Lacustrine mollusc radiations in the Lake Malawi Basin : experiments in a natural laboratory for evolution

In terminal Pliocene-early Pleistocene times, part of the Malawi Basin was occupied by paleo-lake Chiwondo. Molluscan biostratigraphy situates this freshwater lake either in the East African wet phase between 2.7-2.4 Ma or that of 2.0-1.8 Ma. In-lake divergent evolution remained restricted to a few molluscan taxa and was very modest. The lacustrine Chiwondo fauna went extinct at the beginning of the Pleistocene. The modern Lake Malawi malacofauna is depauperate and descends from ubiquistic southeast African taxa and some Malawi basin endemics that invaded the present lake after the Late Pleistocene mega-droughts. The Pleistocene aridity crises caused dramatic changes, affecting the malacofauna of all East African lakes. All lacustrine endemic faunas that had evolved in the Pliocene rift lakes, such as paleo-lake Chiwondo, became extinct. In Lake Tanganyika, the freshwater ecosystem did not crash as in other lakes, but the environmental changes were sufficiently important to trigger a vast radiation. All African endemic lacustrine molluscan clades that are the result of in-lake divergence are hence geologically young, including the vast Lavigeria clade in Lake Tanganyika (ca. 43 species)

On the best constant of Hardy-Sobolev Inequalities

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant

Trace Hardy--Sobolev--Mazy'a inequalities for the half fractional Laplacian

In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in \cite{FMT1}