Computational aspects of the numerical solution of SDEs

In the last 30 to 40 years, many researchers have combined to build the knowledge base of theory and solution techniques that can be applied to the case of differential equations which include the effects of noise. This class of ``noisy'' differential equations is now known as stochastic differential equations (SDEs). Markov diffusion processes are included within the field of SDEs through the drift and diffusion components of the Itô form of an SDE. When these drift and diffusion components are moderately smooth functions, then the processes' transition probability densities satisfy the Fokker-Planck-Kolmogorov (FPK) equation -- an ordinary partial differential equation (PDE). Thus there is a mathematical inter-relationship that allows solutions of SDEs to be determined from the solution of a noise free differential equation which has been extensively studied since the 1920s. The main numerical solution technique employed to solve the FPK equation is the classical Finite Element Method (FEM). The FEM is of particular importance to engineers when used to solve FPK systems that describe noisy oscillators. The FEM is a powerful tool but is limited in that it is cumbersome when applied to multidimensional systems and can lead to large and complex matrix systems with their inherent solution and storage problems. I show in this thesis that the stochastic Taylor series (TS) based time discretisation approach to the solution of SDEs is an efficient and accurate technique that provides transition and steady state solutions to the associated FPK equation. The TS approach to the solution of SDEs has certain advantages over the classical techniques. These advantages include their ability to effectively tackle stiff systems, their simplicity of derivation and their ease of implementation and re-use. Unlike the FEM approach, which is difficult to apply in even only two dimensions, the simplicity of the TS approach is independant of the dimension of the system under investigation. Their main disadvantage, that of requiring a large number of simulations and the associated CPU requirements, is countered by their underlying structure which makes them perfectly suited for use on the now prevalent parallel or distributed processing systems. In summary, l will compare the TS solution of SDEs to the solution of the associated FPK equations using the classical FEM technique. One, two and three dimensional FPK systems that describe noisy oscillators have been chosen for the analysis. As higher dimensional FPK systems are rarely mentioned in the literature, the TS approach will be extended to essentially infinite dimensional systems through the solution of stochastic PDEs. In making these comparisons, the advantages of modern computing tools such as computer algebra systems and simulation software, when used as an adjunct to the solution of SDEs or their associated FPK equations, are demonstrated.

A novel bio-kinematic encoder for human exercise representation and decomposition - Part 1 : Indexing and modelling

Current bio-kinematic encoders use velocity, acceleration and angular information to encode human exercises. However, in exercise physiology there is a need to distinguish between the shape of the trajectory and its execution dynamics. In this paper we propose such a two-component model and explore how best to compute these components of an action. In particular, we show how a new spatial indexing scheme, derived directly from the underlying differential geometry of curves, provides robust estimates of the shape and dynamics compared to standard temporal indexing schemes.

Reduced order functional observers with application to partial state estimation of linear systems with input delay

Reduced order multi-functional observer design for multi-input multi-utput (MIMO) linear time-invariant (LTI) systems with constant delayed inputs is studied. This research is useful in the input estimation of LTI systems with actuator delay, as well as system monitoring and fault detection of these systems. Two approaches for designing an asymptotically stable functional observer for the system are proposed: delay-dependent and delay-free. The delay-dependent observer is infinite-dimensional, while the delay-free structure is finite-dimensional. Moreover, since the delay-free observer does not require any information on the time delay, it is more practical in real applications. However, the delay-dependent observer contains less restrictive assumptions and covers more variety of systems. The proposed observer design schemes are novel, simple to implement, and have improved numerical features compared to some of the other available approaches to design (unknown-input) functional observers. In addition, the proposed observers usually possess lower order than ordinary Luenberger observers, and the design schemes do not need the observability or detectability requirements of the system. The necessary and sufficient conditions of the existence of an asymptoticobserver in each scenario are explored. The extensions of the proposed observers to systems with multiple delayed-inputs are also discussed. Several numerical examples and simulation results are employed to support our theories.

A new indexing method for high dimensional dataset

Indexing high dimensional datasets has attracted extensive attention from many researchers in the last decade. Since R-tree type of index structures are known as suffering curse of dimensionality problems, Pyramid-tree type of index structures, which are based on the B-tree, have been proposed to break the curse of dimensionality. However, for high dimensional data, the number of pyramids is often insufficient to discriminate data points when the number of dimensions is high. Its effectiveness degrades dramatically with the increase of dimensionality. In this paper, we focus on one particular issue of curse of dimensionality; that is, the surface of a hypercube in a high dimensional space approaches 100% of the total hypercube volume when the number of dimensions approaches infinite. We propose a new indexing method based on the surface of dimensionality. We prove that the Pyramid tree technology is a special case of our method. The results of our experiments demonstrate clear priority of our novel method.

Scope for further analytical solutions for constant flux infiltration into a semi-infinite soil profile or redistribution in a finite soil profile.

We attempt to generate new solutions for the moisture content form of the one-dimensional Richards' [1931] equation using the Lisle [1992] equivalence mapping. This mapping is used as no more general set of transformations exists for mapping the one-dimensional Richards' equation into itself. Starting from a given solution, the mapping has the potential to generate an infinite number of new solutions for a series of nonlinear diffusivity and hydraulic conductivity functions. We first seek new analytical solutions satisfying Richards' equation subject to a constant flux surface boundary condition for a semi-infinite dry soil, starting with the Burgers model. The first iteration produces an existing solution, while subsequent iterations are shown to endlessly reproduce this same solution. Next, we briefly consider the problem of redistribution in a finite-length soil. In this case, Lisle's equivalence mapping is generalized to account for arbitrary initial conditions. As was the case for infiltration, however, it is found that new analytical solutions are not generated using the equivalence mapping, although existing solutions are recovered.

Gaussian vault geometry : digital design and fabrication of scaled prototypes

This paper reports on three approaches to the translation of Gaussian surface models into scaled physical prototype models. Using the geometry of Eladio Dieste's Gaussian Vaults, the paper reports on the aspects encountered in the process of digital to physical prototype fabrication. The primary focus of the paper is on exploring the design geometry, investigating methods for preparing the geometry for fabrication and constructing physical prototypes. Three different approaches in the translation from digital to physical models are investigated: rapid prototyping, two dimensional surface models in paper and structural component models using Computer Numerical Controlled (CNC) fabrication. The three approaches identify a body of knowledge in the design and prototyping of Gaussian vaults. Finally the paper discusses the digital to. fabrication translation processes with regards to the characteristics, benefits and limitations of the three approaches of prototyping the ruled surface geometry of Gaussian Vaults.

Three dimensional stability charts for slopes based on limit analysis methods

This paper uses finite element upper and lower bound limit analysis to produce chart solutions for three-dimensional (3D) natural slopes for both short- and long-term stability. The presented chart solutions are convenient tools that can be used for preliminary design purposes. The rigorous limit analysis results in this paper were found to bracket the true factor of safety within ±10% or better, which can be used as a benchmark for the solutions from other methods. The depth of the slip surfaces is observed to be generally shallow for most analyzed cases, particularly for the long-term slope stability problem. In addition, it was found that using a two-dimensional (2D) analysis may lead to significant differences in estimating safety factors, which can differ by 2%–60% depending on the slope geometry and soil properties. Therefore, great care and judgement are required when applying 2D analyses to 3D slope problems.

Effects of variation in modulator temperature during cryogenic modulation in comprehensive two-dimensional gas chromatography

Many modulation systems in comprehensive 2D GC (GC×GC) are based on cryogenic methods. High trapping temperatures in these systems can result in ineffective trapping of the more volatile compounds, whilst temperatures that are too low can prevent efficient remobilisation of some compounds. To better understand the trapping and release of compounds over a wide range of volatilities, we have investigated a number of different constant temperature modulator settings, and have also examined a constant temperature differential between the cryo-trap and the chromatographic oven. These investigations have led us to modify the temperature regulation capabilities of the longitudinally modulated cryogenic system (LMCS). In contrast to the current system, where the user sets a constant temperature for the cooling chamber, the user now sets the temperature difference between the cryo-trap and the chromatographic oven. In this configuration, the cooling chamber temperature increases during the chromatographic run, tracking the oven temperature ramp. This produces more efficient, volatility-dependent modulation, and increases the range of volatile compounds that can be analysed under optimal trap-and-release conditions within a single analytical run. This system also reduces cryogenic fluid consumption.

Computation and construction of vault geometry prototypes

Physical models and scaled prototypes of architecture play an important role in design. They enable architects and designers to investigate the formal, functional, and material attributes of the design. Understanding digital processes of realizing scaled prototypes is a significant problem confronting design practice. This paper reports on three approaches to the translation of Gaussian surface models into scaled physical prototype models. Based on the geometry of Eladio Dieste’s Gaussian Vaults, the paper reports on the aspects encountered in the process of digital to physical construction using scaled prototypes. The primary focus of the paper is on computing the design geometry, investigating methods for preparing the geometry for fabrication and physical construction. Three different approaches in the translation from digital to physical models are investigated: rapid prototyping, two-dimensional surface models in paper and structural component models using CNC fabrication. The three approaches identify a body of knowledge in the design and prototyping of Gaussian vaults. Finally the paper discusses the digital to fabrication translation processes with regards to the characteristics, benefits and limitations of the three approaches of prototyping the ruled surface geometry of Gaussian Vaults. The results of each of three fabrication processes allowed for a better understanding of the digital to physical translation process. The use of rapid prototyping permits the production of form models that provide a representation of the physical characteristics such as size, shape and proportion of the Gaussian Vault.

Super-resolution of a 3-dimensional scene from novel viewpoints

Super-resolution is a method of post-processing image enhancement that increases the spatial resolution of video or images. Existing super-resolution techniques apply only to images captured of a planar scene. This paper aims to extend super-resolution concepts from the 2D domain to the 3D domain, drawing on ideas from both superresolution and multi-view geometry, two fields of research that until now have predominantly been studied in isolation. 2D super-resolution methods are not without their complexities and limitations. However, once multiple views of a scene are considered within a super-resolution framework, a new range of issues arise that must also be resolved. For example, when input images of a scene with variation in depth are considered, it is no longer clear how and where the images should be registered. This paper describes the use of sparse 3D reconstruction in order to ‘register’ the input images, which are then transferred to a novel image plane and combined to increase the perceived detail in the scene. Experimental results using real images captured from generally positioned input cameras are presented.

Teaching 3-D geometry - the multi representational way

Sonja Kalbitzer and Esther Loong provide an excellent range of activities that promote geometric thinking through the exploration of three-dimensional objects. They also provide some discussion on assessing the tasks and providing student feedback.

Anti-metastatic and differential effects on protein expression of epigallocatechin-3-gallate in HCCLM6 hepatocellular carcinoma cells

Hepatocellular carcinoma (HCC) is the fifth most common cancer worldwide and the third highest cause of cancer-related mortality in humans. Epigallocatechin-3-gallate (EGCG) has been shown to inhibit the metastatic activity of certain cancer cells. The aim of this study was to determine the effects and molecular mechanism(s) of action of EGCG in human HCC cells. A migration and invasion assay for the metastatic behavior of HCCLM6 cells was performed. The anti-metastatic effects of EGCG were investigated by RT-PCR and gelatin zymography. A total cellular protein profile was obtained using 2-dimensional gel electrophoresis (2-DE), followed by matrix-assisted laser desorption/ionization-time of flight mass spectrometry (MALDI-TOF-MS) analyses of proteins with significant differences in expression following treatment with EGCG. The results revealed that EGCG induced apoptosis and inhibited the metastasis of HCCLM6 cells. The anti-metastatic effects of EGCG were associated with the inhibition of matrix metalloproteinase (MMP)-2 and MMP-9 activity. The expression levels of far upstream element (FUSE) binding protein 1 (FUBP1), heat shock protein beta 1 (HSPB1), heat shock 60 kDa protein 1 (chaperonin) (CH60) and nucleophosmin (NPM) proteins, which are associated with metastasis, were significantly altered in the EGCG-treated HCCLM6 cells. The data from the present study suggest that EGCG has potential as a therapeutic agent for the treatment of HCC.

Geometry of compensatory feeding and water consumption in Drosophila melanogaster

Feeding behaviour is an expression of an animal’s underlying nutritional strategy. The study of feeding decisions can hence delineate nutritional strategies. Studies of Drosophila melanogaster feeding behaviour have yielded conflicting accounts, and little is known about how nutrients affect feeding patterns in this important model species. Here, we conducted two experiments to characterize nutrient prioritization and regulation. In a choice experiment, we allowed female flies to self-regulate their intake of yeast, sucrose and water by supplying individual flies with three microcapillary tubes: one containing only yeast of varying concentrations, another with just sucrose of varying concentrations, and the last with just water. Flies tightly regulated yeast and sucrose to a constant ratio at the expense of excess water intake, indicating that flies prioritize macronutrient regulation over excess water consumption. To determine the relative importance of yeast and sucrose, in a no-choice experiment, we provided flies with two microcapillary tubes: the first with one of the 28 diets varying in yeast and sucrose content and the other with only water. Flies increased total water intake in relation to yeast consumption but not sucrose consumption. Additionally, flies increased diet intake as diet concentration decreased and as the ratio of sugar to yeast equalized. Using a geometric scaling approach, we found that the patterns of diet intake can be explained by flies prioritizing protein and carbohydrates equally and by the lack of substitutability between the nutrients. We conclude by illustrating how our results harmonize conflicting results in the literature once viewed in a two-dimensional diet landscape.

Greater association of peak neuromuscular performance with cortical bone geometry, bone mass and bone strength than bone density : a study in 417 older women

BACKGROUND: We evaluated which aspects of neuromuscular performance are associated with bone mass, density, strength and geometry. METHODS: 417 women aged 60-94years were examined. Countermovement jump, sit-to-stand test, grip strength, forearm and calf muscle cross-sectional area, areal bone mineral content and density (aBMC and aBMD) at the hip and lumbar spine via dual X-ray absorptiometry, and measures of volumetric vBMC and vBMD, bone geometry and section modulus at 4% and 66% of radius length and 4%, 38% and 66% of tibia length via peripheral quantitative computed tomography were performed. The first principal component of the neuromuscular variables was calculated to generate a summary neuromuscular variable. Percentage of total variance in bone parameters explained by the neuromuscular parameters was calculated. Step-wise regression was also performed. RESULTS: At all pQCT bone sites (radius, ulna, tibia, fibula), a greater percentage of total variance in measures of bone mass, cortical geometry and/or bone strength was explained by peak neuromuscular performance than for vBMD. Sit-to-stand performance did not relate strongly to bone parameters. No obvious differential in the explanatory power of neuromuscular performance was seen for DXA aBMC versus aBMD. In step-wise regression, bone mass, cortical morphology, and/or strength remained significant in relation to the first principal component of the neuromuscular variables. In no case was vBMD positively related to neuromuscular performance in the final step-wise regression models. CONCLUSION: Peak neuromuscular performance has a stronger relationship with leg and forearm bone mass and cortical geometry as well as proximal forearm section modulus than with vBMD.