoden zum Lösen gekoppelter Probleme. Ein Schwerpunkt in beiden genannten Themen ist die Beziehung zwischen der Topologie eines Schaltkreises und mathematischen Eigenschaften der zugehörigen DAE. Der Analyse-Teil umfasst eine detaillierte Beschreibung einer Normalform für Schaltkreis DAEs vom Index 2 und Abschätzungen, die für die Sensitivität des Schaltkreises bezüglich seiner Input-Quellen folgen. Es wird gezeigt, wie diese Abschätzungen wesentlich von der topologischen Position der Input-Quellen im Schaltkreis abhängen. Die zunehmend komplexen Schaltkreise in technologischen Geräten erfordern oftmals eine Modellierung als gekoppeltes System. Waveform relaxation (WR) empfiehlt sich zur Lösung solch gekoppelter Probleme, da sie auf die Subprobleme angepasste Lösungsmethoden und Schrittweiten ermöglicht. Es ist bekannt, dass WR zwar bei Anwendung auf gewöhnliche Differentialgleichungen konvergiert, falls diese eine Lipschitz-Bedingung erfüllen, selbiges jedoch bei DAEs nicht ohne Hinzunahme eines Kontraktivitätskriteriums sichergestellt werden kann. Wir beschreiben allgemeine Konvergenzkriterien für WR auf DAEs vom Index 2. Für den Fall von Schaltkreisen, die entweder mit anderen Schaltkreisen oder mit elektromagnetischen Feldern verkoppelt sind, leiten wir außerdem hinreichende topologische Konvergenzkriterien her, die anhand von Beispielen veranschaulicht werden. Weiterhin werden die Konvergenzraten des Jacobi WR Verfahrens und des Gauss-Seidel WR Verfahrens verglichen. Simulationen von einfachen Beispielsystemen zeigen drastische Unterschiede des WR-Konvergenzverhaltens, abhängig davon, ob die Konvergenzbedingungen erfüllt sind oder nicht. / The main topics of this thesis are firstly a thorough analysis of nonlinear differential-algebraic equations (DAEs) of index 2 which arise from the modified nodal analysis (MNA) for electrical circuits and secondly the derivation of convergence criteria for waveform relaxation (WR) methods on coupled problems. In both topics, a particular focus is put on the relations between a circuit's topology and the mathematical properties of the corresponding DAE. The analysis encompasses a detailed description of a normal form for circuit DAEs of index 2 and consequences for the sensitivity of the circuit with respect to its input source terms. More precisely, we provide bounds which describe how strongly changes in the input sources of the circuit affect its behaviour. Crucial constants in these bounds are determined in terms of the topological position of the input sources in the circuit. The increasingly complex electrical circuits in technological devices often call for coupled systems modelling. Allowing for each subsystem to be solved by dedicated numerical solvers and time scales, WR is an adequate method in this setting. It is well-known that while WR converges on ordinary differential equations if a Lipschitz condition is satisfied, an additional convergence criterion is required to guarantee convergence on DAEs. We present general convergence criteria for WR on higher index DAEs. Furthermore, based on our results of the analysis part, we derive topological convergence criteria for coupled circuit/circuit problems and field/circuit problems. Examples illustrate how to practically check if the criteria are satisfied. If a sufficient convergence criterion holds, we specify at which rate of convergence the Jacobi and Gauss-Seidel WR methods converge. Simulations of simple benchmark systems illustrate the drastically different convergence behaviour of WR depending on whether or not the circuit topological convergence conditions are satisfied.

二次世界大戰時，美國擊敗日本，使朝鮮半島脫離殖民統治，然而隨後的美蘇冷戰，使得朝鮮半島分裂成南北兩韓，而美國與南韓簽訂條約，成立美韓同盟(U.S.–South Korea Alliance)，成為繼日本之後，美國在亞洲的另一個戰略同盟。冷戰與後冷戰期間，美韓關係雖有波折，但不影響美韓同盟的強度。直到2000年美國小布希就任，其強硬的北韓政策與南韓金大中的陽光政策形成對比，成了美韓關係不協調的開端。而後連任的小布希延續其北韓政策，南韓繼任的盧武鉉將陽光政策擴大實施，推出和平繁榮政策，美韓兩國的北韓政策再度不同調，兩國關係持續跌宕起伏至2008年。本研究試圖以攻勢現實主義分析美國此時期的北韓政策；以新自由制度主義檢視南韓的交往政策，透過理論交鋒研究兩國利益的差異，並檢視外部因素如中國、日本、俄羅斯的影響，進而解釋此時期美韓關係不協調的原因。 / In 1945, the U.S. defeated Japan. The Korean peninsula was liberated from Japanese colonization at the end of World War II. However, the confrontation between the U.S. and the Soviet Union left two Koreas separated by the Demilitarized Zone from the Cold War to the present. In addition, based on the Mutual Defense Treaty Between the United States of America and the Republic of Korea, the U.S.–South Korea Alliance was established. During the Cold War and Post Cold War era, the U.S.-South Korea relations faced hard times, but the alliance remained strong. When George W. Bush became the president of the U.S. in the year 2000, his hardline policy toward North Korea collided with South Korea’s Sunshine Policy, which was made by the president Kim Dae-jung. These different policies toward the North caused tensions to the U.S.-South Korea relations. Then the re-elected Bush continued hardline policy against North Korea, but South Korea’s new president—Roh Moo-hyun—decided to inherit the sunshine policy and develop Peace and Prosperity Policy. Washington and Seoul still failed to reach a consensus on how to deal with Pyongyang. The U.S.-South Korea relations continued to fluctuate until 2008. This study tries to analyze the U.S. policy toward North Korea through offensive realism and examine South Korean engagement policy through neoliberal institutionalism from 2000 to 2008. Besides, this thesis also considers exogenous factors such as China, Japan, and Russia, trying to explain the inconstancy of the U.S.-South Korea relations.

Die Anwenung von Operator-Splitting-Methoden auf gewöhnliche Differentialgleichungen ist gut etabliert. Für Differential-algebraische Gleichungen und partielle Differential-algebraische Gleichungen unterliegt sie jedoch vielen Einschränkungen aufgrund des Vorhandenseins von Nebenbedingungen. Die räumliche Diskretisierung reduziert PDAEs und lenkt unseren Fokus auf das Konzept der DAEs. Um eine reibungslose Übertragung des Operator-Splittings von ODEs auf DAEs durchzuführen, ist es wichtig, eine geeignete entkoppelte Struktur für das gewünschte Differential-algebraische System zu haben. In dieser Arbeit betrachten wir ein Modell, das partielle Differentialgleichungen für elektromagnetische Bauelemente - modelliert durch die Maxwell-Gleichungen - mit Differential-algebraischen Gleichungen koppelt, die die elementaren Schaltungselemente beschreiben. Nach der räumlichen Diskretisierung der klassischen Formulierung der Maxwell-Gleichungen mit Hilfe der finiten Integrationstechnik formulieren wir das resultierende gekoppelte System als Differential-algebraische Gleichung. Um eine geeignete Entkopplung zu bekommen, verwenden wir den zweigorientierten Loop-Cutset-Ansatz für die Schaltungsmodellierung. Daraus folgt, dass wir in der Lage sind, eine geeignete Operatorzerlegung so zu konstruieren, dass wir eine natürliche topologisch entkoppelte Port-Hamiltonsche DAE-Struktur erhalten. Wir schlagen einen Operator-Splitting-Ansatz für die Schaltungs-DAEs und gekoppelten Feld-Schaltungs-DAEs in entkoppelter Form vor und analysieren seine numerischen Eigenschaften. Darüber hinaus nutzen wir das Hamiltonsche Verhalten der inhärenten gewöhnlichen Differentialgleichung durch die Verwendung expliziter und energieerhaltender Zeitintegrations-methoden. Schließlich führen wir numerische Tests, um das mathematische Modell zu illustrieren und die Konvergenzergebnisse für das vorgeschlagene DAE-Operator-Splitting zu demonstrieren. / Le equazioni algebriche differenziali e algebriche alle derivate parziali hanno avuto un enorme successo come modelli di sistemi dinamici vincolati. Nella modellazione matem- atica, spesso si desidera catturare diversi aspetti di una situazione come le leggi di conservazione della fisica, il trasporto convettivo o la diffusione. Queste aspetti si riflettono nel sistema di equazioni del modello come operatori diversi. La tecnica dell’Operator Splitting si è rivelata una strategia di successo per affrontare problemi così complicati. L’applicazione dei metodi di Operator Splitting alle equazioni differenziali ordinarie (ODE) è ormai una tecnologia ben consolidata. Tuttavia, per equazioni algebriche differenziali (DAE) e algebriche differenziali parziali (PDAE), l’approccio è soggetto a molte restrizioni dovute alla presenza di vincoli e alla proprietà di indice. La discretizzazione spaziale riduce le PDAE e indirizza la nostra attenzione al concetto di DAE. Le DAE emergono in problemi dinamici vincolati come circuiti elettrici o reti di trasporto di energia. Al fine di generalizzare agevolmente la tecnica dell’Operator Splitting dalle ODE alle DAE, è importante avere una struttura disaccoppiata adeguata per il sistema algebrico differenziale desiderato. In questa tesi, consideriamo un modello che accoppia equazioni differenziali alle derivate parziali per dispositivi elettromagnetici -modellati dalle equazioni di Maxwell- con equazioni algebriche differenziali che descrivono gli elementi base del circuito. Dopo aver discretizzato spazialmente la formulazione classica delle equazioni di Maxwell usando la tecnica di integrazione finita, formuliamo il sistema accoppiato risultante come una equazione algebrica differenziale. Interpretando il dispositivo elettromagnetico come un elemento capacitivo, l’indice dell’intero sistema di circuito e campo accoppiato può essere specificato utilizzando le proprietà topologiche del circuito e non supera il valore di due. Per eseguire un disaccoppiamento appropriato, utilizziamo l’approccio loop-cutset per la modellazione dei circuiti. In tal modo siamo in grado di costruire una opportuna decomposizione dell’operatore tale da ottenere una naturale struttura disaccoppiata port-Hamiltonian DAE. Proponiamo un approccio di suddivisione dell’operatore per i DAE a circuito disaccoppiato e a circuito di campo accoppiato utilizzando gli algoritmi di divisione Lie-Trotter e Strang e per analizzare le proprietà numeriche di questi sistemi. Inoltre, sfruttiamo il comportamento hamiltoniano del sistema di equazioni differenziali ordinarie mediante l’utilizzo di metodi di integrazione temporale con esatta conservazione dell’energia. Poggiando sull’analisi di convergenza del metodo di suddivisione dell’operatore ODE, deriviamo i risultati di convergenza per l’approccio proposto che dipendono dall’indice delsistema e quindi dalla sua struttura topologica. Infine, eseguiamo prove numeriche di sistemi circuitali, nonchè sistemi accoppiati a circuito di campo, per testare il modello matematico e dimostrare i risultati di convergenza per la proposta Operator Splitting DAE. / The application of operator splitting methods to ordinary differential equations (ODEs) is well established. However, for differential-algebraic equations (DAEs) and partial differential-algebraic equations (PDAEs), it is subjected to many restrictions due to the presence of constraints. In constrained dynamical problems as electrical circuits or energy transport networks, DAEs arise. In order to perform a smooth transfer of the operator splitting from ODEs to DAEs, it is important to have a suitable decoupled structure for the desired differential-algebraic system. In this thesis, we consider a model which couples partial differential equations for electro- magnetic devices -modeled by Maxwell’s equations- with differential-algebraic equations describing the basic circuit elements. After spatially discretizing the classical formulation of Maxwell’s equations using the finite integration technique, we formulate the resulting coupled system as a differential-algebraic equation. To perform an appropriate decoupling, we use the branch oriented loop-cutset approach for circuit modeling. It follows that we are able to construct a suitable operator decomposition such that we obtain a natural topologically decoupled port-Hamiltonian DAE structure. We propose an operator splitting approach for the decoupled circuit and coupled field- circuit DAEs using the Lie-Trotter and Strang splitting algorithms and analyze its numerical properties. Furthermore, we exploit the Hamiltonian behavior of the system’s inherent ordinary differential equation by the utilization of explicit and energy-preserving time integration methods. Based on the convergence analysis of the ODE operator splitting method, we derive convergence results for the proposed approach that depends on the index of the system and thus on its topological structure. Finally, we perform numerical tests, to underline the mathematical model and to demonstrate the convergence results for the proposed DAE operator splitting.