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algebra angles apply approximations arbitrary assume asymptotics belongs boundary bounded braid closed coefficients commutative complete components condition cone Consequently consider constant construction contains continuous convergence Corollary corresponding curvature curve defined definition denote depend differential dimension domain element equal equation estimate example exists fact field finite formula function geodesic given Hence homomorphism identity implies inequality integral interval lattice Lemma limit Math Mathematical matrix means measure method metric minimal Moreover norm Note obtain operator pair particular periodic points positive present problem projection Proof Proposition prove refinable regular relation Remark respectively root satisfies sequence side similar smooth solution space spectrum statement subgroups Subsection suffices Suppose surface Theorem theory transformation triangle true unique values vectors weight zero