Phénomène de Gibbs | EPFL Graph Search

En mathématiques, lors de l'étude des séries de Fourier et des transformées de Fourier, il apparaît parfois une déformation du signal, connue sous le nom de phénomène de Gibbs. Ce phénomène est un effet de bord qui se produit à proximité d'une discontinuité, lors de l'analyse d'une fonction dérivable par morceaux. Histoire Le phénomène fut mis pour la première fois en évidence en 1848 par Henry Wilbraham, mais cette découverte ne connut guère d'écho. En 1898, Albert Michelson développa un système mécanique capable de calculer et sommer la série de Fourier d'un signal donné en entrée. Il observa alors un effet d'amplification des discontinuités, qui persistait malgré l'augmentation du nombre de coefficients calculés. Alors que Michelson soupçonnait un défaut dans la fabrication de son engin, Josiah Willard Gibbs montra que le phénomène était d'origine mathématique et se produisait dans des conditions très générales. En 1906, Maxime Bôcher donna la première interprétatio

Modeling spatially dependent functional data via regression with differential regularization

Eleonora Arnone, Fabio Nobile

We propose a method for modelling spatially dependent functional data, based on regression with differential regularization. The regularizing term enables to include problem-specific information about the spatio-temporal variation of phenomenon under study, formalized in terms of a time-dependent partial differential equation. The method is implemented using a discretization based on finite elements in space and finite differences in time. This non-tensor product basis allows to effciently handle data distributed over complex domains and where the shape of the domain influences the phenomenon behavior. Moreover, the method can comply with specific conditions at the boundary of the domain of interest. Simulation studies compare the proposed model to available techniques for spatio-temporal data. The method is also illustrated via an application to the study of blood-flow velocity field in a carotid artery affected by atherosclerosis, starting from echo-color doppler and magnetic resonance imaging data.

2019

MATHICSE Technical Report : Modelling spatially dependent functional data via regression with differential regularization

Eleonora Arnone, Fabio Nobile

We propose a method for modelling spatially dependent functional data, based on regression with differential regularization. The regularizing term enables to include problem-specific information about the spatio-temporal variation of phenomenon under study, formalized in terms of a time-dependent partial differential equation. The method is implemented using a discretization based on finite elements in space and finite differences in time. This non-tensor product basis allows to efficiently handle data distributed over complex domains and where the shape of the domain influences the phenomenon behavior. Moreover, the method can comply with specific conditions at the boundary of the domain of interest. Simulation studies compare the proposed model to available techniques for spatio-temporal data. The method is also illustrated via an application to the study of blood-flow velocity field in a carotid artery affected by atherosclerosis, starting from echo-color doppler and magnetic resonance imaging data.

MATHICSE2018

Marine and freshwater micropearls: biomineralization producing strontium-rich amorphous calcium carbonate inclusions is widespread in the genus Tetraselmis (Chlorophyta)

Michaël Bensimon, Montserrat Filella

Unicellular algae play important roles in the biogeochemical cycles of numerous elements, particularly through the biomineralization capacity of certain species (e.g., coccolithophores greatly contributing to the "organic carbon pump" of the oceans), and unidentified actors of these cycles are still being discovered. This is the case of the unicellular alga Tetraselmis cordiformis (Chlorophyta) that was recently discovered to form intracellular mineral inclusions, called micropearls, which had been previously overlooked. These intracellular inclusions of hydrated amorphous calcium carbonates (ACCs) were first described in Lake Geneva (Switzerland) and are the result of a novel biomineralization process. The genus Tetraselmis includes more than 30 species that have been widely studied since the description of the type species in 1878. The present study shows that many other Tetraselmis species share this biomineralization capacity: 10 species out of the 12 tested contained micropearls, including T. chui, T. convolutae, T. levis, T. subcordiformis, T. suecica and T. tetrathele. Our results indicate that micropearls are not randomly distributed inside the Tetraselmis cells but are located preferentially under the plasma membrane and seem to form a definite pattern, which differs among species. In Tetraselmis cells, the biomineralization process seems to systematically start with a rod-shaped nucleus and results in an enrichment of the micropearls in Sr over Ca (the Sr/Ca ratio is more than 200 times higher in the micropearls than in the surrounding water or growth medium). This concentrating capacity varies among species and may be of interest for possible bioremediation techniques regarding radioactive Sr-90 water pollution. The Tetraselmis species forming micropearls live in various habitats, indicating that this novel biomineralization process takes place in different environments (marine, brackish and freshwater) and is therefore a widespread phenomenon.

COPERNICUS GESELLSCHAFT MBH2018

Marine and freshwater micropearls: biomineralization producing strontium-rich amorphous calcium carbonate inclusions is widespread in the genus Tetraselmis (Chlorophyta)

Michaël Bensimon, Montserrat Filella

COPERNICUS GESELLSCHAFT MBH2018