Croissance et ensemble nodal de fonctions propres du laplacien sur des surfaces

Dans cette thèse, nous étudions les fonctions propres de l'opérateur de Laplace-Beltrami - ou simplement laplacien - sur une surface fermée, c'est-à-dire une variété riemannienne lisse, compacte et sans bord de dimension 2. Ces fonctions propres satisfont l'équation $\Delta_g \phi_\lambda + \lambda \phi_\lambda = 0$ et les valeurs propres forment une suite infinie. L'ensemble nodal d'une fonction propre du laplacien est celui de ses zéros et est d'intérêt depuis les expériences de plaques vibrantes de Chladni qui remontent au début du 19ème siècle et, plus récemment, dans le contexte de la mécanique quantique. La taille de cet ensemble nodal a été largement étudiée ces dernières années, notamment par Donnelly et Fefferman, Colding et Minicozzi, Hezari et Sogge, Mangoubi ainsi que Sogge et Zelditch. L'étude de la croissance de fonctions propres n'est pas en reste, avec entre autres les récents travaux de Donnelly et Fefferman, Sogge, Toth et Zelditch, pour ne nommer que ceux-là. Notre thèse s'inscrit dans la foulée du travail de Nazarov, Polterovich et Sodin et relie les propriétés de croissance des fonctions propres avec la taille de leur ensemble nodal dans l'asymptotique $\lambda \nearrow \infty$. Pour ce faire, nous considérons d'abord les exposants de croissance, qui mesurent la croissance locale de fonctions propres et qui sont obtenus à partir de la norme uniforme de celles-ci. Nous construisons ensuite la croissance locale moyenne d'une fonction propre en calculant la moyenne sur toute la surface de ces exposants de croissance, définis sur de petits disques de rayon comparable à la longueur d'onde. Nous montrons alors que la taille de l'ensemble nodal est contrôlée par le produit de cette croissance locale moyenne et de la fréquence $\sqrt{\lambda}$. Ce résultat permet une reformulation centrée sur les fonctions propres de la célèbre conjecture de Yau, qui prévoit que la mesure de l'ensemble nodal croît au rythme de la fréquence. Notre travail renforce également l'intuition répandue selon laquelle une fonction propre se comporte comme un polynôme de degré $\sqrt{\lambda}$. Nous généralisons ensuite nos résultats pour des exposants de croissance construits à partir de normes $L^q$. Nous sommes également amenés à étudier les fonctions appartenant au noyau d'opérateurs de Schrödinger avec petit potentiel dans le plan. Pour de telles fonctions, nous obtenons deux résultats qui relient croissance et taille de l'ensemble nodal.

Cosmo and the magic sneeze.

La historia de un gatito blanco y negro llamado Cosmo y sus padres, India y Mephisto. Mephisto es un gato de brujas, un gato que puede utilizar su magia para ayudar a estornudar a una bruja en sus hechizos, mientras que India es un gato doméstico normal. Cosmo tiene que pasar una prueba especial para ver si es más como Mephisto o India. Cuando pasa la prueba especial, es capaz de ayudar a estornudar a la bruja Sybil y llevar a cabo la magia estornudo que añade el ingrediente extra-especial al caldero. Sybil y su madre, Eufemia, están haciendo una poción secreta para un hechizo que produce estatuas de oro y Scarlett, Bunty y Cosmo tienen que averiguar cuáles son los componentes de la poción, antes de que sea demasiado tarde. Sobre todo porque las estatuas tienen forma de gatitos.

From the assimilation to the acceptance of the other? : educational policy for indigenous peoples in Latin America and the 'intercultural bilingual education' policy in Chile. '¿De la asimilación a la aceptación del otro? : política educativa para pueblos indígenas en América Latina y la política de 'educación intercultural bilingüe' en Chile'.

Monográfico con el título: 'Desde la diversidad hacia la desigualdad: ¿destino inexorable de la globalización?'. Resumen basado en el de la publicación

As Relações entre Filhos/as de Santo e Espíritos no Cosmo Religioso Umbandista : uma abordagem a partir do contexto de Viçosa/AL

Contundentemente, espíritos do cosmo religioso umbandista estão e interferem na vida dos filhos/as de santo ou como proclama a religiosidade umbandista, nos seus ‘cavalos’. Este trabalho pesquisa oito terreiros de Umbanda inseridos no município de Viçosa/Alagoas, por meio das relações existentes entre filhos/as de santo e espíritos neste cosmo religioso ímpar e multifacetado. Para o desenvolvimento desta pesquisa, fez-se uso de um aparato metodológico que possibilitou compreender fenomenologicamente este processo análogo circunscrito em discursos sobre ‘marcas’ ou ‘sofrimentos’ resultantes das atitudes negligenciais dos filhos/as de santo que constituem a memória umbandista viçosense. Pretendeu-se mostrar também que o imbricamento dos aspectos relacionais como: reverência, temor, devoção, respeito, esperança, fidelidade, dedicação e obediência às entidades cultuadas, configuram-se nessa dinâmica relacional. Este cosmo religioso traz em seu arcabouço existencial discursos e situações vivenciadas em todas as esferas da vida desses filhos/as de santo, principalmente religiosa. O presente trabalho apontou, portanto, a relação de interdependência existente como fruto dessa religiosidade marcada por fé e mistérios.

The prediction of extratropical storm tracks by the ECMWF and NCEP ensemble prediction systems

The prediction of extratropical cyclones by the European Centre for Medium Range Weather Forecasts (ECMWF) and the National Centers for Environmental Prediction (NCEP) Ensemble Prediction Systems (EPS) has been investigated using an objective feature tracking methodology to identify and track the cyclones along the forecast trajectories. Overall the results show that the ECMWF EPS has a slightly higher level of skill than the NCEP EPS in the northern hemisphere (NH). However in the southern hemisphere (SH), NCEP has higher predictive skill than ECMWF for the intensity of the cyclones. The results from both EPS indicate a higher level of predictive skill for the position of extratropical cyclones than their intensity and show that there is a larger spread in intensity than position. Further analysis shows that the predicted propagation speed of cyclones is generally too slow for the ECMWF EPS and show a slight bias for the intensity of the cyclones to be overpredicted. This is also true for the NCEP EPS in the SH. For the NCEP EPS in the NH the intensity of the cyclones is underpredicted. There is small bias in both the EPS for the cyclones to be displaced towards the poles. For each ensemble forecast of each cyclone, the predictive skill of the ensemble member that best predicts the cyclones position and intensity was computed. The results are very encouraging showing that the predictive skill of the best ensemble member is significantly higher than that of the control forecast in terms of both the position and intensity of the cyclones. The prediction of cyclones before they are identified as 850 hPa vorticity centers in the analysis cycle was also considered. It is shown that an indication of extratropical cyclones can be given by at least 1 ensemble member 7 days before they are identified in the analysis. Further analysis of the ECMWF EPS shows that the ensemble mean has a higher level of skill than the control forecast, particularly for the intensity of the cyclones, 2 from day 3 of the forecast. There is a higher level of skill in the NH than the SH and the spread in the SH is correspondingly larger. The difference between the ensemble mean and spread is very small for the position of the cyclones, but the spread of the ensemble is smaller than the ensemble mean error for the intensity of the cyclones in both hemispheres. Results also show that the ECMWF control forecast has ½ to 1 day more skill than the perturbed members, for both the position and intensity of the cyclones, throughout the forecast.

Storm tracking and ensemble prediction

The prediction of extratropical cyclones by the European Centre for Medium Range Weather Forecasts (ECMWF) and the National Centers for Environmental Prediction (NCEP) Ensemble Prediction Systems (EPS) is investigated using a storm-tracking forecast verifica-tion methodology. The cyclones are identified and tracked along the forecast trajectories so that statistics can be generated to determine the rate at which the position and intensity of the forecasted cyclones diverge from the corresponding analysed cyclones with forecast time. Overall the ECMWF EPS has a slightly higher level of performance than the NCEP EPS. However, in the southern hemisphere the NCEP EPS has a slightly higher level of skill for the intensity of the storms. The results from both EPS indicate a higher level of predictive skill for the position of extratropical cyclones than their intensity and show that there is a larger spread in intensity than position. The results also illustrate several benefits an EPS can offer over a deterministic forecast.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Data assimilation for a coastal area morphodynamic model: Morecambe Bay

This paper investigates the use of data assimilation in coastal area morphodynamic modelling using Morecambe Bay as a study site. A simple model of the bay has been enhanced with a data assimilation scheme to better predict large-scale changes in bathymetry observed in the bay over a 3-year period. The 2DH decoupled morphodynamic model developed for the work is described, as is the optimal interpolation scheme used to assimilate waterline observations into the model run. Each waterline was acquired from a SAR satellite image and is essentially a contour of the bathymetry at some level within the inter-tidal zone of the bay. For model parameters calibrated against validation observations, model performance is good, even without data assimilation. However the use of data assimilation successfully compensates for a particular failing of the model, and helps to keep the model bathymetry on track. It also improves the ability of the model to predict future bathymetry. Although the benefits of data assimilation are demonstrated using waterline observations, any observations of morphology could potentially be used. These results suggest that data assimilation should be considered for use in future coastal area morphodynamic models.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state.We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.