On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

Wave activity for large-amplitude disturbances described by the primitive equations on the sphere

Pseudomomentum and pseudoenergy are both measures of wave activity for disturbances in a fluid, relative to a notional background state. Together they give information on the propagation, growth, and decay of disturbances. Wave activity conservation laws are most readily derived for the primitive equations on the sphere by using isentropic coordinates. However, the intersection of isentropic surfaces with the ground (and associated potential temperature anomalies) is a crucial aspect of baroclinic wave evolution. A new expression is derived for pseudoenergy that is valid for large-amplitude disturbances spanning isentropic layers that may intersect the ground. The pseudoenergy of small-amplitude disturbances is also obtained by linearizing about a zonally symmetric background state. The new expression generalizes previous pseudoenergy results for quasigeostrophic disturbances on the β plane and complements existing large-amplitude results for pseudomomentum. The pseudomomentum and pseudoenergy diagnostics are applied to an extended winter from the European Centre for Medium-Range Weather Forecasts Interim Re-Analysis data. The time series identify distinct phenomena such as a baroclinic wave life cycle where the wave activity in boundary potential temperature saturates nonlinearly almost two days before the peak in wave activity near the tropopause. The coherent zonal propagation speed of disturbances at tropopause level, including distinct eastward, westward, and stationary phases, is shown to be dictated by the ratio of total hemispheric pseudoenergy to pseudomomentum. Variations in the lower-boundary contribution to pseudoenergy dominate changes in propagation speed; phases of westward progression are associated with stronger boundary potential temperature perturbations.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

A semi-distributed ntegrated itrogen model for multiple source assessment in tchments (INCA): Part I — model structure and process equations

A new model has been developed for assessing multiple sources of nitrogen in catchments. The model (INCA) is process based and uses reaction kinetic equations to simulate the principal mechanisms operating. The model allows for plant uptake, surface and sub-surface pathways and can simulate up to six land uses simultaneously. The model can be applied to catchment as a semi-distributed simulation and has an inbuilt multi-reach structure for river systems. Sources of nitrogen can be from atmospheric deposition, from the terrestrial environment (e.g. agriculture, leakage from forest systems etc.), from urban areas or from direct discharges via sewage or intensive farm units. The model is a daily simulation model and can provide information in the form of time series at key sites, or as profiles down river systems or as statistical distributions. The process model is described and in a companion paper the model is applied to the River Tywi catchment in South Wales and the Great Ouse in Bedfordshire.

Elevated atmospheric CO2 and humidity delay leaf fall in Betula pendula, but not in Alnus glutinosa or Populus tremula × tremuloides

Context: Anthropogenic activity has increased the level of atmospheric CO2, which is driving an increase of global temperatures and associated changes in precipitation patterns. At Northern latitudes, one of the likely consequences of global warming is increased precipitation and air humidity. Aims: In this work, the effects of both elevated atmospheric CO2 and increased air humidity on trees commonly growing in northern European forests were assessed. Methods: The work was carried out under field conditions by using Free Air Carbon dioxide Enrichment (FACE) and Free Air Humidity Manipulation (FAHM) systems. Leaf litter fall was measured over 4 years (FACE) or 5 years (FAHM) to determine the effects of FACE and FAHM on leaf phenology. Results: Increasing air humidity delayed leaf litter fall in Betula pendula, but not in Populus tremula × tremuloides. Similarly, under elevated atmospheric CO2, leaf litter fall was delayed in Betula pendula, but not in Alnus glutinosa. Increased CO2 appeared to interact with periods of low precipitation in summer and high ozone levels during these periods to effect leaf fall. Conclusions: This work shows that increased CO2 and humidity delay leaf fall, but this effect is species specific.

Development and validation of a parent report measure for detection of cognitive delay in infancy

Aim To develop a brief, parent-completed instrument (‘ERIC’) for detection of cognitive delay in 10-24 month-olds born preterm, or with low birth weight, or with perinatal complications, and to establish its diagnostic properties. Method Scores were collected from parents of 317 children meeting ≥1 inclusion criteria (birth weight <1500g; gestational age <34 completed weeks; 5-minute Apgar <7; presence of hypoxic-ischemic encephalopathy) and meeting no exclusion criteria. Children were assessed for cognitive delay using a criterion score on the Bayley Scales of Infant and Toddler Development Cognitive Scale III1 <80. Items were retained according to their individual associations with delay. Sensitivity, specificity, Positive and Negative Predictive Values were estimated and a truncated ERIC was developed for use <14 months. Results ERIC detected 17 out of 18 delayed children in the sample, with 94.4% sensitivity (95% CI [confidence interval] 83.9-100%), 76.9% specificity (72.1-81.7%), 19.8% positive predictive value (11.4-28.2%); 99.6% negative predictive value (98.7-100%); 4.09 likelihood ratio positive; and 0.07 likelihood ratio negative; the associated Area under the Curve was .909 (.829-.960). Interpretation ERIC has potential value as a quickly-administered diagnostic instrument for the absence of early cognitive delay in preterm or premature infants of 10-24 months, and as a screen for cognitive delay. Further research may be needed before ERIC can be recommended for wide-scale use.

Curl free pressure gradients over orography in a solution of the fully compressible Euler equations with implicit treatment of acoustic and gravity waves

Steep orography can cause noisy solutions and instability in models of the atmosphere. A new technique for modelling flow over orography is introduced which guarantees curl free gradients on arbitrary grids, implying that the pressure gradient term is not a spurious source of vorticity. This mimetic property leads to better hydrostatic balance and better energy conservation on test cases using terrain following grids. Curl-free gradients are achieved by using the co-variant components of velocity over orography rather than the usual horizontal and vertical components. In addition, gravity and acoustic waves are treated implicitly without the need for mean and perturbation variables or a hydrostatic reference profile. This enables a straightforward description of the implicit treatment of gravity waves. Results are presented of a resting atmosphere over orography and the curl-free pressure gradient formulation is advantageous. Results of gravity waves over orography are insensitive to the placement of terrain-following layers. The model with implicit gravity waves is stable in strongly stratified conditions, with N∆t up to at least 10 (where N is the Brunt-V ̈ais ̈al ̈a frequency). A warm bubble rising over orography is simulated and the curl free pressure gradient formulation gives much more accurate results for this test case than a model without this mimetic property.

Simple M-factor algorithm for improved estimation of the basic maximum usable frequency of radio waves reflected from the ionospheric F-region

The equations of Milsom are evaluated, giving the ground range and group delay of radio waves propagated via the horizontally stratified model ionosphere proposed by Bradley and Dudeney. Expressions for the ground range which allow for the effects of the underlying E- and F1-regions are used to evaluate the basic maximum usable frequency or M-factors for single F-layer hops. An algorithm for the rapid calculation of the M-factor at a given range is developed, and shown to be accurate to within 5%. The results reveal that the M(3000)F2-factor scaled from vertical-incidence ionograms using the standard URSI procedure can be up to 7.5% in error. A simple addition to the algorithm effects a correction to ionogram values to make these accurate to 0.5%.

Design of delay-tolerant space–time codes with limited feedback

In cooperative communication networks, owing to the nodes' arbitrary geographical locations and individual oscillators, the system is fundamentally asynchronous. Such a timing mismatch may cause rank deficiency of the conventional space-time codes and, thus, performance degradation. One efficient way to overcome such an issue is the delay-tolerant space-time codes (DT-STCs). The existing DT-STCs are designed assuming that the transmitter has no knowledge about the channels. In this paper, we show how the performance of DT-STCs can be improved by utilizing some feedback information. A general framework for designing DT-STC with limited feedback is first proposed, allowing for flexible system parameters such as the number of transmit/receive antennas, modulated symbols, and the length of codewords. Then, a new design method is proposed by combining Lloyd's algorithm and the stochastic gradient-descent algorithm to obtain optimal codebook of STCs, particularly for systems with linear minimum-mean-square-error receiver. Finally, simulation results confirm the performance of the newly designed DT-STCs with limited feedback.

Accurate delay analysis in prioritised wireless sensor networks for generalized packet arrival

This letter presents an accurate delay analysis in prioritised wireless sensor networks (WSN). The analysis is an enhancement of the existing analysis proposed by Choobkar and Dilmaghani, which is only applicable to the case where the lower priority nodes always have packets to send in the empty slots of the higher priority node. The proposed analysis is applicable for any pattern of packet arrival, which includes the general case where the lower priority nodes may or may not have packets to send in the empty slots of the higher priority nodes. Evaluation of both analyses showed that the proposed delay analysis has better accuracy over the full range of loads and provides an excellent match to simulation results.

Anticipatory engineering: anticipation in sensory-motor systems of human

In visual tracking experiments, distributions of the relative phase be-tween target and tracer showed positive relative phase indicating that the tracer precedes the target position. We found a mode transition from the reactive to anticipatory mode. The proposed integrated model provides a framework to understand the antici-patory behaviour of human, focusing on the integration of visual and soma-tosensory information. The time delays in visual processing and somatosensory feedback are explicitly treated in the simultaneous differential equations. The anticipatory behaviour observed in the visual tracking experiments can be ex-plained by the feedforward term of target velocity, internal dynamics, and time delay in somatosensory feedback.