Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

The philosophy of W. Ross Ashby and its relationship to 'The Matrix'

Ashby was a keen observer of the world around him, as per his technological and psychiatrical developments. Over the years, he drew numerous philosophical conclusions on the nature of human intelligence and the operation of the brain, on artificial intelligence and the thinking ability of computers and even on science in general. In this paper, the quite profound philosophy espoused by Ashby is considered as a whole, in particular in terms of its relationship with the world as it stands now and even in terms of scientific predictions of where things might lead. A meaningful comparison is made between Ashby's comments and the science fiction concept of 'The Matrix' and serious consideration is given as to how much Ashby's ideas lay open the possibility of the matrix becoming a real world eventuality.

Comparison of the computational cost of a Monte Carlo and deterministic algorithm for computing bilinear forms of matrix powers

In this paper we consider bilinear forms of matrix polynomials and show that these polynomials can be used to construct solutions for the problems of solving systems of linear algebraic equations, matrix inversion and finding extremal eigenvalues. An almost Optimal Monte Carlo (MAO) algorithm for computing bilinear forms of matrix polynomials is presented. Results for the computational costs of a balanced algorithm for computing the bilinear form of a matrix power is presented, i.e., an algorithm for which probability and systematic errors are of the same order, and this is compared with the computational cost for a corresponding deterministic method.

A molecular T-matrix approach to calculating low-energy electron diffusion intensities for ordered molecular adsorbates

We present a novel approach to calculating Low-Energy Electron Diffraction (LEED) intensities for ordered molecular adsorbates. First, the intra-molecular multiple scattering is computed to obtain a non-diagonal molecular T-matrix. This is then used to represent the entire molecule as a single scattering object in a conventional LEED calculation, where the Layer Doubling technique is applied to assemble the different layers, including the molecular ones. A detailed comparison with conventional layer-type LEED calculations is provided to ascertain the accuracy of this scheme of calculation. Advantages of this scheme for problems involving ordered arrays of molecules adsorbed on surfaces are discussed.

Solving surface structures from normal incidence X-ray standing wave data

A program is provided to determine structural parameters of atoms in or adsorbed on surfaces by refinement of atomistic models towards experimentally determined data generated by the normal incidence X-ray standing wave (NIXSW) technique. The method employs a combination of Differential Evolution Genetic Algorithms and Steepest Descent Line Minimisations to provide a fast, reliable and user friendly tool for experimentalists to interpret complex multidimensional NIXSW data sets.

The consumption of milk and dairy foods and the incidence of vascular disease and diabetes: An overview of the evidence

The health effects of milk and dairy food consumption would best be determined in randomised controlled trials. No adequately powered trial has been reported and none is likely because of the numbers required. The best evidence comes, therefore, from prospective cohort studies with disease events and death as outcomes. Medline was searched for prospective studies of dairy food consumption and incident vascular disease and Type 2 diabetes, based on representative population samples. Reports in which evaluation was in incident disease or death were selected. Meta-analyses of the adjusted estimates of relative risk for disease outcomes in these reports were conducted. Relevant case–control retrospective studies were also identified and the results are summarised in this article. Meta-analyses suggest a reduction in risk in the subjects with the highest dairy consumption relative to those with the lowest intake: 0.87 (0.77, 0.98) for all-cause deaths, 0.92 (0.80, 0.99) for ischaemic heart disease, 0.79 (0.68, 0.91) for stroke and 0.85 (0.75, 0.96) for incident diabetes. The number of cohort studies which give evidence on individual dairy food items is very small, but, again, there is no convincing evidence of harm from consumption of the separate food items. In conclusion, there appears to be an enormous mis-match between the evidence from long-term prospective studies and perceptions of harm from the consumption of dairy food items.

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.

Selective biodegradation of keratin matrix in feather rachis reveals classic bioengineering

Flight necessitates that the feather rachis is extremely tough and light. Yet, the crucial filamentous hierarchy of the rachis is unknown—study hindered by the tight chemical bonding between the filaments and matrix. We used novel microbial biodegradation to delineate the fibres of the rachidial cortex in situ. It revealed the thickest keratin filaments known to date (factor >10), approximately 6 µm thick, extending predominantly axially but with a small outer circumferential component. Near-periodic thickened nodes of the fibres are staggered with those in adjacent fibres in two- and three-dimensional planes, creating a fibre–matrix texture with high attributes for crack stopping and resistance to transverse cutting. Close association of the fibre layer with the underlying ‘spongy’ medulloid pith indicates the potential for higher buckling loads and greater elastic recoil. Strikingly, the fibres are similar in dimensions and form to the free filaments of the feather vane and plumulaceous and embryonic down, the syncitial barbules, but, identified for the first time in 140+ years of study in a new location—as a major structural component of the rachis. Early in feather evolution, syncitial barbules were consolidated in a robust central rachis, definitively characterizing the avian lineage of keratin.

Highly accurate detection of ovarian cancer using CA125 but limited improvement with serum matrix-assisted laser desorption/ionization time-of-flight mass spectrometry profiling

Objectives: Our objective was to test the performance of CA125 in classifying serum samples from a cohort of malignant and benign ovarian cancers and age-matched healthy controls and to assess whether combining information from matrix-assisted laser desorption/ionization (MALDI) time-of-flight profiling could improve diagnostic performance. Materials and Methods: Serum samples from women with ovarian neoplasms and healthy volunteers were subjected to CA125 assay and MALDI time-of-flight mass spectrometry (MS) profiling. Models were built from training data sets using discriminatory MALDI MS peaks in combination with CA125 values and tested their ability to classify blinded test samples. These were compared with models using CA125 threshold levels from 193 patients with ovarian cancer, 290 with benign neoplasm, and 2236 postmenopausal healthy controls. Results: Using a CA125 cutoff of 30 U/mL, an overall sensitivity of 94.8% (96.6% specificity) was obtained when comparing malignancies versus healthy postmenopausal controls, whereas a cutoff of 65 U/mL provided a sensitivity of 83.9% (99.6% specificity). High classification accuracies were obtained for early-stage cancers (93.5% sensitivity). Reasons for high accuracies include recruitment bias, restriction to postmenopausal women, and inclusion of only primary invasive epithelial ovarian cancer cases. The combination of MS profiling information with CA125 did not significantly improve the specificity/accuracy compared with classifications on the basis of CA125 alone. Conclusions: We report unexpectedly good performance of serum CA125 using threshold classification in discriminating healthy controls and women with benign masses from those with invasive ovarian cancer. This highlights the dependence of diagnostic tests on the characteristics of the study population and the crucial need for authors to provide sufficient relevant details to allow comparison. Our study also shows that MS profiling information adds little to diagnostic accuracy. This finding is in contrast with other reports and shows the limitations of serum MS profiling for biomarker discovery and as a diagnostic tool

The influence of dietary fatty acid composition on N-ethyl-Nnitrosourea-induced mammary tumour incidence in the rat and on the composition of inositol- and ethanolamine-phospholipids of normal and tumour mammary tissue

This study has investigated the influence of dietary fatty acid composition on mammary tumour incidence in N-ethyl-N-nitrosourea (ENU)-treated rats and has compared the susceptibility to dietary fatty acid modification of the membrane phospholipids phosphatidyliuositol (PI) and phosphatidylethanolamine (PE) from normal and tumour tissue of rat mammary gland. The incidence of mammary tumours was significantly lower in fish oil- (29%), compared with olive oil- (75%; P < 0.04) but not maize oil- (63%; P < 0.1) fed animals. No differences in PI fatty acid composition were found in normal or tumour tissue between rats fed on maize oil, olive oil or fish oil in diets from weaning. When normal and tumour tissue PI fatty acids were compared, significantly higher amounts of stearic acid (18:O) were found in tumour than normal tissue in rats given olive oil (P < 0.05). A similar trend was found in animals fed on maize oil, although differences between normal and tumour tissue did not reach a level of statistical significance (P < 0.1). In mammary PE, maize oil-fed control animals had significantly higher levels of linoleic acid (18:2n-6) than either olive oil- or fish oil-fed animals (P < 0.05, both cases) and levels of arachidonic acid were also higher in maize oil- compared with fish oil-fed animals (P < 0.05). In tumourbearing animals no differences in PE fatty acid composition were found between the three dietary groups. When normal and tumour tissue PE fatty acids were compared, significantly lower amounts of liuoleic acid (18:2n-6; P < 0.01) and significantly greater amounts of arachidonic acid (20:4n-6; P < 0.05) were found in tumour than normal tissue of rats fed on maize oil. The present study shows that the fatty acid composition of PI from both normal and tumour tissue of the mammary gland is resistant to dietary fatty acid modification. The PE fraction is more susceptible to dietary modification and in this fraction there is evidence of increased conversion of linoleic acid to arachidonic acid in tumour compared with normal tissue. Lower tumour incidence rates in rats given fish oils may in part be due to alteration in prostanoid metabolism secondary to displacement of arachidonic acid by eicosapentaenoic acid, but PE rather than PI would appear to be the most likely locus for diet-induced alteration in prostanoid synthesis in this tissue. Effects of dietary fatty acids other than on the balance of n-6 and n-3 fatty acids, and on prostanoid metabolism, should also be considered. The significance of increased stearic acid content of PI in tumours of olive oil-fed animals and the possible influence of dietary fatty acids on the capacity for stearic acid accumulation requires further study.

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.