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Mukai and Sakai proved that given a vector bundle$E$of rank$n$on a connected smooth projective curve of genus$g$and any$r$∈[1,$n$], there is subbundle$S$of rank$r$such that deg Hom($S, E/S$)≤$r$($n−r$)$g$. We prove a generalization of this bound for equivariant principal bundles. Our proof even simplifies the one given by Holla and Narasimhan for usual principal bundles.

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold of$T$^{*}$R$^{$n$+1}-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball$B$⊂$C$^{$n$}with its relative logarithmic capacity in$C$^{$n$}with respect to the same ball$B$. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of$C$^{$n$}is also proved.

Let$f$be meromorphic in the open unit disc$D$and strongly normal; that is, $$(1 - |z|^2 )f^\# (z) \to 0as|z| \to 1,$$