Nystrom Methods in the RKQ Algorithm for Initial-value Problems

We incorporate explicit Nystrom methods into the RKQ algorithm for stepwise global error control in numerical solutions of initial-value problems. The initial-value problem is transformed into an explicitly second-order problem, so as to be suitable for Nystrom integration. The Nystrom methods used are fourth-order, fifth-order and 10th-order. Two examples demonstrate the effectiveness of the algorithm.Comment: This is an extension of ideas published in J. Math. Res. (open access); see refs [1] and [2

An Euler-type method for Volterra integro-differential equations

We describe an algorithm, based on Euler's method, for solving Volterra integro-differential equations. The algorithm approximates the relevant integral by means of the composite Trapezium Rule, using the discrete nodes of the independent variable as the required nodes for the integration variable. We have developed an error control device, using Richardson extrapolation, and we have achieved accuracy better than 1e-12 for all numerical examples considered.Comment: 11 page

Stability analysis of an implicit and explicit numerical method for Volterra integro-differential equations with kernel K(x,y(t),t)

We present implicit and explicit versions of a numerical algorithm for solving a Volterra integro-differential equation. These algorithms are an extension of our previous work, and cater for a kernel of general form. We use an appropriate test equation to study the stability of both algorithms,, numerically deriving stability regions. The region for the implicit method appears to be unbounded, while the explicit has a bounded region close to the origin. We perform a few calculations to demonstrate our results.Comment: 10 pages, 1 Figur

Error propagation in an explicit and an implicit numerical method for Volterra integro-differential equations

We study error propagation in both an explicit and an implicit method for solving Volterra integro-differential equations. We determine the relationship between local and global errors. We derive upper bounds for the global error, and show that the global order for both methods is expected to be first-order. A few numerical examples illustrate our results.Comment: 14p, 5 fig

Range and Domain Partitioning in Piecewise Polynomial Approximation

Abstract: Error control in piecewise polynomial interpolation of a smooth univariate function f requires that the interval of approximation be subdivided into many subintervals, on each of which an interpolating polynomial is determined. The number of such subintervals is often over- estimated through the use of a high-order derivative of f . We report on a partitioning algorithm, in which we attempt to reduce the number of subintervals required, by imposing conditions on f and its relevant higher derivative. One of these conditions facilitates a distinction between the need for absolute or relative error control. Two examples demonstrate the effectiveness of this partitioning algorithm. Key Words: Piecewise Polynomial; Range Partitioning; Domain Partitioning; Error Contro

Transformation of stimulus correlations by the retina

Redundancies and correlations in the responses of sensory neurons seem to waste neural resources but can carry cues about structured stimuli and may help the brain to correct for response errors. To assess how the retina negotiates this tradeoff, we measured simultaneous responses from populations of ganglion cells presented with natural and artificial stimuli that varied greatly in correlation structure. We found that pairwise correlations in the retinal output remained similar across stimuli with widely different spatio-temporal correlations including white noise and natural movies. Meanwhile, purely spatial correlations tended to increase correlations in the retinal response. Responding to more correlated stimuli, ganglion cells had faster temporal kernels and tended to have stronger surrounds. These properties of individual cells, along with gain changes that opposed changes in effective contrast at the ganglion cell input, largely explained the similarity of pairwise correlations across stimuli where receptive field measurements were possible.Comment: author list corrected in metadat

Causal relationships vs. emergent patterns in the global controls of fire frequency

Global controls on month-by-month fractional burnt area (2000–2005) were investigated by fitting a generalised linear model (GLM) to Global Fire Emissions Database (GFED) data, with 11 predictor variables representing vegetation, climate, land use and potential ignition sources. Burnt area is shown to increase with annual net primary production (NPP), number of dry days, maximum temperature, grazing-land area, grass/shrub cover and diurnal temperature range, and to decrease with soil moisture, cropland area and population density. Lightning showed an apparent (weak) negative influence, but this disappeared when pure seasonal-cycle effects were taken into account. The model predicts observed geographic and seasonal patterns, as well as the emergent relationships seen when burnt area is plotted against each variable separately. Unimodal relationships with mean annual temperature and precipitation, population density and gross domestic product (GDP) are reproduced too, and are thus shown to be secondary consequences of correlations between different controls (e.g. high NPP with high precipitation; low NPP with low population density and GDP). These findings have major implications for the design of global fire models, as several assumptions in current models – most notably, the widely assumed dependence of fire frequency on ignition rates – are evidently incorrect

Luminosity distributions of Type Ia Supernovae

We have assembled a dataset of 165 low redshift, $z<$0.06, publicly available type Ia supernovae (SNe Ia). We produce maximum light magnitude ($M_{B}$ and $M_{V}$) distributions of SNe Ia to explore the diversity of parameter space that they can fill. Before correction for host galaxy extinction we find that the mean $M_{B}$ and $M_{V}$ of SNe Ia are $-18.58\pm0.07$mag and $-18.72\pm0.05$mag respectively. Host galaxy extinction is corrected using a new method based on the SN spectrum. After correction, the mean values of $M_{B}$ and $M_{V}$ of SNe Ia are $-19.10\pm0.06$ and $-19.10\pm0.05$mag respectively. After correction for host galaxy extinction, `normal' SNeIa ($\Delta m_{15}(B)<1.6$mag) fill a larger parameter space in the Width-Luminosity Relation (WLR) than previously suggested, and there is evidence for luminous SNe Ia with large $\Delta m_{15}(B)$. We find a bimodal distribution in $\Delta m_{15}(B)$, with a pronounced lack of transitional events at $\Delta m_{15}(B)$=1.6 mag. We confirm that faster, low-luminosity SNe tend to come from passive galaxies. Dividing the sample by host galaxy type, SNe Ia from star-forming (S-F) galaxies have a mean $M_{B}=-19.20 \pm 0.05$ mag, while SNe Ia from passive galaxies have a mean $M_{B}=-18.57 \pm 0.24$ mag. Even excluding fast declining SNe, `normal' ($M_{B}<-18$ mag) SNe Ia from S-F and passive galaxies are distinct. In the $V$-band, there is a difference of 0.4$\pm$0.13 mag between the median ($M_{V}$) values of the `normal' SN Ia population from passive and S-F galaxies. This is consistent with ($\sim 15 \pm$10)% of `normal' SNe Ia from S-F galaxies coming from an old stellar population