Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,\Lambda,E)$ for which 1 is an admissible function and Jacobi quotient manifolds of $M$. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.Comment: 18 page

On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms

We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providing also some examples of Courant-Jacobi algebroid morphisms.Comment: 14 pages, to appear in Journal of Geometry and Physic

Twisted Jacobi manifolds, twisted Dirac-Jacobi structures and quasi-Jacobi bialgebroids

We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and we prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant-Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted Dirac-Jacobi structures are presented

On quasi-Jacobi and Jacobi-quasi bialgebroids

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissonization

Europa, que futuro?

As Europe’s future seems suspended, Tarizzo’s text invites us to reflect upon the conditions of possibility of leading the original European project. In this paper we offer a critical account on the economic and political European dimensions, and its relationship. Our goal is to show that the challenge today in Europe is not only to reconceptualize the relationship between economy and democracy; rather, it is imperative to look for new conditions for the reinvention of politics.Num momento em que o futuro da Europa parece estar suspenso, o texto de Tarizzo convida-nos a refletir sobre as condições de possibilidade para levar a cabo o projeto europeu. Neste artigo oferecemos uma leitura crítica acerca das dimensões económica e política europeias e da relação entre elas. O nosso objectivo é mostrar que o desafio na Europa hoje não é só de repensar a relação entre economia, por um lado, e democracia, por outro, mas também buscar as condições para a reinvenção da própria política

Turbulence in the sea ice impacted Southern Ocean and its implications for primary production and carbon export

The sea-ice impacted Southern Ocean, south of the Antarctic Circumpolar Current, is one of the most important regions on earth for the cycling of carbon and distribution of heat and freshwater around the globe. Here, along-isopycnal upwelling of warm, carbon-rich circumpolar deep water coincides with the annual growth and melt of Antarctic sea ice that represents one of the world's largest surface water transformations. The air-sea-ice buoyancy exchanges and biological processes that change the surface water properties therefore have global consequences, as they set the properties of downwelling intermediate waters that enter the upper branch of the global thermohaline circulation. The region hosts some of the largest uncertainties in global climate models. The reason for this stems from two sources. Firstly, the spatio-temporal resolution of global climate models is limited by computational constraints such that smaller scale processes need to be parameterized. Secondly, the challenges associated with making observations in or near sea ice and in the harsh and remote conditions of the Southern Ocean means that the region is sparsely sampled, and as such, the parameterizations of the small scale and turbulent terms in global climate models are validated based only on a few in situ samples. This thesis concerns the observation and interpretation of (sub)meso- to micro scale turbulence and its implications in the sea ice impacted Southern Ocean. I aimed to understand the 0.01-1 km scale physical and biological processes that drive changes in the properties of the upper ocean following sea ice melt, using groundbreaking sustained high temporal and spatial resolution observations made by gliders. There are three main findings. Firstly, we find that sea ice melt by introducing a lateral freshwater gradient enhances stirring of submesoscale flows (0.1-10 km) and therefore lateral variability in the upper ocean, but simultaneously constrains vertical fluxes between the ocean interior and surface by enhancing stratification. Secondly, turbulent diapycnal mixing and double diffusive convection (0.1-1 m scales) drive the warming of the subsurface winter water, therefore mediating fluxes between the ocean interior and surface. Finally, phytoplankton respond favourably to larger volume sea ice that enhances winter mixing of nutrients from the deep reservoir and to upper ocean stratification in the summer. The preliminary evidence from this study suggests that the resultant higher intensity phytoplankton bloom translates to enhanced short term carbon export but not necessarily long term export. Overall, we show, using observations, that the variability and transport of heat and freshwater flux in the sea ice impacted Southern Ocean are sensitive to sea ice, with downstream impacts on phytoplankton, the biological carbon pump and ultimately the upper cell of the meridional overturning circulation

Jacobi quasi-Nijenhuis Algebroids

In this paper, for a Jacobi algebroid $A$, by introducing the notion of Jacobi quasi-Nijenhuis algebroids, which is a generalization of Poisson quasi-Nijenhuis manifolds introduced by Sti\'{e}non and Xu, we study generalized complex structures on the Courant-Jacobi algebroid $A\oplus A^*$, which unifies generalized complex (contact) structures on an even(odd)-dimensional manifold.Comment: 17 pages, no figur