Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

A hypoplastic model of skeletal muscle development displaying reduced foetal myoblast cell numbers, increased oxidative myofibres and improved specific tension capacity

The major component of skeletal muscle is the myofibre. Genetic intervention inducing over-enlargement of myofibres beyond a certain threshold through acellular growth causes a reduction in the specific tension generating capacity of the muscle. However the physiological parameters of a genetic model that harbours reduced skeletal muscle mass have yet to be analysed. Genetic deletion of Meox2 in mice leads to reduced limb muscle size and causes some patterning defects. The loss of Meox2 is not embryonically lethal and a small percentage of animals survive to adulthood making it an excellent model with which to investigate how skeletal muscle responds to reductions in mass. In this study we have performed a detailed analysis of both late foetal and adult muscle development in the absence of Meox2. In the adult, we show that the loss of Meox2 results in smaller limb muscles that harbour reduced numbers of myofibres. However, these fibres are enlarged. These myofibres display a molecular and metabolic fibre type switch towards a more oxidative phenotype that is induced through abnormalities in foetal fibre formation. In spite of these changes, the muscle from Meox2 mutant mice is able to generate increased levels of specific tension compared to that of the wild type.

Optimal control of a class of discrete–continuous non-linear systems — decomposition and hierarchical structure

A technique is derived for solving a non-linear optimal control problem by iterating on a sequence of simplified problems in linear quadratic form. The technique is designed to achieve the correct solution of the original non-linear optimal control problem in spite of these simplifications. A mixed approach with a discrete performance index and continuous state variable system description is used as the basis of the design, and it is shown how the global problem can be decomposed into local sub-system problems and a co-ordinator within a hierarchical framework. An analysis of the optimality and convergence properties of the algorithm is presented and the effectiveness of the technique is demonstrated using a simulation example with a non-separable performance index.

A combined SMOTE and PSO based RBF classifier for two-class imbalanced problems

This contribution proposes a powerful technique for two-class imbalanced classification problems by combining the synthetic minority over-sampling technique (SMOTE) and the particle swarm optimisation (PSO) aided radial basis function (RBF) classifier. In order to enhance the significance of the small and specific region belonging to the positive class in the decision region, the SMOTE is applied to generate synthetic instances for the positive class to balance the training data set. Based on the over-sampled training data, the RBF classifier is constructed by applying the orthogonal forward selection procedure, in which the classifier's structure and the parameters of RBF kernels are determined using a PSO algorithm based on the criterion of minimising the leave-one-out misclassification rate. The experimental results obtained on a simulated imbalanced data set and three real imbalanced data sets are presented to demonstrate the effectiveness of our proposed algorithm.

A sequential two meal challenge reveals abnormalities in postprandial TAG but not glucose in men with increasing numbers of metabolic syndrome components

Objective To examine the impact of increasing numbers of metabolic syndrome (MetS) components on postprandial lipaemia. Methods Healthy men (n = 112) underwent a sequential meal postprandial investigation, in which blood samples were taken at regular intervals after a test breakfast (0 min) and lunch (330 min). Lipids and glucose were measured in the fasting sample, with triacylglycerol (TAG), non-esterified fatty acids and glucose analysed in the postprandial samples. Results Subjects were grouped according to the number of MetS components regardless of the combinations of components (0/1, 2, 3 and 4/5). As expected, there was a trend for an increase in body mass index, blood pressure, fasting TAG, glucose and insulin, and a decrease in fasting high-density lipoprotein cholesterol with increasing numbers of MetS components (P≤0.0004). A similar trend was observed for the summary measures of the postprandial TAG and glucose responses. For TAG, the area under the curve (AUC) and maximum concentration (maxC) were significantly greater in men with ≥ 3 than < 3 components (P < 0.001), whereas incremental AUC was greater in those with 3 than 0/1 and 2, and 4/5 compared with 2 components (P < 0.04). For glucose, maxC after the test breakfast (0-330 min) and total AUC (0-480 min) were higher in men with ≥ 3 than < 3 components (P≤0.001). Conclusions Our data analysis has revealed a linear trend between increasing numbers of MetS components and magnitude (AUC) of the postprandial TAG and glucose responses. Furthermore, the two meal challenge discriminated a worsening of postprandial lipaemic control in subjects with ≥ 3 MetS components.