An iterative method for a Cauchy problem for the heat equation

An iterative method for reconstruction of the solution to a parabolic initial boundary value problem of second order from Cauchy data is presented. The data are given on a part of the boundary. At each iteration step, a series of well-posed mixed boundary value problems are solved for the parabolic operator and its adjoint. The convergence proof of this method in a weighted L2-space is included.

Fokker-Planck equation approach to the description of soliton statistics in optical fiber transmission systems

We derive rigorously the Fokker-Planck equation that governs the statistics of soliton parameters in optical transmission lines in the presence of additive amplifier spontaneous emission. We demonstrate that these statistics are generally non-Gaussian. We present exact marginal probability-density functions for soliton parameters for some cases. A WKB approach is applied to describe the tails of the probability-density functions. © 2005 Optical Society of America.

Impacts of flood hazards on small and medium companies:strategies for property level protection and business continuity

Worldwide floods have become one of the costliest weather-related hazards, causing large-scale human, economic, and environmental damage during the recent past. Recent years have seen a large number of such flood events around the globe, with Europe and the United Kingdom being no exception. Currently, about one in six properties in England is at risk of flooding (EA, 2009), and the risk is expected to further increase in the future (Evans et al., 2004). Although public spending on community-level flood protection has increased and some properties are protected by such protection schemes, many properties at risk of flooding may still be left without adequate protection. As far as businesses are concerned, this has led to an increased need for implementing strategies for property-level flood protection and business continuity, in order to improve their capacity to survive a flood hazard. Small and medium-sized enterprises (SMEs) constitute a significant portion of the UK business community. In the United Kingdom, more than 99% of private sector enterprises fall within the category of SMEs (BERR, 2008). They account for more than half of employment creation (59%) and turnover generation (52%) (BERR, 2008), and are thus considered the backbone of the UK economy. However, they are often affected disproportionately by natural hazards when compared with their larger counterparts (Tierney and Dahlhamer, 1996; Webb, Tierney, and Dahlhamer, 2000; Alesch et al., 2001) due to their increased vulnerability. Previous research reveals that small businesses are not adequately prepared to cope with the risk of natural hazards and to recover following such events (Tierney and Dahlhamer, 1996; Alesch et al., 2001; Yoshida and Deyle, 2005; Crichton, 2006; Dlugolecki, 2008). For instance, 90% of small businesses do not have adequate insurance coverage for their property (AXA Insurance UK, 2008) and only about 30% have a business continuity plan (Woodman, 2008). Not being adequately protected by community-level flood protection measures as well as property- and business-level protection measures threatens the survival of SMEs, especially those located in flood risk areas. This chapter discusses the potential effects of flood hazards on SMEs and the coping strategies that the SMEs can undertake to ensure the continuity of their business activities amid flood events. It contextualizes this discussion within a survey conducted under the Engineering and Physical Sciences Research Council (EPSRC) funded research project entitled “Community Resilience to Extreme Weather — CREW”.

The verbal transformation effect and the perceptual organization of speech:influence of formant transitions and F0-contour continuity

This study explored the role of formant transitions and F0-contour continuity in binding together speech sounds into a coherent stream. Listening to a repeating recorded word produces verbal transformations to different forms; stream segregation contributes to this effect and so it can be used to measure changes in perceptual coherence. In experiment 1, monosyllables with strong formant transitions between the initial consonant and following vowel were monotonized; each monosyllable was paired with a weak-transitions counterpart. Further stimuli were derived by replacing the consonant-vowel transitions with samples from adjacent steady portions. Each stimulus was concatenated into a 3-min-long sequence. Listeners only reported more forms in the transitions-removed condition for strong-transitions words, for which formant-frequency discontinuities were substantial. In experiment 2, the F0 contour of all-voiced monosyllables was shaped to follow a rising or falling pattern, spanning one octave. Consecutive tokens either had the same contour, giving an abrupt F0 change between each token, or alternated, giving a continuous contour. Discontinuous sequences caused more transformations and forms, and shorter times to the first transformation. Overall, these findings support the notion that continuity cues provided by formant transitions and the F0 contour play an important role in maintaining the perceptual coherence of speech.

Travelling waves in a complex Ginzburg-Landau equation with time-delay feedback

One of the simplest ways to create nonlinear oscillations is the Hopf bifurcation. The spatiotemporal dynamics observed in an extended medium with diffusion (e.g., a chemical reaction) undergoing this bifurcation is governed by the complex Ginzburg-Landau equation, one of the best-studied generic models for pattern formation, where besides uniform oscillations, spiral waves, coherent structures and turbulence are found. The presence of time delay terms in this equation changes the pattern formation scenario, and different kind of travelling waves have been reported. In particular, we study the complex Ginzburg-Landau equation that contains local and global time-delay feedback terms. We focus our attention on plane wave solutions in this model. The first novel result is the derivation of the plane wave solution in the presence of time-delay feedback with global and local contributions. The second and more important result of this study consists of a linear stability analysis of plane waves in that model. Evaluation of the eigenvalue equation does not show stabilisation of plane waves for the parameters studied. We discuss these results and compare to results of other models.

The Birmingham relationship continuity measure:the development and evaluation of a measure of the perceived continuity of spousal relationships in dementia

Background: Qualitative research has suggested that spousal carers of someone with dementia differ in terms of whether they perceive their relationship with that person as continuous with the premorbid relationship or as radically different, and that a perception of continuity may be associated with more person-centered care and the experience of fewer of the negative emotions associated with caring. The aim of the study was to develop and evaluate a quantitative measure of the extent to which spousal carers perceive the relationship to be continuous. Methods: An initial pool of 42 questionnaire items was generated on the basis of the qualitative research about relationship continuity. These were completed by 51 spousal carers and item analysis was used to reduce the pool to 23 items. The retained items, comprising five subscales, were then administered to a second sample of 84 spousal carers, and the questionnaire's reliability, discriminative power, and validity were evaluated. Results: The questionnaire showed good reliability: Cronbach's α for the full scale was 0.947, and test-retest reliability was 0.932. Ferguson's δ was 0.987, indicating good discriminative power. Evidence of construct validity was provided by predicted patterns of subscale correlations with the Closeness and Conflict Scale and the Marwit-Meuser Caregiver Grief Inventory. Conclusion: Initial psychometric evaluation of the measure was encouraging. The measure provides a quantitative means of investigating ideas from qualitative research about the role of relationship continuity in influencing how spousal carers provide care and how they react emotionally to their caring role. © 2012 International Psychogeriatric Association.

Free energy functional expansion as the generalized approach to the equation of state of dense fluids

A version of the thermodynamic perturbation theory based on a scaling transformation of the partition function has been applied to the statistical derivation of the equation of state in a highpressure region. Two modifications of the equations of state have been obtained on the basis of the free energy functional perturbation series. The comparative analysis of the experimental PV T- data on the isothermal compression for the supercritical fluids of inert gases has been carried out. © 2012.

Conditional probability calculations for the nonlinear Schrödinger equation with additive noise

The method for the computation of the conditional probability density function for the nonlinear Schrödinger equation with additive noise is developed. We present in a constructive form the conditional probability density function in the limit of small noise and analytically derive it in a weakly nonlinear case. The general theory results are illustrated using fiber-optic communications as a particular, albeit practically very important, example.

Green's function for paraxial equation

The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Eating behaviours of preadolescent children over time:stability, continuity and the moderating role of perceived parental feeding practices

The links between childhood eating behaviours and parental feeding practices are well-established in younger children, but there is a lack of research examining these variables in a preadolescent age group, particularly from the child's perspective, and longitudinally. This study firstly aimed to examine the continuity and stability of preadolescent perceptions of their parents' controlling feeding practices (pressure to eat and restriction) over a 12 month period. The second aim was to explore if perceptions of parental feeding practices moderated the relationship between preadolescents' eating behaviours longitudinally. Two hundred and twenty nine preadolescents (mean age at recruitment 8.73 years) completed questionnaires assessing their eating behaviours and their perceptions of parental feeding practices at two time points, 12 months apart (T1 and T2). Preadolescents' perceptions of their parental feeding practices remained stable. Perceptions of restriction and pressure to eat were continuous. Perceptions of parental pressure to eat and restriction significantly moderated the relationships between eating behaviours at T1 and T2. The findings from this study suggest that in a preadolescent population, perceptions of parental pressure to eat and restriction of food may exacerbate the development of problematic eating behaviours.

Inverse space-dependent force problems for the wave equation

The determination of the displacement and the space-dependent force acting on a vibrating structure from measured final or time-average displacement observation is thoroughly investigated. Several aspects related to the existence and uniqueness of a solution of the linear but ill-posed inverse force problems are highlighted. After that, in order to capture the solution a variational formulation is proposed and the gradient of the least-squares functional that is minimized is rigorously and explicitly derived. Numerical results obtained using the Landweber method and the conjugate gradient method are presented and discussed illustrating the convergence of the iterative procedures for exact input data. Furthermore, for noisy data the semi-convergence phenomenon appears, as expected, and stability is restored by stopping the iterations according to the discrepancy principle criterion once the residual becomes close to the amount of noise. The present investigation will be significant to researchers concerned with wave propagation and control of vibrating structures.