In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.

The purpose of the current study was to determine the efficiency of an electronic data collection method compared to a traditional paper-based method in the educational field, in terms of the accuracy of data collected and the time required to do it. In addition, data were collected to assess users' preference and system usability. The study included a sample of 20 preschool special educators from the Mailman Segal Center's Baudhuin Preschool and Autism Institute located in Davie County, Florida, who conduct daily data collection and analysis. The study used both quantitative and qualitative methods to determine answers to five research questions. These were, (1) to what extent is electronic data collection faster than traditional paper-based data collection, (2) to what extent does electronic data collection aid special education teachers and paraprofessionals to collect more accurate data than traditional paper-based data collection, (3) to what extent is the use of electronic data collection result in significant time savings relative to traditional paper-based data collection during data graphing, (4) to what extent do specialists prefer either data collection method and for what reasons, and (5) to what extent do specialists rate the usability of the handheld device used for electronic data collection. Results suggested that both formats are comparable in terms of data collection time F(1, 18) = 3.53, p = .077 and accuracy, F(1, 18) = .928, p = .348 but that electronic data graphing is faster (M = 40.4, SD = 2.17) than paper-based graphing (M = 80.4, SD = 52.61). A higher percentage of participants (60%) preferred the electronic-based data collection method due to its graphing capability and better organization of data. The electronic data collection system used in this study was found to be more usable than 86.8% of all products tested using the System Usability Scale (SUS) and, therefore, could be considered a "C" or at an "acceptable" level or "good" relative to the other 200+ systems tested using the SUS by Bangor, Kortum, and Miller (2009). The electronic-based data collection system could also be considered an "A-" based on Sauro and Lewis' (2012) scale.

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem. In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a triangulation of each r-stable (n,k)-hypersimplex within. We then use this triangulation to compute the facet-defining inequalities of these polytopes. In the k=2 case, we use shelling techniques to devise a combinatorial interpretation of the coefficients of the h*-polynomials in terms of independent sets of certain graphs. From this, we then extract some results on unimodality. We also characterize the Gorenstein r-stable (n,k)-hypersimplices, and we conclude that these also have unimodal h*-polynomials. In the second case, for a graph G on p vertices we consider the closure of the cone of concentration matrices of G. The extreme rays of this cone, and their associated ranks, have applications in maximum likelihood estimation for the undirected Gaussian graphical model associated to G. Consequently, the extreme ranks of this cone have been well-studied. Yet, there are few graph classes for which all the possible extreme ranks are known. We show that the facet-normals of the cut polytope of G can serve to identify extreme rays of this nonpolyhedral cone. We see that for graphs without K5 minors each facet-normal of the cut polytope identifies an extreme ray in the cone, and we determine the rank of this extreme ray. When the graph is also series-parallel, we find that all possible extreme ranks arise in this fashion, thereby extending the collection of graph classes for which all the possible extreme ranks are known.

Thesis (MScEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Production of graphite dust inside the Pebble Bed Modular Reactor (PBMR) influences the reactor operation negatively. Graphite is used as a moderator in the reactor core and the formation and transportation of graphite dust away from the reactor core decreases the amount of moderator which in turn has a negative impact on the reactor operation. High levels of radioactive dust may also contaminate reactor components which may pose a health risk to maintenance personnel. In this study a pressure vessel was designed and used to measure the wear of a graphite pebble in helium at elevated temperatures. By means of a multi-linear regression analysis a proper mathematical function was established in order to relate graphite wear to certain tribological parameters. These parameters were identified through a literature study. Discrete Element Modelling (DEM) was used to simulate the gravitational flow of graphite pebbles through the reactor core. The experimentally determined mathematical function was incorporated into the DEM simulation to estimate the annual mass of graphite dust to be produced by the PBMR pebble bed as a result of pebble-pebble interaction and pebble-wall interaction during refuelling. / AFRIKAANSE OPSOMMING: Die vorming van grafiet stof binne die korrelbed-modulêre reaktor (PBMR) beïnvloed die werking daarvan negatief. Grafiet word gebruik as 'n moderator in die reaktor kern en die vorming en vervoer van grafietstof weg van die reaktor kern lei tot 'n afname in die hoeveelheid moderator en dit het 'n negatiewe impak op die werking van die reaktor. Hoë vlakke van radioaktiewe grafietstof kontamineer ook reaktorkomponente wat 'n gesondheidsrisiko vir onderhoudspersoneel inhou. In hierdie studie was 'n drukvat ontwerp en gebruik om die slytasie van 'n grafietkorrel in helium by verhoogde temperature te meet. 'n Multi-lineêre regressie analise is dan gebruik om 'n wiskundige funksie daar te stel wat die verband tussen grafietslytasie en die eksperimentele parameters vas stel. Hierdie parameters was met behulp van 'n literatuurstudie geïdentifiseer. Diskrete Element Modellering (DEM) was gebruik om die gravitasionele vloei van grafietkorrels in die reaktor te modelleer. Die eksperimenteel bepaalde wiskundige funksie word in die DEM simulasie ge-inkorporeer om 'n skatting te maak van die jaarlikse massa grafietstof wat gevorm sal word in die PBMR korrelbed as 'n gevolg van korrel-korrel interaksie en korrel-wand interaksie gedurende hersirkulasie.

Thesis (MScEng (Process Engineering))--Stellenbosch University, 2008. / The general objective for this thesis is to assess the ability of cellular automata to model relatively complex processes or phenomena, in particular thermodynamic scenarios. The mass transfer in packing materials of distillation columns was selected as an example due to the sufficient level of complexity in the distillation process, and its importance in a wide range of applications. A literature survey on cellular automata that summarizes the information currently available in formal publications and the internet is included to provide a general overview on the basic theoretical principles and the application of cellular automata models in the process engineering industry. The literature study was also used to identify potential requirements for the new research project. The study objective includes the construction of a cellular automata model that is able to represent transition of solutes from the fluid on the micro-surfaces of packing materials to the by-passing vapour stream, as well as the steady-state equilibrium between evaporation and condensation. Iterated model parameters sufficient for the realistic modelling of mass transfer as a result of thermodynamic driving forces, are required to meet this objective. The model behaviour was compared and the parameters subsequently adjusted according to the behaviour that is theoretically expected from the system being simulated. Qualitative (although sometimes in a quantitative format) rather than quantitative observations and comparisons were made seeing that the model has not yet been calibrated. The model that has been developed to date is not able to simulate the individual effects of chemical and thermodynamic properties although a realistic simulation of the cumulative effect exerted by these factors, or change thereof, on a system has been achieved. The accuracy of the results that have been obtained by using iterated parameters cannot be guaranteed for scenarios that deviate too much from the systems that have already been modelled successfully. The trade-off between the ability of the model to incorporate the effect of polarization, its ability to represent separation, in particular the condensation of hydrophilic substances, for strong hydrophilic packing materials and its ability to incorporate a large number of species limits the range of scenarios that can be successfully modelled. The model is able to represent the effect of a declining driving force (difference between the component vapour pressure of the gas phase and that of the liquid phase) that is typical of a system which is allowed to reach equilibrium after an initial disturbance. The model is also able to represent an additional driving force for separation caused by the effect of intermolecular forces. The model also displays the potential ability to represent the effect of different surface structures of the packing material on the extent of separation achieved at steady state as well as the rate at which such steady state conditions have been achieved. The model must be correctly scaled to minimize inaccurate results. Although several adjustments are needed to eliminate some limitations, the model has proven itself worthy of further development due to its capability to represent the basic characteristics of mass transfer in packing materials.

A low-order discrete dynamical system (DDS) for finite-rate chemistry of H2-air combustion is derived in 3D. Fourier series with a single wavevector are employed to represent dependent variables of subgrid-scale (SGS) behaviors for applications to large-eddy simulation (LES). A Galerkin approximation is applied to the governing equations for comprising the DDS. Regime maps are employed to aid qualitative determination of useful values for bifurcation parameters of the DDS. Both isotropic and anisotropic assumptions are employed when constructing regime maps and studying bifurcation parameters sequences. For H2-air reactions, two reduced chemical mechanisms are studied via the DDS. As input to the DDS, physical quantities from experimental turbulent flow are used. Numerical solutions consisting of time series of velocities, species mass fractions, temperature, and the sum of mass fractions are analyzed. Numerical solutions are compared with experimental data at selected spatial locations within the experimental flame to check whether this model is suitable for an entire flame field. The comparisons show the DDS can mimic turbulent combustion behaviors in a qualitative sense, and the time-averaged computed results of some species are quantitatively close to experimental data.

The move towards integrating products and services has increased significantly. As a result, some business models, such as Product Service Systems (PSS) have been developed. PSS emphasises the sale of use of the product rather than the sale of the product itself. In this case, product ownership lies with the manufacturers/suppliers. Customers will be provided with a capable and available product for their use. In PSS, manufacturers/suppliers are penalised for any down time of their product according to the PSS contract. This has formed a pressure on the service providers (maintenance teams) to assure the availability of their products in use. This pressure increases as the products are scattered in remote places (customer locations). Authors have urged that different product monitoring levels are applied to enable service providers to monitor their products remotely allowing maintenance to be performed accordingly. They claim that by adopting these monitoring levels, the product performance will increase. Their claim is based on reasoning, not on experimental/empirical methods. Therefore, further experimental research is required to observe the effect of such monitoring levels on complex maintenance operations systems as a whole which includes e.g. product location, different types of failure, labour and their skills and locations, travel times, spare part inventory, etc. In the literature, monitoring levels have been classified as Reactive, Diagnostics, and Prognostics. This research aims to better understand and evaluate the complex maintenance operations of a product in use with different levels of product monitoring strategies using a Discrete Event Simulation (DES) approach. A discussion of the suitability of DES over other techniques has been provided. DES has proven its suitability to give a better understanding of the product monitoring levels on the wider maintenance system. The requirements for simulating a complex maintenance operation have been identified and documented. Two approaches are applied to gather these generic requirements. The first is to identify those requirements of modelling complex maintenance operations in a literature review. This is followed by conducting interviews with academics and industrial practitioners to find out more requirements that were not captured in the literature. As a result, a generic conceptual model is assimilated. A simulation module is built through the Witness software package to represent different product monitoring levels (Reactive, Diagnostics, and Prognostics). These modules are then linked with resources (e.g. labour, tools, and spare parts). To ensure the ease of use and rapid build of such a complex maintenance system through these modules, an Excel interface is developed and named as Product Monitoring Levels Simulation (PMLS). The developed PMLS tool needed to be demonstrated and tested for tool validation purposes. Three industrial case studies are presented and different experimentations are carried out to better understand the effect of different product monitoring levels on the complex maintenance operations. Face to face validation with case companies is conducted followed by an expert validation workshop. This work presents a novel Discrete Event Simulation (DES) approach which is developed to support maintenance operations decision makers in selecting the appropriate product monitoring level for their particular operation. This unique approach provides numerical evidence and proved that the higher product monitoring level does not always guarantee higher product availability.

Healthcare problems are complex; they exhibit both detail and dynamic complexity. It has been argued that Discrete Event Simulation (DES), with its ability to capture detail, is ideal for problems exhibiting this type of complexity. On the other hand, System Dynamics (SD) with its focus on feedback and nonlinear relationships lends itself naturally to comprehend dynamic complexity. Although these modelling paradigms provide valuable insights, neither of them are proficient in capturing both detail and dynamic complexity to the same extent. It has been argued in literature that a hybrid approach, wherein SD and DES are integrated symbiotically, will provide more realistic picture of complex systems with fewer assumptions and less complexity. In spite of wide recognition of healthcare as a complex multi- dimensional system, there has not been any reported study which utilises hybrid simulation. This could be attributed to the fact that due to fundamental differences, the mixing of methodologies is quite challenging. In order to overcome these challenges a generic theoretical framework for hybrid simulation is required. However, there is presently no such generic framework which provides guidance about integration of SD and DES to form hybrid models. This research has attempted to provide such a framework for hybrid simulation which can be utilised in healthcare domain. On the basis of knowledge induced from literature, three requirements for the generic framework have been established. It is argued that the framework for hybrid simulation should be able to provide answers to Why (why hybrid simulation is required), What (what information is exchanged between SD and DES models) and How (how SD and DES models are going to interact with each other over the time to exchange information) within the context of implementation of hybrid simulation to different problem scenarios. In order to meet these requirements, a three-phase generic framework for hybrid simulation has been proposed. Each phase of the framework is mapped to an established requirement and provides guidelines for addressing that requirement. The proposed framework is then evaluated theoretically based on its ability to meet these requirements by using multiple cases, and accordingly modified. It is further evaluated empirically with a single case study comprising of Accident and Emergency department of a London district general hospital. The purpose of this empirical evaluation is to identify the limitations of the framework with regard to the implementation of hybrid models. It is realised during implementation that the modified framework has certain limitations pertaining to the exchange of information between SD and DES models. These limitations are reflected upon and addressed in the final framework. The main contribution of this thesis is the generic framework for hybrid simulation which has been applied within healthcare context. Through an extensive review of existing literature in hybrid simulation, the thesis has also contributed to knowledge in multi-method approaches. A further contribution is that this research has attempted to quantify the impact of intangible benefits of information systems into tangible business process improvements. It is expected that this work will encourage those engaged in simulation (e.g., researchers, practitioners, decision makers) to realise the potential of cross-fertilisation of the two simulation paradigms.

Thesis (MScEng)--Stellenbosch University, 2012 / ENGLISH ABSTRACT: The class-D audio power amplifier has found widespread use in both the consumer and professional audio industry for one reason: efficiency. A higher efficiency leads to a smaller and cheaper design, and in the case of mobile devices, a longer battery life. Unfortunately, the basic class-D amplifier has some serious drawbacks. These include high distortion levels, a load dependent frequency response and the potential to radiate EMI. Except for EMI, the aforementioned issues can be mitigated by the proper implementation of global negative feedback. Negative feedback also has the potential to indirectly reduce EMI, since the timing requirements of the output devices can be relaxed. This thesis discusses the design of a clocked analogue controlled pulse-width modulated class-D audio amplifier with global negative feedback. The analogue control loop is converted to the z-domain by modelling the PWM comparator as a sampling operation. A method is implemented that improves clip recovery and ensures stability during over-modulation. Loop gain is shaped to provide a high gain across the audio band, and ripple compensation is implemented to minimize the negative effect of ripple feedback. Experimental results are presented. / AFRIKAANSE OPSOMMING: Die klas-D klankversterker geniet wydverspreide gebruik in beide die verbruiker en professionele oudio industrie vir een rede: benuttingsgraad. ’n Hoër benuttingsgraad lei tot ’n kleiner en goedkoper ontwerp, en in die geval van draagbare toestelle, tot langer batterylewe. Ongelukkig het die basiese klas-D klankversterker ernstige tekortkominge, naamlik hoë distorsievlakke, ’n lasafhanklike frekwensierespons en die vermoë om EMI te genereer. Behalwe vir EMI kan hierdie kwessies deur die korrekte toepassing van globale negatiewe terugvoer aangespreek word. Negatiewe terugvoer het ook die potensiaal om EMI indirek te verminder, aangesien die tydvereistes van die skakel stadium verlaag kan word. Hierdie tesis bespreek die ontwerp van ’n geklokte analoog-beheerde pulswydte-modulerende klas-D klankversterker met globale negatiewe terugvoer. Die analoogbeheerlus word omgeskakel na die z-vlak deur die PWM vlakvergelyker as ’n monster operasie te modelleer. ’n Metode word geïmplementeer wat die stabiliteit van die lus verseker tydens oormodulasie. Die lusaanwins word gevorm om ’n hoë aanwins in die oudioband te verseker en riffelkompensasie word geïmplementeer om die negatiewe effek van terugvoerriffel teen te werk. Eksperimentele resultate word voorgelê.

This study investigates the self-assembly of ligands consisting of sterically hindered backbones. Ligands synthesised in this study took advantage of the twisted nature of benzil-2,3-dihydrazone brought about by the steric repulsion between two neighbouring phenyl groups. Synthesis of all ligands used a simple imine condensation reaction, where benzil-2,3-dihydrazone was combined with a chosen aromatic aldehyde. Coordination studies were conducted using first row transition metals and described in text are the solid state structures of twelve new discrete supramolecular complexes, characterised by single crystal X-ray crystallography. Chapter 1 outlines an introduction to the topics discussed in this text, while providing examples drawn from literature and nature. Chapter 2 details the modified synthesis of bis-2-pyridyliminohydrazono-1,2-diphenylethane, L1, and the in-depth background of previous studies with the ligand. The complexes of L1 with Mn(II), Fe(II), Co(II), Cu(I) and Zn(II) determined by single X-ray crystallography are presented, noting the structural effects different metal ion sources have upon L1. In chapter 3 the modified synthesis and characterisation of bis-2-imidazolyliminohydrazono-1,2-diphenylethane, L2, is presented. Crystallographically characterised complexes from the combination of L2 with Mn(II), Ni(II), Co(II) and Zn(II) show the formation of interesting topologies which are compared to structures previously determined. Chapter 4 begins with a brief introduction to clusters, followed by the synthesis and characterisation of a recently synthesised ligand, bis-2-salicyliminohydrazono-1,2-diphenylethane, L3,. The formation of a Ni(II) cluster with L3 is further discussed in detail, and outlines the future scope for the work with these ligands. The synthesis and characterisation of L4, and the discussion of mononuclear complex formed between L4 and Cu(II) is also introduced. The chapter concludes with a discussion with the potential future direction of this work with these ligands and their related compounds. Chapter 5 concludes the results and discussion with a brief summary of these results. Chapter 6 outlines the synthesis and characterisation of all ligands and complexes used in this study as well as potential ligands for future studies.