
Mathematics
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October 21st, 2015, 11:00 a.m., Science conference room.
Title: A conformal geometric inequality on the Klein bottle
Speaker: Dr. Chady El Mir (Lebanese University, North Lebanon)
Abstract: The systole of a compact non simply connected Riemannian manifold is the smallest length of a noncontractible closed curve. In 1986, C. Bavard proved an optimal lower bound on the area of the Klein bottle (denoted by K) in terms of the square of the systole. More precisely, he showed that (K), where the equality is attained for a spherical metric outside a singular line. In our talk, we will prove an optimal conformal inequality on the Klein bottle in the same spirit of Bavard's inequality. We will give an optimal conformal lower bound on the area of the Klein bottle in terms of the product of the least lengths of two homotopy classes of curves that generate the fundamental group of K i.e. the class of a glide reflection and the class of a vertical translation in . These two classes are potential candidates to realize the systole of K. A similar result on the 2dimensional torus was proved by L. Keen in 1967.
November 25th, 2015, 11:00 a.m., Science Conference Room.
Title: Convergence analysis of two numerical schemes applied to a nonlinear elliptic problem
Speaker: Dr. Tony Sayah (USJ, Mansourieh)
Abstract: For a given nonlinear problem discretized by standard finite elements, we propose two Iterative schemes to solve the discrete problem. We prove the wellposedness of the corresponding problems and their convergence. Next, we construct error indicators and prove optimal a posteriori estimates where we treat separately the discretization and linearization errors. Some numerical experiments confirm the validity of the schemes and allow us to compare them.
December 16th, 2015, 11:00 a.m., Science Conference Room.
Speaker: Dr. Leila Issa (LAU, Beirut).
Title: Modeling Surface Currents in the Eastern Levantine Mediterranean
Abstract: We consider the problem of reconstructing the mesoscale features of the currents in the Eastern Levantine Mediterranean from combining insitu and satellite altimetry data. Mathematically, this is an inverse problem where the objective is to invert Lagrangian trajectories, which are positions of drifters launched at sea, in order to improve the coarse Eulerian velocity, provided by the altimetric satellite measurements. We shall use a variational assimilation approach, whereby the Eulerian velocity correction is obtained by minimizing the distance between the simulated position from a velocity background and actual observations. One important property of our approach is that it is model free, so that it is inexpensive and can be easily cast into realtime oceanic operational products. Our method is first validated with twin experiments, where we conduct sensitivity analysis to parameters such as number of drifters, assimilation time window and spatial filter length. The approach is next validated with past and present data from the Levantine Mediterranean by correcting velocity fields derived from altimetry by assimilating drifters’ data. The drifters’ data used here were collected in the context of the MedSVP program and more recently by the National Lebanese Marine Center (CNSM) in September 2013. The CNSM with its boat CANA has developed an important activity of data collection along the Lebanese coast so far and this activity will permit it to extend its collaborations further by integrating the modeling and data assimilation methods for reconstructing the surface currents.
