Promotion carryover as a missing data problem

An important feature of agribusiness promotion programs is their lagged impact on consumption. Efficient investment in advertising requires reliable estimates of these lagged responses and it is desirable from both applied and theoretical standpoints to have a flexible method for estimating them. This note derives an alternative Bayesian methodology for estimating lagged responses when investments occur intermittently within a time series. The method exploits a latent-variable extension of the natural-conjugate, normal-linear model, Gibbs sampling and data augmentation. It is applied to a monthly time series on Turkish pasta consumption (1993:5-1998:3) and three, nonconsecutive promotion campaigns (1996:3, 1997:3, 1997:10). The results suggest that responses were greatest to the second campaign, which allocated its entire budget to television media; that its impact peaked in the sixth month following expenditure; and that the rate of return (measured in metric tons additional consumption per thousand dollars expended) was around a factor of 20.

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.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere

The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjørtoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered.

The (il)logical problem of heritage speaker bilingualism and incomplete acquisition

This Forum challenges and problematizes the term incomplete acquisition, which has been widely used to describe the state of competence of heritage speaker (HS) bilinguals for well over a decade (see, e.g., Montrul, 2008). It is suggested and defended that HS competence, while often different from monolingual peers, is in fact not incomplete (given any reasonable definition by the word incomplete), but simply distinct for reasons related to the realities of their environment.

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

The 'child of Babylon' and the problem of paternity in Medieval French Alexander romances

A study of the use of hybrid physical appearance both to signal and to explore the disputed paternity of Alexander the Great throughout its vernacular French tradition. The article compares the 'child of Babylon' portent and Alexander's son Alior in the twelfth-century French "Roman d'Alexandre" poem cycle, and a fifteenth-century prose adaptation of it.

Problem-based learning approaches in meteorology

Problem-Based Learning, despite recent controversies about its effectiveness, is used extensively as a teaching method throughout higher education. In meteorology, there has been little attempt to incorporate Problem-Based Learning techniques into the curriculum. Motivated by a desire to enhance the reflective engagement of students within a current field course module, this project describes the implementation of two test Problem-Based Learning activities and testing and improvement using several different and complementary means of evaluation. By the end of a 2-year program of design, implementation, testing, and reflection and re-evaluation, two robust, engaging activities have been developed that provide an enhanced and diverse learning environment in the field course. The results suggest that Problem-Based Learning techniques would be a useful addition to the meteorology curriculum and suggestions for courses and activities that may benefit from this approach are included in the conclusions.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

The dependent variable problem in quantitative studies of Active Labour Market Programmes: uncovering hidden dynamics?

The question of what explains variation in expenditures on Active Labour Market Programs (ALMPs) has attracted significant scholarship in recent years. Significant insights have been gained with respect to the role of employers, unions and dual labour markets, openness, and partisanship. However, there remain significant disagreements with respects to key explanatory variables such the role of unions or the impact of partisanship. Qualitative studies have shown that there are both good conceptual reasons as well as historical evidence that different ALMPs are driven by different dynamics. There is little reason to believe that vastly different programs such as training and employment subsidies are driven by similar structural, interest group or indeed partisan dynamics. The question is therefore whether different ALMPs have the same correlation with different key explanatory variables identified in the literature? Using regression analysis, this paper shows that the explanatory variables identified by the literature have different relation to distinct ALMPs. This refinement adds significant analytical value and shows that disagreements are at least partly due to a dependent variable problem of ‘over-aggregation’.

Antarctic ocean and sea ice response to ozone depletion: a two timescale problem

The response of the Southern Ocean to a repeating seasonal cycle of ozone loss is studied in two coupled climate models and found to comprise both fast and slow processes. The fast response is similar to the inter-annual signature of the Southern Annular Mode (SAM) on Sea Surface Temperature (SST), on to which the ozone-hole forcing projects in the summer. It comprises enhanced northward Ekman drift inducing negative summertime SST anomalies around Antarctica, earlier sea ice freeze-up the following winter, and northward expansion of the sea ice edge year-round. The enhanced northward Ekman drift, however, results in upwelling of warm waters from below the mixed layer in the region of seasonal sea ice. With sustained bursts of westerly winds induced by ozone-hole depletion, this warming from below eventually dominates over the cooling from anomalous Ekman drift. The resulting slow-timescale response (years to decades) leads to warming of SSTs around Antarctica and ultimately a reduction in sea-ice cover year-round. This two-timescale behavior - rapid cooling followed by slow but persistent warming - is found in the two coupled models analysed, one with an idealized geometry, the other a complex global climate model with realistic geometry. Processes that control the timescale of the transition from cooling to warming, and their uncertainties are described. Finally we discuss the implications of our results for rationalizing previous studies of the effect of the ozone-hole on SST and sea-ice extent. %Interannual variability in the Southern Annular Mode (SAM) and sea ice covary such that an increase and southward shift in the surface westerlies (a positive phase of the SAM) coincides with a cooling of Sea Surface Temperature (SST) around 70-50$^\circ$S and an expansion of the sea ice cover, as seen in observations and models alike. Yet, in modeling studies, the Southern Ocean warms and sea ice extent decreases in response to sustained, multi-decadal positive SAM-like wind anomalies driven by 20th century ozone depletion. Why does the Southern Ocean appear to have disparate responses to SAM-like variability on interannual and multidecadal timescales? Here it is demonstrated that the response of the Southern Ocean to ozone depletion has a fast and a slow response. The fast response is similar to the interannual variability signature of the SAM. It is dominated by an enhanced northward Ekman drift, which transports heat northward and causes negative SST anomalies in summertime, earlier sea ice freeze-up the following winter, and northward expansion of the sea ice edge year round. The enhanced northward Ekman drift causes a region of Ekman divergence around 70-50$^\circ$S, which results in upwelling of warmer waters from below the mixed layer. With sustained westerly wind enhancement in that latitudinal band, the warming due to the anomalous upwelling of warm waters eventually dominates over the cooling from the anomalous Ekman drift. Hence, the slow response ultimately results in a positive SST anomaly and a reduction in the sea ice cover year round. We demonstrate this behavior in two models: one with an idealized geometry and another, more detailed, global climate model. However, the models disagree on the timescale of transition from the fast (cooling) to the slow (warming) response. Processes that controls this transition and their uncertainties are discussed.