A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

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.

Vegetation and soils feedbacks on the response of the African monsoon response to orbital forcing in early to middle Holocene

Fossil pollen, ancient lake sediments and archaeological evidence from Africa indicate that the Sahel and Sahara regions were considerably wetter than today during the early to middle Holocene period, about 12,000 to 5,000 years ago1–4. Vegetation associated with the modern Sahara/Sahel boundary was about 5° farther north, and there were more and larger lakes between 15 and 30° N. Simulations with climate models have shown that these wetter conditions were probably caused by changes in Earth's orbital parameters that increased the amplitude of the seasonal cycle of solar radiation in the Northern Hemisphere, enhanced the land-ocean temperature contrast, and thereby strengthened the African summer monsoon5–7. However, these simulations underestimated the consequent monsoon enhancement as inferred from palaeorecords4. Here we use a climate model to show that changes in vegetation and soil may have increased the climate response to orbital forcing. We find that replacing today's orbital forcing with that of the mid-Holocene increases summer precipitation by 12% between 15 and 22° N. Replacing desert with grassland, and desert soil with more loamy soil, further enhances the summer precipitation (by 6 and 10% respectively), giving a total precipitation increase of 28%. When the simulated climate changes are applied to a biome model, vegetation becomes established north of the current Sahara/Sahel boundary, thereby shrinking the area of the Sahara by 11% owing to orbital forcing alone, and by 20% owing to the combined influence of orbital forcing and the prescribed vegetation and soil changes. The inclusion of the vegetation and soil feedbacks thus brings the model simulations and palaeovegetation observations into closer agreement.

Palaeoenvironmental evidence for the impact of the crusades on the local and regional environment of medieval (13th–16th century) northern Latvia, eastern Baltic

This paper evaluates the impact of the crusades on the landscape and environment of northern Latvia between the 13th–16th centuries (medieval Livonia). The crusades replaced tribal societies in the eastern Baltic with a religious state (Ordenstaat) run by the military orders and their allies, accompanied by significant social, cultural and economic developments. These changes have previously received little consideration in palaeoenvironmental studies of past land use in the eastern Baltic region, but are fundamental to understanding the development and expansion of a European Christian identity. Sediment cores from Lake Trikāta, located adjacent to a medieval castle and settlement, were studied using pollen, macrofossils, loss-on-ignition and magnetic susceptibility. Our results show that despite continuous agricultural land use from 500 BC, the local landscape was still densely wooded until the start of the crusades in AD 1198 when a diversified pattern of pasture, meadow and arable land use was established. Colonisation followed the crusades, although in Livonia this occurred on a much smaller scale than in the rest of the Ordenstaat; Trikāta is atypical showing significant impact following the crusades with many other palaeoenvironmental studies only revealing more limited impact from the 14th century and later. Subsequent wars and changes in political control in the post-medieval period had little apparent effect on agricultural land use.