Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Microstructural characterization of electron beam evaporated tungsten oxide films for gas sensing applications

Tungsten trioxide is one of the potential semiconducting materials used for sensing NH3, CO, CH4 and acetaldehyde gases. The current research aims at development, microstructural characterization and gas sensing properties of thin films of Tungsten trioxide (WO3). In this paper, we intend to present the microstructural characterization of these films as a function of post annealing heat treatment. Microstructural and elemental analysis of electron beam evaporated WO3 thin films and iron doped WO3 films (WO3:Fe) have been carried out using analytical techniques such as Transmission electron microscopy, Rutherford Backscattered Spectroscopy and XPS analysis. TEM analysis revealed that annealing at 300oC for 1 hour improves cyrstallinity of WO3 film. Both WO3 and WO3:Fe films had uniform thickness and the values corresponded to those measured during deposition. RBS results show a fairly high concentration of oxygen at the film surface as well as in the bulk for both films, which might be due to adsorption of oxygen from atmosphere or lattice oxygen vacancy inherent in WO3 structure. XPS results indicate that tungsten exists in 4d electronic state on the surface but at a depth of 10 nm, both 4d and 4f electronic states were observed. Atomic force microscopy reveals nanosize particles and porous structure of the film. This study shows e-beam evaporation technique produces nanoaparticles and porous WO3 films suitable for gas sensing applications and doping with iron decreases the porosity and particle size which can help improve the gas selectivity.

Integrity determination of RTK solutions in precision farming applications

Integrity of Real Time Kinematic (RTK) positioning solutions relates to the confidential level that can be placed in the information provided by the RTK system. It includes the ability of the RTK system to provide timely valid warnings to users when the system must not be used for the intended operation. For instance, in the controlled traffic farming (CTF) system that controls traffic separates wheel beds and root beds, RTK positioning error causes overlap and increases the amount of soil compaction. The RTK system’s integrity capacity can inform users when the actual positional errors of the RTK solutions have exceeded Horizontal Protection Levels (HPL) within a certain Time-To-Alert (TTA) at a given Integrity Risk (IR). The later is defined as the probability that the system claims its normal operational status while actually being in an abnormal status, e.g., the ambiguities being incorrectly fixed and positional errors having exceeded the HPL. The paper studies the required positioning performance (RPP) of GPS positioning system for PA applications such as a CTF system, according to literature review and survey conducted among a number of farming companies. The HPL and IR are derived from these RPP parameters. A RTK-specific rover autonomous integrity monitoring (RAIM) algorithm is developed to determine the system integrity according to real time outputs, such as residual square sum (RSS), HDOP values. A two-station baseline data set is analyzed to demonstrate the concept of RTK integrity and assess the RTK solution continuity, missed detection probability and false alarm probability.

Effects of switching time on output voltages of a multilevel inverter used in high frequency applications

Employing multilevel inverters is a proper solution to reduce harmonic content of output voltage and electromagnetic interference in high power electronic applications. In this paper, a new pulse width modulation method for multilevel inverters is proposed in which power devices’ on-off switching times have been considered. This method can be surveyed in order to analyse the effect of switching time on harmonic contents of output voltage in high frequency applications when a switching time is not negligible compared to a switching cycle. Fast Fourier transform calculation and analysis of output voltage waveforms and harmonic contents with regard to switching time variation are presented in this paper for a single phase (3, 5)-level inverters used in high voltage and high frequency converters. Mathematical analysis and MATLAB simulation results have been carried out to validate the proposed method.

A new family of multi-output DC-DC converter topologies to supply an asymmetrical four-level diode-clamped inverter

Purpose Multi-level diode-clamped inverters have the challenge of capacitor voltage balancing when the number of DC-link capacitors is three or more. On the other hand, asymmetrical DC-link voltage sources have been applied to increase the number of voltage levels without increasing the number of switches. The purpose of this paper is to show that an appropriate multi-output DC-DC converter can resolve the problem of capacitor voltage balancing and utilize the asymmetrical DC-link voltages advantages. Design/methodology/approach A family of multi-output DC-DC converters is presented in this paper. The application of these converters is to convert the output voltage of a photovoltaic (PV) panel to regulate DC-link voltages of an asymmetrical four-level diode-clamped inverter utilized for domestic applications. To verify the versatility of the presented topology, simulations have been directed for different situations and results are presented. Some related experiments have been developed to examine the capabilities of the proposed converters. Findings The three-output voltage-sharing converters presented in this paper have been mathematically analysed and proven to be appropriate to improve the quality of the residential application of PV by means of four-level asymmetrical diode-clamped inverter supplying highly resistive loads. Originality/value This paper shows that an appropriate multi-output DC-DC converter can resolve the problem of capacitor voltage balancing and utilize the asymmetrical DC-link voltages advantages and that there is a possibility of operation at high-modulation index despite reference voltage magnitude and power factor variations.

Synthesis and modifications of metal oxide nanostructures and their applications

Transition metal oxides are functional materials that have advanced applications in many areas, because of their diverse properties (optical, electrical, magnetic, etc.), hardness, thermal stability and chemical resistance. Novel applications of the nanostructures of these oxides are attracting significant interest as new synthesis methods are developed and new structures are reported. Hydrothermal synthesis is an effective process to prepare various delicate structures of metal oxides on the scales from a few to tens of nanometres, specifically, the highly dispersed intermediate structures which are hardly obtained through pyro-synthesis. In this thesis, a range of new metal oxide (stable and metastable titanate, niobate) nanostructures, namely nanotubes and nanofibres, were synthesised via a hydrothermal process. Further structure modifications were conducted and potential applications in catalysis, photocatalysis, adsorption and construction of ceramic membrane were studied. The morphology evolution during the hydrothermal reaction between Nb2O5 particles and concentrated NaOH was monitored. The study demonstrates that by optimising the reaction parameters (temperature, amount of reactants), one can obtain a variety of nanostructured solids, from intermediate phases niobate bars and fibres to the stable phase cubes. Trititanate (Na2Ti3O7) nanofibres and nanotubes were obtained by the hydrothermal reaction between TiO2 powders or a titanium compound (e.g. TiOSO4·xH2O) and concentrated NaOH solution by controlling the reaction temperature and NaOH concentration. The trititanate possesses a layered structure, and the Na ions that exist between the negative charged titanate layers are exchangeable with other metal ions or H+ ions. The ion-exchange has crucial influence on the phase transition of the exchanged products. The exchange of the sodium ions in the titanate with H+ ions yields protonated titanate (H-titanate) and subsequent phase transformation of the H-titanate enable various TiO2 structures with retained morphology. H-titanate, either nanofibres or tubes, can be converted to pure TiO2(B), pure anatase, mixed TiO2(B) and anatase phases by controlled calcination and by a two-step process of acid-treatment and subsequent calcination. While the controlled calcination of the sodium titanate yield new titanate structures (metastable titanate with formula Na1.5H0.5Ti3O7, with retained fibril morphology) that can be used for removal of radioactive ions and heavy metal ions from water. The structures and morphologies of the metal oxides were characterised by advanced techniques. Titania nanofibres of mixed anatase and TiO2(B) phases, pure anatase and pure TiO2(B) were obtained by calcining H-titanate nanofibres at different temperatures between 300 and 700 °C. The fibril morphology was retained after calcination, which is suitable for transmission electron microscopy (TEM) analysis. It has been found by TEM analysis that in mixed-phase structure the interfaces between anatase and TiO2(B) phases are not random contacts between the engaged crystals of the two phases, but form from the well matched lattice planes of the two phases. For instance, (101) planes in anatase and (101) planes of TiO2(B) are similar in d spaces (~0.18 nm), and they join together to form a stable interface. The interfaces between the two phases act as an one-way valve that permit the transfer of photogenerated charge from anatase to TiO2(B). This reduces the recombination of photogenerated electrons and holes in anatase, enhancing the activity for photocatalytic oxidation. Therefore, the mixed-phase nanofibres exhibited higher photocatalytic activity for degradation of sulforhodamine B (SRB) dye under ultraviolet (UV) light than the nanofibres of either pure phase alone, or the mechanical mixtures (which have no interfaces) of the two pure phase nanofibres with a similar phase composition. This verifies the theory that the difference between the conduction band edges of the two phases may result in charge transfer from one phase to the other, which results in effectively the photogenerated charge separation and thus facilitates the redox reaction involving these charges. Such an interface structure facilitates charge transfer crossing the interfaces. The knowledge acquired in this study is important not only for design of efficient TiO2 photocatalysts but also for understanding the photocatalysis process. Moreover, the fibril titania photocatalysts are of great advantage when they are separated from a liquid for reuse by filtration, sedimentation, or centrifugation, compared to nanoparticles of the same scale. The surface structure of TiO2 also plays a significant role in catalysis and photocatalysis. Four types of large surface area TiO2 nanotubes with different phase compositions (labelled as NTA, NTBA, NTMA and NTM) were synthesised from calcination and acid treatment of the H-titanate nanotubes. Using the in situ FTIR emission spectrescopy (IES), desorption and re-adsorption process of surface OH-groups on oxide surface can be trailed. In this work, the surface OH-group regeneration ability of the TiO2 nanotubes was investigated. The ability of the four samples distinctively different, having the order: NTA > NTBA > NTMA > NTM. The same order was observed for the catalytic when the samples served as photocatalysts for the decomposition of synthetic dye SRB under UV light, as the supports of gold (Au) catalysts (where gold particles were loaded by a colloid-based method) for photodecomposition of formaldehyde under visible light and for catalytic oxidation of CO at low temperatures. Therefore, the ability of TiO2 nanotubes to generate surface OH-groups is an indicator of the catalytic activity. The reason behind the correlation is that the oxygen vacancies at bridging O2- sites of TiO2 surface can generate surface OH-groups and these groups facilitate adsorption and activation of O2 molecules, which is the key step of the oxidation reactions. The structure of the oxygen vacancies at bridging O2- sites is proposed. Also a new mechanism for the photocatalytic formaldehyde decomposition with the Au-TiO2 catalysts is proposed: The visible light absorbed by the gold nanoparticles, due to surface plasmon resonance effect, induces transition of the 6sp electrons of gold to high energy levels. These energetic electrons can migrate to the conduction band of TiO2 and are seized by oxygen molecules. Meanwhile, the gold nanoparticles capture electrons from the formaldehyde molecules adsorbed on them because of gold’s high electronegativity. O2 adsorbed on the TiO2 supports surface are the major electron acceptor. The more O2 adsorbed, the higher the oxidation activity of the photocatalyst will exhibit. The last part of this thesis demonstrates two innovative applications of the titanate nanostructures. Firstly, trititanate and metastable titanate (Na1.5H0.5Ti3O7) nanofibres are used as intelligent absorbents for removal of radioactive cations and heavy metal ions, utilizing the properties of the ion exchange ability, deformable layered structure, and fibril morphology. Environmental contamination with radioactive ions and heavy metal ions can cause a serious threat to the health of a large part of the population. Treatment of the wastes is needed to produce a waste product suitable for long-term storage and disposal. The ion-exchange ability of layered titanate structure permitted adsorption of bivalence toxic cations (Sr2+, Ra2+, Pb2+) from aqueous solution. More importantly, the adsorption is irreversible, due to the deformation of the structure induced by the strong interaction between the adsorbed bivalent cations and negatively charged TiO6 octahedra, and results in permanent entrapment of the toxic bivalent cations in the fibres so that the toxic ions can be safely deposited. Compared to conventional clay and zeolite sorbents, the fibril absorbents are of great advantage as they can be readily dispersed into and separated from a liquid. Secondly, new generation membranes were constructed by using large titanate and small ã-alumina nanofibres as intermediate and top layers, respectively, on a porous alumina substrate via a spin-coating process. Compared to conventional ceramic membranes constructed by spherical particles, the ceramic membrane constructed by the fibres permits high flux because of the large porosity of their separation layers. The voids in the separation layer determine the selectivity and flux of a separation membrane. When the sizes of the voids are similar (which means a similar selectivity of the separation layer), the flux passing through the membrane increases with the volume of the voids which are filtration passages. For the ideal and simplest texture, a mesh constructed with the nanofibres 10 nm thick and having a uniform pore size of 60 nm, the porosity is greater than 73.5 %. In contrast, the porosity of the separation layer that possesses the same pore size but is constructed with metal oxide spherical particles, as in conventional ceramic membranes, is 36% or less. The membrane constructed by titanate nanofibres and a layer of randomly oriented alumina nanofibres was able to filter out 96.8% of latex spheres of 60 nm size, while maintaining a high flux rate between 600 and 900 Lm–2 h–1, more than 15 times higher than the conventional membrane reported in the most recent study.

Students’ conceptual knowledge about workplace pedagogies and applications to learning in the workplace

The development of effective workplace pedagogies is integral to work-integrated and work-based learning. The workplace pedagogies that facilitate and support learning include: activities in which individuals engage such as daily work practices, questioning; observing, and listening; interactions with more experienced workers through coaching and modelling; and referencing documented procedures. Each of these dimensions is significant in enhancing processes of workplace learning. Learning can be optimised when these pedagogies are appropriately embedded in the context of and process of participating in normal work activities. Yet learners need to understand the nature of these pedagogies, and how and for what purposes to use these to achieve a range of learning outcomes. This is because it is the worker-learners who play the key roles in the process and outcomes of learning through work. A pilot study was conducted on students’ conceptions of how each of these dimensions of workplace pedagogy help their learning, by providing examples of learning from these sources; and stating their preferences for learning in the workplace. A sample of seventeen students, enrolled in the second year of a Diploma in Nursing course at a Technical and Further Education institution, participated in a survey intended to capture these conceptions and the importance attached to each of them. The findings indicate that these students have basic understanding of how each of seven workplace pedagogic practices can contribute to their learning. They reported relying mostly on daily practices, observing and listening to others, modelling, coaching, and other workers. Their selection of these contributions emphasise significant opportunities for guided learning by others, yet suggest fewer student-initiated interactions, less intensity in interactions, and likelihood that learning is more passive. The data also suggests that these students rely mostly on using academic learning skill, and limited workplace learning skills. It is proposed, therefore, that the knowledge and understandings about workplace learning and pedagogies might be best embedded in the full curriculum and not become add-on shortly before students go on work placement. This approach will allow students to appreciate the significance and use of workplace pedagogies for learning.

Infrared spectroscopic investigation of the effects of titania photocatalyst on the degradation of linear low density polyethylene film for commercial applications

There is a need in industry for a commodity polyethylene film with controllable degradation properties that will degrade in an environmentally neutral way, for applications such as shopping bags and packaging film. Additives such as starch have been shown to accelerate the degradation of plastic films, however control of degradation is required so that the film will retain its mechanical properties during storage and use, and then degrade when no longer required. By the addition of a photocatalyst it is hoped that polymer film will breakdown with exposure to sunlight. Furthermore, it is desired that the polymer film will degrade in the dark, after a short initial exposure to sunlight. Research has been undertaken into the photo- and thermo-oxidative degradation processes of 25 ìm thick LLDPE (linear low density polyethylene) film containing titania from different manufacturers. Films were aged in a suntest or in an oven at 50 °C, and the oxidation product formation was followed using IR spectroscopy. Degussa P25, Kronos 1002, and various organic-modified and doped titanias of the types Satchleben Hombitan and Hunstsman Tioxide incorporated into LLDPE films were assessed for photoactivity. Degussa P25 was found to be the most photoactive with UVA and UVC exposure. Surface modification of titania was found to reduce photoactivity. Crystal phase is thought to be among the most important factors when assessing the photoactivity of titania as a photocatalyst for degradation. Pre-irradiation with UVA or UVC for 24 hours of the film containing 3% Degussa P25 titania prior to aging in an oven resulted in embrittlement in ca. 200 days. The multivariate data analysis technique PCA (principal component analysis) was used as an exploratory tool to investigate the IR spectral data. Oxidation products formed in similar relative concentrations across all samples, confirming that titania was catalysing the oxidation of the LLDPE film without changing the oxidation pathway. PCA was also employed to compare rates of degradation in different films. PCA enabled the discovery of water vapour trapped inside cavities formed by oxidation by titania particles. Imaging ATR/FTIR spectroscopy with high lateral resolution was used in a novel experiment to examine the heterogeneous nature of oxidation of a model polymer compound caused by the presence of titania particles. A model polymer containing Degussa P25 titania was solvent cast onto the internal reflection element of the imaging ATR/FTIR and the oxidation under UVC was examined over time. Sensitisation of 5 ìm domains by titania resulted in areas of relatively high oxidation product concentration. The suitability of transmission IR with a synchrotron light source to the study of polymer film oxidation was assessed as the Australian Synchrotron in Melbourne, Australia. Challenges such as interference fringes and poor signal-to-noise ratio need to be addressed before this can become a routine technique.

Influence of smectite suspension structure on sheet orientation in dry sediments : XRD and AFM applications

The structure-building phenomena within clay aggregates are governed by forces acting between clay particles. Measurements of such forces are important to understand in order to manipulate the aggregate structure for applications such as dewatering of mineral processing tailings. A parallel particle orientation is required when conducting XRD investigation on the oriented samples and conduct force measurements acting between basal planes of clay mineral platelets using at. force microscopy (AFM). To investigate how smectite clay platelets were oriented on silicon wafer substrate when dried from suspension range of methods like SEM, XRD and AFM were employed. From these investigations, we conclude that high clay concns. and larger particle diams. (up to 5 μm) in suspension result in random orientation of platelets in the substrate. The best possible laminar orientation in the clay dry film, represented in the XRD 0 0 1/0 2 0 intensity ratio of 47 was obtained by drying thin layers from 0.02 wt.% clay suspensions of the natural pH. Conducted AFM investigations show that smectite studied in water based electrolytes show very long-range repulsive forces lower in strength than electrostatic forces from double-layer repulsion. It was suggested that these forces may have structural nature. Smectite surface layers rehydrate in water environment forms surface gel with spongy and cellular texture which cushion approaching AFM probe. This structural effect can be measured in distances larger than 1000 nm from substrate surface and when probe penetrate this gel layer, structural linkages are forming between substrate and clay covered probe. These linkages prevent subsequently smooth detachments of AFM probe on way back when retrieval. This effect of tearing new formed structure apart involves larger adhesion-like forces measured in retrieval. It is also suggested that these effect may be enhanced by the nano-clay particles interaction.