Approximate solution of u'=-A2u using a relation between exp(-tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u(t) = -A(2)u. Using a representation of the semigroup exp(-A(2)t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w(t) = W-yy, with initial values v solving the initial value problem for v(y) = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2(nd) order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4(th) order equation for u to that of the 2(nd) order equation for v, followed by the solution of the heat equation in one space variable.

El ni??o exp??sito en el Despotismo Ilustrado : su crianza y su educaci??n

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Age analysis, aragonite and high magnesium calcite content from different IODP Holes from Exp 310

The widespread occurrence of microbialites in the last deglacial reef frameworks (16-6 Ka BP) implies that the accurate study of their development patterns is of prime importance to unravel the evolution of reef architecture through time and to reconstruct the reef response to sea-level variations and environmental changes. The present study is based on the sedimentological and chronological analysis (14C AMS dating) of drill cores obtained during the IODP Expedition #310 "Tahiti Sea Level" on the successive terraces which typify the modern reef slopes from Tahiti. It provides a comprehensive data base to investigate the microbialite growth patterns (i.e. growth rates and habitats), to analyze their roles in reef frameworks and to reconstruct the evolution of the reef framework architecture during sea-level rise. The last deglacial reefs from Tahiti are composed of two distinctive biological communities: (1) the coralgal communities including seven assemblages characterized by various growth forms (branching, robust branching, massive, tabular and encrusting) that form the initial frameworks and (2) the microbial communities developed in the primary cavities of those frameworks, a few meters (1.5 to 6 m) below the living coral reef surface, where they heavily encrusted the coralgal assemblages to form microbialite crusts. The dating results demonstrate the occurrence of two distinctive generations of microbialites: the "reefal microbialites" which developed a few hundred years after coralgal communities in shallow-water environments, whereas the "slope microbialites" grew a few thousands of years later in significantly deeper water conditions after the demise of coralgal communities. The development of microbialites was controlled by the volume and the shape of the primary cavities of the initial reef frameworks determined by the morphology and the packing of coral colonies. The most widespread microbialite development occurred in frameworks dominated by branching, thin encrusting, tabular and robust branching coral colonies which built loose and open frameworks typified by a high porosity (> 50%). In contrast, their growth was minimal in compact coral frameworks formed by massive and thick encrusting corals where primary cavities yielded a low porosity (~ 30%) and could not host a significant microbialite expansion.