Abstract. These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
Collisions with buildings are a major source of mortality for wild birds, but these instantaneous events are difficult to observe. As a result, the mechanistic causes of collision mortality are poorly understood. Here, we evaluate whether sensory and biomechanical traits can explain why some species are more collision-prone than others. We first examined concordance of species vulnerability estimates across previous North American studies to determine whether these estimates are repeatable, and whether vulnerability is more similar among closely-related species. We found moderate concordance and phylogenetic signal, indicating that some bird species are consistently more collision-prone than others. We next tested whether morphological traits related to flight performance and sensory guidance explain these differences among species. Our comparative analysis shows that two traits primarily predict collision vulnerability within passerines: relative beak length and relative wing length. Small passerine species with relatively short wings and those with relatively long beaks are more collision-prone, suggesting that greater maneuverability and obstructed vision contribute to risk. Together, these findings can help inform mitigation strategies and predict which species will be most at risk in other regions.
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