Fast numerical algorithms for the computation of invariant tori in Hamiltonian systems

In this paper, we develop numerical algorithms that use small requirements of storage and operations for the computation of invariant tori in Hamiltonian systems (exact symplectic maps and Hamiltonian vector fields). The algorithms are based on the parameterization method and follow closely the proof of the KAM theorem given in [LGJV05] and [FLS07]. They essentially consist in solving a functional equation satisfied by the invariant tori by using a Newton method. Using some geometric identities, it is possible to perform a Newton step using little storage and few operations. In this paper we focus on the numerical issues of the algorithms (speed, storage and stability) and we refer to the mentioned papers for the rigorous results. We show how to compute efficiently both maximal invariant tori and whiskered tori, together with the associated invariant stable and unstable manifolds of whiskered tori. Moreover, we present fast algorithms for the iteration of the quasi-periodic cocycles and the computation of the invariant bundles, which is a preliminary step for the computation of invariant whiskered tori. Since quasi-periodic cocycles appear in other contexts, this section may be of independent interest. The numerical methods presented here allow to compute in a unified way primary and secondary invariant KAM tori. Secondary tori are invariant tori which can be contracted to a periodic orbit. We present some preliminary results that ensure that the methods are indeed implementable and fast. We postpone to a future paper optimized implementations and results on the breakdown of invariant tori.

Agronomic and commercial performance of four Spanish carob cultivars

The carob tree (Ceratonia siliqua) shows interesting prospects for some coastal Mediterranean growing areas and is widely used for industrial, agricultural, and ornamental purposes. It can be an alternative crop adapted to part-time farming and can also be used to regenerate vegetation in areas with a mild climate and erosion problems. Four Spanish carob cultivars were examined (Banya de Cabra, Duraio, Matalafera, and Rojal) to determine the one that performed the best for planting new orchards in northeastern Spain (Catalonia). The trees in this rain-fed trial (average rainfall of 500 mm) were planted in 1986 using seedling rootstocks that were budded in 1987. The trees were trained using the free-vase system and were spaced 8 x 9 m (138 trees/ha including 12% pollinators). The results showed that ‘Rojal’ was the earliest bearing cultivar. However, no significant differences were observed for cumulative pod production 18 years after budding. With respect to cumulative seed yield, ‘Duraio’ had the highest production (95 kg/tree). The lowest tree vigor (trunk cross-section) was observed in ‘Matalafera’. ‘Rojal’ trees produced the largest pods (average fruit weight of 18.9 g) and lowest seed content (11.8%), while ‘Banya de Cabra’ and ‘Duraio’ produced the smallest fruit (weighing 15.3 and 16.2 g, respectively) with the highest seed content (15.2% and 17.3%, respectively). Gum content, expressed as a percentage of the dry weight, was highest in ‘Duraio’ (56.9%) and was lowest in ‘Rojal’ (54.1%). Thus, in terms of kernel and pod production, ‘Duraio’ appeared to be the best-performing female cultivar for planting new carob orchards

Impacte visual aeri a Argentona

Treball de recerca realitzat per un alumne d'ensenyament secundari i guardonat amb un Premi CIRIT per fomentar l'esperit cientí­fic del Jovent l'any 2009. Aquest treball té com a finalitat millorar l'entorn aeri d'Argentona, un poble situat a la comarca del Maresme. Els elements que creen més impacte visual aeri són les antenes en desús i el cablejat no soterrat. S'han buscat propostes per canviar aquesta situació: la retirada de les antenes, aprofitant l'arribada de la TDT, i el soterrament del cablejat aeri. Donat que ambdues accions afectarien al municipi i a la seva població, s'ha considerat necessari incloure una enquesta, per a conèixer l'opinió dels argentonins, i dues entrevistes, a l'alcalde i a l'enginyer municipal, per conèixer la postura oficial.