A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

An improved sum-product estimate for general finite fields

An improved sum-product estimate for subsets of a finite field whose order is not prime is provided. It is shown, under certain conditions, that max{∣∣∣A+A∣∣∣,∣∣∣A⋅A∣∣∣}≫∣∣A∣∣12/11(log2∣∣A∣∣)5/11. This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev

A survey for planetary transits in the field of NGC 7789

We present results from 30 nights of observations of the open cluster NGC 7789 with the Wide Field Camera on the Isaac Newton Telescope, La Palma. From ~900 epochs, we obtained light curves and Sloan r'-i' colours for ~33000 stars, with ~2400 stars having better than 1 per cent precision. We expected to detect ~2 transiting hot Jupiter planets if 1 per cent of stars host such a companion and a typical hot Jupiter radius is ~1.2R_J. We find 24 transit candidates, 14 of which we can assign a period. We rule out the transiting planet model for 21 of these candidates using various robust arguments. For two candidates, we are unable to decide on their nature, although it seems most likely that they are eclipsing binaries as well. We have one candidate exhibiting a single eclipse, for which we derive a radius of 1.81+0.09-0.00R_J. Three candidates remain that require follow-up observations in order to determine their nature.

Achieving market segmentation from B2B sectorisation

Purpose – The creation of a target market strategy is integral to developing an effective business strategy. The concept of market segmentation is often cited as pivotal to establishing a target market strategy, yet all too often business-to-business marketers utilise little more than trade sectors or product groups as the basis for their groupings of customers, rather than customers' characteristics and buying behaviour. The purpose of this paper is to offer a solution for managers, focusing on customer purchasing behaviour, which evolves from the organisation's existing criteria used for grouping its customers. Design/methodology/approach – One of the underlying reasons managers fail to embrace best practice market segmentation is their inability to manage the transition from how target markets in an organisation are currently described to how they might look when based on customer characteristics, needs, purchasing behaviour and decision-making. Any attempt to develop market segments should reflect the inability of organisations to ignore their existing customer group classification schemes and associated customer-facing operational practices, such as distribution channels and sales force allocations. Findings – A straightforward process has been derived and applied, enabling organisations to practice market segmentation in an evolutionary manner, facilitating the transition to customer-led target market segments. This process also ensures commitment from the managers responsible for implementing the eventual segmentation scheme. This paper outlines the six stages of this process and presents an illustrative example from the agrichemicals sector, supported by other cases. Research implications – The process presented in this paper for embarking on market segmentation focuses on customer purchasing behaviour rather than business sectors or product group classifications - which is true to the concept of market segmentation - but in a manner that participating managers find non-threatening. The resulting market segments have their basis in the organisation's existing customer classification schemes and are an iteration to which most managers readily buy-in. Originality/value – Despite the size of the market segmentation literature, very few papers offer step-by-step guidance for developing customer-focused market segments in business-to-business marketing. The analytical tool for assessing customer purchasing deployed in this paper originally was created to assist in marketing planning programmes, but has since proved its worth as the foundation for creating segmentation schemes in business marketing, as described in this paper.

Efficient group communication for large-scale parallel clustering

Global communication requirements and load imbalance of some parallel data mining algorithms are the major obstacles to exploit the computational power of large-scale systems. This work investigates how non-uniform data distributions can be exploited to remove the global communication requirement and to reduce the communication cost in iterative parallel data mining algorithms. In particular, the analysis focuses on one of the most influential and popular data mining methods, the k-means algorithm for cluster analysis. The straightforward parallel formulation of the k-means algorithm requires a global reduction operation at each iteration step, which hinders its scalability. This work studies a different parallel formulation of the algorithm where the requirement of global communication can be relaxed while still providing the exact solution of the centralised k-means algorithm. The proposed approach exploits a non-uniform data distribution which can be either found in real world distributed applications or can be induced by means of multi-dimensional binary search trees. The approach can also be extended to accommodate an approximation error which allows a further reduction of the communication costs.

Complex-valued B-spline neural networks for modeling and inverting Hammerstein systems

Many communication signal processing applications involve modelling and inverting complex-valued (CV) Hammerstein systems. We develops a new CV B-spline neural network approach for efficient identification of the CV Hammerstein system and effective inversion of the estimated CV Hammerstein model. Specifically, the CV nonlinear static function in the Hammerstein system is represented using the tensor product from two univariate B-spline neural networks. An efficient alternating least squares estimation method is adopted for identifying the CV linear dynamic model’s coefficients and the CV B-spline neural network’s weights, which yields the closed-form solutions for both the linear dynamic model’s coefficients and the B-spline neural network’s weights, and this estimation process is guaranteed to converge very fast to a unique minimum solution. Furthermore, an accurate inversion of the CV Hammerstein system can readily be obtained using the estimated model. In particular, the inversion of the CV nonlinear static function in the Hammerstein system can be calculated effectively using a Gaussian-Newton algorithm, which naturally incorporates the efficient De Boor algorithm with both the B-spline curve and first order derivative recursions. The effectiveness of our approach is demonstrated using the application to equalisation of Hammerstein channels.

Ecophysiology of seed dormancy and the control of germination in early spring-flowering Galanthus nivalis and Narcissus pseudonarcissus (Amaryllidaceae)

Seed dormancy induction and alleviation in the winter-flowering moist temperate woodland species Galanthus nivalis and Narcissus pseudonarcissus are complex and poorly understood. Temperature, light and desiccation were investigated to elucidate their role in the germination ecophysiology of these species. Outdoor and laboratory experiments simulating different seasonal temperatures, seasonal durations, and temperature fluctuations; the presence of light during different seasons; and intermittent drying (during the summer period) over several ‘years’ investigated the importance of these factors in germination. Warm summer-like temperatures (20°C) were necessary for germination at subsequent cooler autumn-like temperatures (greatest at 15°C in G. nivalis and 10°C in N. pseudonarcissus). As the warm temperature duration increased so did germination at subsequent cooler temperatures; further germination occurred in subsequent ‘years’ at cooler temperatures following a second, and also third, warm period. Germination was significantly greater in darkness, particularly in G. nivalis. Dormancy increased with seed maturation period in G. nivalis, because seeds extracted from green capsules germinated more readily than those from yellow. Desiccation increased dormancy in an increasing proportion of N. pseudonarcissus seeds the later they were dried in ‘summer’. Seed viability was only slightly reduced by desiccation in N. pseudonarcissus but was poor and variable in G. nivalis. Shoot formation occurred both at the temperature at which germination was greatest and also if 5°C cooler. In summary, continuous hydration of seeds of both species during warm summer-like temperatures results in the gradual release of seed dormancy; thereafter, darkness and cooler temperatures promote germination. Cold temperatures, increased seed maturity (G. nivalis), and desiccation (N. pseudonarcissus) increase dormancy while light inhibits germination.

Analysis of incoherent scatter radar data from non-thermal F-region plasma

A procedure is presented for fitting incoherent scatter radar data from non-thermal F-region ionospheric plasma, using theoretical spectra previously predicted. It is found that values of the shape distortion factor D∗, associated with deviations of the ion velocity distribution from a Maxwellian distribution, and ion temperatures can be deduced (the results being independent of the path of iteration) if the angle between the line-of-sight and the geomagnetic field is larger than about 15–20°. The procedure can be used with one or both of two sets of assumptions. These concern the validity of the adopted model for the line-of-sight ion velocity distribution in the one case or for the full three-dimensional ion velocity distribution function in the other. The distribution function employed was developed to describe the line-of-sight velocity distribution for large aspect angles, but both experimental data and Monte Carlo simulations indicate that the form of the field-perpendicular distribution can also describe the distribution at more general aspect angles. The assumption of this form for the line-of-sight velocity distribution at a general aspect angle enables rigorous derivation of values of the one-dimensional, line-of-sight ion temperature. With some additional assumptions (principally that the field-parallel distribution is always Maxwellian and there is a simple relationship between the ion temperature anisotropy and the distortion of the field-perpendicular distribution from a Maxwellian), fits to data for large aspect angles enable determination of line-of-sight temperatures at all aspect angles and hence, of the average ion temperature and the ion temperature anisotropy. For small aspect angles, the analysis is restricted to the determination of the line-of-sight ion temperature because the theoretical spectrum is insensitive to non-thermal effects when the plasma is viewed along directions almost parallel to the magnetic field. This limitation is expected to apply to any realistic model of the ion velocity distribution function and its consequences are discussed. Fit strategies which allow for mixed ion composition are also considered. Examples of fits to data from various EISCAT observing programmes are presented.