Computationsl modelling of dimethyl ether separation and steam reforming in fluidized bed reactors

This study presents a computational fluid dynamic (CFD) study of Dimethyl Ether (DME) gas adsorptive separation and steam reforming (DME-SR) in a large scale Circulating Fluidized Bed (CFB) reactor. The CFD model is based on Eulerian-Eulerian dispersed flow and solved using commercial software (ANSYS FLUENT). Hydrogen is currently receiving increasing interest as an alternative source of clean energy and has high potential applications, including the transportation sector and power generation. Computational fluid dynamic (CFD) modelling has attracted considerable recognition in the engineering sector consequently leading to using it as a tool for process design and optimisation in many industrial processes. In most cases, these processes are difficult or expensive to conduct in lab scale experiments. The CFD provides a cost effective methodology to gain detailed information up to the microscopic level. The main objectives in this project are to: (i) develop a predictive model using ANSYS FLUENT (CFD) commercial code to simulate the flow hydrodynamics, mass transfer, reactions and heat transfer in a large scale dual fluidized bed system for combined gas separation and steam reforming processes (ii) implement a suitable adsorption models in the CFD code, through a user defined function, to predict selective separation of a gas from a mixture (iii) develop a model for dimethyl ether steam reforming (DME-SR) to predict hydrogen production (iv) carry out detailed parametric analysis in order to establish ideal operating conditions for future industrial application. The project has originated from a real industrial case problem in collaboration with the industrial partner Dow Corning (UK) and jointly funded by the Engineering and Physical Research Council (UK) and Dow Corning. The research examined gas separation by adsorption in a bubbling bed, as part of a dual fluidized bed system. The adsorption process was simulated based on the kinetics derived from the experimental data produced as part of a separate PhD project completed under the same fund. The kinetic model was incorporated in FLUENT CFD tool as a pseudo-first order rate equation; some of the parameters for the pseudo-first order kinetics were obtained using MATLAB. The modelling of the DME adsorption in the designed bubbling bed was performed for the first time in this project and highlights the novelty in the investigations. The simulation results were analysed to provide understanding of the flow hydrodynamic, reactor design and optimum operating condition for efficient separation. Bubbling bed validation by estimation of bed expansion and the solid and gas distribution from simulation agreed well with trends seen in the literatures. Parametric analysis on the adsorption process demonstrated that increasing fluidizing velocity reduced adsorption of DME. This is as a result of reduction in the gas residence time which appears to have much effect compared to the solid residence time. The removal efficiency of DME from the bed was found to be more than 88%. Simulation of the DME-SR in FLUENT CFD was conducted using selected kinetics from literature and implemented in the model using an in-house developed user defined function. The validation of the kinetics was achieved by simulating a case to replicate an experimental study of a laboratory scale bubbling bed by Vicente et al [1]. Good agreement was achieved for the validation of the models, which was then applied in the DME-SR in the large scale riser section of the dual fluidized bed system. This is the first study to use the selected DME-SR kinetics in a circulating fluidized bed (CFB) system and for the geometry size proposed for the project. As a result, the simulation produced the first detailed data on the spatial variation and final gas product in such an industrial scale fluidized bed system. The simulation results provided insight in the flow hydrodynamic, reactor design and optimum operating condition. The solid and gas distribution in the CFB was observed to show good agreement with literatures. The parametric analysis showed that the increase in temperature and steam to DME molar ratio increased the production of hydrogen due to the increased DME conversions, whereas the increase in the space velocity has been found to have an adverse effect. Increasing temperature between 200 oC to 350 oC increased DME conversion from 47% to 99% while hydrogen yield increased substantially from 11% to 100%. The CO2 selectivity decreased from 100% to 91% due to the water gas shift reaction favouring CO at higher temperatures. The higher conversions observed as the temperature increased was reflected on the quantity of unreacted DME and methanol concentrations in the product gas, where both decreased to very low values of 0.27 mol% and 0.46 mol% respectively at 350 °C. Increasing the steam to DME molar ratio from 4 to 7.68 increased the DME conversion from 69% to 87%, while the hydrogen yield increased from 40% to 59%. The CO2 selectivity decreased from 100% to 97%. The decrease in the space velocity from 37104 ml/g/h to 15394 ml/g/h increased the DME conversion from 87% to 100% while increasing the hydrogen yield from 59% to 87%. The parametric analysis suggests an operating condition for maximum hydrogen yield is in the region of 300 oC temperatures and Steam/DME molar ratio of 5. The analysis of the industrial sponsor’s case for the given flow and composition of the gas to be treated suggests that 88% of DME can be adsorbed from the bubbling and consequently producing 224.4t/y of hydrogen in the riser section of the dual fluidized bed system. The process also produces 1458.4t/y of CO2 and 127.9t/y of CO as part of the product gas. The developed models and parametric analysis carried out in this study provided essential guideline for future design of DME-SR at industrial level and in particular this work has been of tremendous importance for the industrial collaborator in order to draw conclusions and plan for future potential implementation of the process at an industrial scale.

Sufficient Conditions of Optimality for Control Pproblem Governed by Variational Inequalities

* This work was completed while the author was visiting the University of Limoges. Support from the laboratoire “Analyse non-linéaire et Optimisation” is gratefully acknowledged.

Evaluation of 100G DP-QPSK long-haul transmission performance using second order co-pumped Raman laser based amplification

We present, for the first time, a detailed investigation of the impact of second order co-propagating Raman pumping on long-haul 100G WDM DP-QPSK coherent transmission of up to 7082 km using Raman fibre laser based configurations. Signal power and noise distributions along the fibre for each pumping scheme were characterised both numerically and experimentally. Based on these pumping schemes, the Q factor penalties versus co-pump power ratios were experimentally measured and quantified. A significant Q factor penalty of up to 4.15 dB was observed after 1666 km using symmetric bidirectional pumping, compared with counter-pumping only. Our results show that whilst using co-pumping minimises the intra-cavity signal power variation and amplification noise, the Q factor penalty with co-pumping was too great for any advantage to be seen. The relative intensity noise (RIN) characteristics of the induced fibre laser and the output signal, and the intra-cavity RF spectra of the fibre laser are also presented. We attribute the Q factor degradation to RIN induced penalty due to RIN being transferred from the first order fibre laser and second order co-pump to the signal. More importantly, there were two different fibre lasing regimes contributing to the amplification. It was random distributed feedback lasing when using counter-pumping only and conventional Fabry-Perot cavity lasing when using all bidirectional pumping schemes. This also results in significantly different performances due to different laser cavity lengths for these two classes of laser.

RIN mitigation in second-order pumped Raman fibre laser based amplification

We experimentally investigate three Raman fibre laser based amplification techniques with second-order bidirectional pumping. Relatively intensity noise (RIN) being transferred to the signal can be significantly suppressed by reducing first-order reflection near the input end. © 2015 OSA.

Development of a high-frequency surface-wave radar cross section model for icebergs

In this thesis, the first-order radar cross section (RCS) of an iceberg is derived and simulated. This analysis takes place in the context of a monostatic high frequency surface wave radar with a vertical dipole source that is driven by a pulsed waveform. The starting point of this work is a general electric field equation derived previ- ously for an arbitrarily shaped iceberg region surrounded by an ocean surface. The condition of monostatic backscatter is applied to this general field equation and the resulting expression is inverse Fourier transformed. In the time domain the excitation current of the transmit antenna is specified to be a pulsed sinusoid signal. The result- ing electric field equation is simplified and its physical significance is assessed. The field equation is then further simplified by restricting the iceberg's size to fit within a single radar patch width. The power received by the radar is calculated using this electric field equation. Comparing the received power with the radar range equation gives a general expression for the iceberg RCS. The iceberg RCS equation is found to depend on several parameters including the geometry of the iceberg, the radar frequency, and the electrical parameters of both the iceberg and the ocean surface. The RCS is rewritten in a form suitable for simulations and simulations are carried out for rectangularly shaped icebergs. Simulation results are discussed and are found to be consistent with existing research.

Advanced Techniques for Image Quality Assessment of Modern X-ray Computed Tomography Systems

X-ray computed tomography (CT) imaging constitutes one of the most widely used diagnostic tools in radiology today with nearly 85 million CT examinations performed in the U.S in 2011. CT imparts a relatively high amount of radiation dose to the patient compared to other x-ray imaging modalities and as a result of this fact, coupled with its popularity, CT is currently the single largest source of medical radiation exposure to the U.S. population. For this reason, there is a critical need to optimize CT examinations such that the dose is minimized while the quality of the CT images is not degraded. This optimization can be difficult to achieve due to the relationship between dose and image quality. All things being held equal, reducing the dose degrades image quality and can impact the diagnostic value of the CT examination.

A recent push from the medical and scientific community towards using lower doses has spawned new dose reduction technologies such as automatic exposure control (i.e., tube current modulation) and iterative reconstruction algorithms. In theory, these technologies could allow for scanning at reduced doses while maintaining the image quality of the exam at an acceptable level. Therefore, there is a scientific need to establish the dose reduction potential of these new technologies in an objective and rigorous manner. Establishing these dose reduction potentials requires precise and clinically relevant metrics of CT image quality, as well as practical and efficient methodologies to measure such metrics on real CT systems. The currently established methodologies for assessing CT image quality are not appropriate to assess modern CT scanners that have implemented those aforementioned dose reduction technologies.

Thus the purpose of this doctoral project was to develop, assess, and implement new phantoms, image quality metrics, analysis techniques, and modeling tools that are appropriate for image quality assessment of modern clinical CT systems. The project developed image quality assessment methods in the context of three distinct paradigms, (a) uniform phantoms, (b) textured phantoms, and (c) clinical images.

The work in this dissertation used the “task-based” definition of image quality. That is, image quality was broadly defined as the effectiveness by which an image can be used for its intended task. Under this definition, any assessment of image quality requires three components: (1) A well defined imaging task (e.g., detection of subtle lesions), (2) an “observer” to perform the task (e.g., a radiologists or a detection algorithm), and (3) a way to measure the observer’s performance in completing the task at hand (e.g., detection sensitivity/specificity).

First, this task-based image quality paradigm was implemented using a novel multi-sized phantom platform (with uniform background) developed specifically to assess modern CT systems (Mercury Phantom, v3.0, Duke University). A comprehensive evaluation was performed on a state-of-the-art CT system (SOMATOM Definition Force, Siemens Healthcare) in terms of noise, resolution, and detectability as a function of patient size, dose, tube energy (i.e., kVp), automatic exposure control, and reconstruction algorithm (i.e., Filtered Back-Projection– FPB vs Advanced Modeled Iterative Reconstruction– ADMIRE). A mathematical observer model (i.e., computer detection algorithm) was implemented and used as the basis of image quality comparisons. It was found that image quality increased with increasing dose and decreasing phantom size. The CT system exhibited nonlinear noise and resolution properties, especially at very low-doses, large phantom sizes, and for low-contrast objects. Objective image quality metrics generally increased with increasing dose and ADMIRE strength, and with decreasing phantom size. The ADMIRE algorithm could offer comparable image quality at reduced doses or improved image quality at the same dose (increase in detectability index by up to 163% depending on iterative strength). The use of automatic exposure control resulted in more consistent image quality with changing phantom size.

Based on those results, the dose reduction potential of ADMIRE was further assessed specifically for the task of detecting small (<=6 mm) low-contrast (<=20 HU) lesions. A new low-contrast detectability phantom (with uniform background) was designed and fabricated using a multi-material 3D printer. The phantom was imaged at multiple dose levels and images were reconstructed with FBP and ADMIRE. Human perception experiments were performed to measure the detection accuracy from FBP and ADMIRE images. It was found that ADMIRE had equivalent performance to FBP at 56% less dose.

Using the same image data as the previous study, a number of different mathematical observer models were implemented to assess which models would result in image quality metrics that best correlated with human detection performance. The models included naïve simple metrics of image quality such as contrast-to-noise ratio (CNR) and more sophisticated observer models such as the non-prewhitening matched filter observer model family and the channelized Hotelling observer model family. It was found that non-prewhitening matched filter observers and the channelized Hotelling observers both correlated strongly with human performance. Conversely, CNR was found to not correlate strongly with human performance, especially when comparing different reconstruction algorithms.

The uniform background phantoms used in the previous studies provided a good first-order approximation of image quality. However, due to their simplicity and due to the complexity of iterative reconstruction algorithms, it is possible that such phantoms are not fully adequate to assess the clinical impact of iterative algorithms because patient images obviously do not have smooth uniform backgrounds. To test this hypothesis, two textured phantoms (classified as gross texture and fine texture) and a uniform phantom of similar size were built and imaged on a SOMATOM Flash scanner (Siemens Healthcare). Images were reconstructed using FBP and a Sinogram Affirmed Iterative Reconstruction (SAFIRE). Using an image subtraction technique, quantum noise was measured in all images of each phantom. It was found that in FBP, the noise was independent of the background (textured vs uniform). However, for SAFIRE, noise increased by up to 44% in the textured phantoms compared to the uniform phantom. As a result, the noise reduction from SAFIRE was found to be up to 66% in the uniform phantom but as low as 29% in the textured phantoms. Based on this result, it clear that further investigation was needed into to understand the impact that background texture has on image quality when iterative reconstruction algorithms are used.

To further investigate this phenomenon with more realistic textures, two anthropomorphic textured phantoms were designed to mimic lung vasculature and fatty soft tissue texture. The phantoms (along with a corresponding uniform phantom) were fabricated with a multi-material 3D printer and imaged on the SOMATOM Flash scanner. Scans were repeated a total of 50 times in order to get ensemble statistics of the noise. A novel method of estimating the noise power spectrum (NPS) from irregularly shaped ROIs was developed. It was found that SAFIRE images had highly locally non-stationary noise patterns with pixels near edges having higher noise than pixels in more uniform regions. Compared to FBP, SAFIRE images had 60% less noise on average in uniform regions for edge pixels, noise was between 20% higher and 40% lower. The noise texture (i.e., NPS) was also highly dependent on the background texture for SAFIRE. Therefore, it was concluded that quantum noise properties in the uniform phantoms are not representative of those in patients for iterative reconstruction algorithms and texture should be considered when assessing image quality of iterative algorithms.

The move beyond just assessing noise properties in textured phantoms towards assessing detectability, a series of new phantoms were designed specifically to measure low-contrast detectability in the presence of background texture. The textures used were optimized to match the texture in the liver regions actual patient CT images using a genetic algorithm. The so called “Clustured Lumpy Background” texture synthesis framework was used to generate the modeled texture. Three textured phantoms and a corresponding uniform phantom were fabricated with a multi-material 3D printer and imaged on the SOMATOM Flash scanner. Images were reconstructed with FBP and SAFIRE and analyzed using a multi-slice channelized Hotelling observer to measure detectability and the dose reduction potential of SAFIRE based on the uniform and textured phantoms. It was found that at the same dose, the improvement in detectability from SAFIRE (compared to FBP) was higher when measured in a uniform phantom compared to textured phantoms.

The final trajectory of this project aimed at developing methods to mathematically model lesions, as a means to help assess image quality directly from patient images. The mathematical modeling framework is first presented. The models describe a lesion’s morphology in terms of size, shape, contrast, and edge profile as an analytical equation. The models can be voxelized and inserted into patient images to create so-called “hybrid” images. These hybrid images can then be used to assess detectability or estimability with the advantage that the ground truth of the lesion morphology and location is known exactly. Based on this framework, a series of liver lesions, lung nodules, and kidney stones were modeled based on images of real lesions. The lesion models were virtually inserted into patient images to create a database of hybrid images to go along with the original database of real lesion images. ROI images from each database were assessed by radiologists in a blinded fashion to determine the realism of the hybrid images. It was found that the radiologists could not readily distinguish between real and virtual lesion images (area under the ROC curve was 0.55). This study provided evidence that the proposed mathematical lesion modeling framework could produce reasonably realistic lesion images.

Based on that result, two studies were conducted which demonstrated the utility of the lesion models. The first study used the modeling framework as a measurement tool to determine how dose and reconstruction algorithm affected the quantitative analysis of liver lesions, lung nodules, and renal stones in terms of their size, shape, attenuation, edge profile, and texture features. The same database of real lesion images used in the previous study was used for this study. That database contained images of the same patient at 2 dose levels (50% and 100%) along with 3 reconstruction algorithms from a GE 750HD CT system (GE Healthcare). The algorithms in question were FBP, Adaptive Statistical Iterative Reconstruction (ASiR), and Model-Based Iterative Reconstruction (MBIR). A total of 23 quantitative features were extracted from the lesions under each condition. It was found that both dose and reconstruction algorithm had a statistically significant effect on the feature measurements. In particular, radiation dose affected five, three, and four of the 23 features (related to lesion size, conspicuity, and pixel-value distribution) for liver lesions, lung nodules, and renal stones, respectively. MBIR significantly affected 9, 11, and 15 of the 23 features (including size, attenuation, and texture features) for liver lesions, lung nodules, and renal stones, respectively. Lesion texture was not significantly affected by radiation dose.

The second study demonstrating the utility of the lesion modeling framework focused on assessing detectability of very low-contrast liver lesions in abdominal imaging. Specifically, detectability was assessed as a function of dose and reconstruction algorithm. As part of a parallel clinical trial, images from 21 patients were collected at 6 dose levels per patient on a SOMATOM Flash scanner. Subtle liver lesion models (contrast = -15 HU) were inserted into the raw projection data from the patient scans. The projections were then reconstructed with FBP and SAFIRE (strength 5). Also, lesion-less images were reconstructed. Noise, contrast, CNR, and detectability index of an observer model (non-prewhitening matched filter) were assessed. It was found that SAFIRE reduced noise by 52%, reduced contrast by 12%, increased CNR by 87%. and increased detectability index by 65% compared to FBP. Further, a 2AFC human perception experiment was performed to assess the dose reduction potential of SAFIRE, which was found to be 22% compared to the standard of care dose.

In conclusion, this dissertation provides to the scientific community a series of new methodologies, phantoms, analysis techniques, and modeling tools that can be used to rigorously assess image quality from modern CT systems. Specifically, methods to properly evaluate iterative reconstruction have been developed and are expected to aid in the safe clinical implementation of dose reduction technologies.

Extended nonlinear analytical models of compliant parallelogram mechanisms: third-order models

This paper proposes extended nonlinear analytical models, third-order models, of compliant parallelogram mechanisms. These models are capable of capturing the accurate effects from the very large axial force within the transverse motion range of 10% of the beam length through incorporating the terms associated with the high-order (up to third-order) axial force. Firstly, the free-body diagram method is employed to derive the nonlinear analytical model for a basic compliant parallelogram mechanism based on load-displacement relations of a single beam, geometry compatibility conditions, and load-equilibrium conditions. The procedures for the forward solutions and inverse solutions are described. Nonlinear analytical models for guided compliant multi-beam parallelogram mechanisms are then obtained. A case study of the compound compliant parallelogram mechanism, composed of two basic compliant parallelogram mechanisms in symmetry, is further implemented. This work intends to estimate the internal axial force change, the transverse force change, and the transverse stiffness change with the transverse motion using the proposed third-order model in comparison with the first-order model proposed in the prior art. In addition, FEA (finite element analysis) results validate the accuracy of the third-order model for a typical example. It is shown that in the case study the slenderness ratio affects the result discrepancy between the third-order model and the first-order model significantly, and the third-order model can illustrate a non-monotonic transverse stiffness curve if the beam is thin enough.

Produção de quitosana a partir de resíduos de camarão: análise do processo e da operação de secagem

A quitina é encontrada principalmente nos exoesqueletos de crustáceos, insetos e na parede celular de fungos. O biopolímero quitosana é obtido através da hidrólise alcalina da quitina. A despolimerização da quitosana é realizada para se obter um produto com valores baixos de massa molecular. O uso da quitosana em diversas áreas é diretamente relacionada com a massa molecular e o grau de desacetilação do polímero. Os objetivos deste trabalho foram o estudo da cinética de secagem de quitina em camada delgada utilizando um modelo difusivo, considerando a resistência externa à transferência de massa; a determinação do comportamento da massa molecular média viscosimétrica da quitosana, durante a secagem convectiva, em camada delgada; a otimização das etapas de desacetilação e despolimerização da quitosana. A quitina foi obtida de resíduos de camarão. Os experimentos da secagem de quitina e da quitosana foram em secador de bandejas, a 60°C, sendo que para a quitina foram utilizadas duas velocidades do ar de 0,5 e 1,5 m/s. A estimativa da viscosidade intrínseca foi através da equação de Huggins e a massa molecular da quitosana foi calculada pela equação de Mark-Houwink-Sakurada. As otimizações da reação de desacetilação e despolimerização foram realizadas utilizando a metodologia da superfície de resposta. Para a reação de desacetilação foram variados o tempo e a temperatura. Para a reação de despolimerização foram analisados a concentração de ácido clorídrico, a temperatura e o tempo de reação. O modelo difusivo com difusividade efetiva variável, utilizado para analisar a secagem de quitina, apresentou concordância com os dados experimentais, onde foi observado o efeito da resistência externa à transferência de massa, quando utilizada a menor velocidade do ar. A condição ótima da reação de desacetilação para massa molecular foi observada na temperatura de 130°C em 90 min, e correspondeu a massa molecular de 150 kDa e um grau de desacetilação de 90%. A operação de secagem da quitosana causou um aumento na massa molecular média viscosimétrica de 27% e este aumento foi linear com o tempo e a umidade do polímero, apresentando duas regiões. As condições da reação de despolimerização para alcançar 50 kDa foram à temperatura de reação de 65°C, concentração de ácido clorídrico de 35% v/v. Nestas condições a cinética de despolimerização foi de pseudo-primeira ordem, apresentando duas fases.

Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain

This paper reports a case study in the use of proof planning in the context of higher order syntax. Rippling is a heuristic for guiding rewriting steps in induction that has been used successfully in proof planning inductive proofs using first order representations. Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling. Previously Boyer-Moore style automation could not be applied to such domains. We demonstrate that a higher-order extension of the rippling heuristic is sufficient to plan such proofs automatically. Accordingly, ordinal arithmetic has been implemented in lambda-clam, a higher order proof planning system for induction, and standard undergraduate text book problems have been successfully planned. We show the synthesis of a fixpoint for normal ordinal functions which demonstrates how our automation could be extended to produce more interesting results than the textbook examples tried so far.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

An investigation of nucleation events in a coastal urban environment in the southern hemisphere

The occurrence of and conditions favourable to nucleation were investigated at an industrial and commercial coastal location in Brisbane, Australia during five different campaigns covering a total period of 13 months. To identify potential nucleation events, the difference in number concentration in the size range 14-30 nm (N14-30) between consecutive observations was calculated using first-order differencing. The data showed that nucleation events were a rare occurrence, and that in the absence of nucleation the particle number was dominated by particles in the range 30-300 nm. In many instances, total particle concentration declined during nucleation. There was no clear pattern in change in NO and NO2 concentrations during the events. SO2 concentration, in the majority of cases, declined during nucleation but there were exceptions. Most events took place in summer, followed by winter and then spring, and no events were observed for the autumn campaigns. The events were associated with sea breeze and long-range transport. Roadside emissions, in contrast, did not contribute to nucleation, probably due to the predominance of particles in the range 50-100 nm associated with these emissions.

Flow of a micropolar fluid bounded by a stretching sheet

We consider boundary layer flow of a micropolar fluid driven by a porous stretching sheet. A similarity solution is defined, and numerical solutions using Runge-Kutta and quasilinearisation schemes are obtained. A perturbation analysis is also used to derive analytic solutions to first order in the perturbing parameter. The resulting closed form solutions involve relatively complex expressions, and the analysis is made more tractable by a combination of offline and online work using a computational algebra system (CAS). For this combined numerical and analytic approach, the perturbation analysis yields a number of benefits with regard to the numerical work. The existence of a closed form solution helps to discriminate between acceptable and spurious numerical solutions. Also, the expressions obtained from the perturbation work can provide an accurate description of the solution for ranges of parameters where the numerical approaches considered here prove computationally more difficult.