## Annals of Mathematics |

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abelian group affine connection algebraic arbitrary arcwise connected arithmetic mean boundary characteristic values circle coefficients commutator subgroup complementary domain congruences connected consider constant contains converges coordinates correlation correlation ratio corresponding cross-ratios cubic curve cycles cyclic cyclic group defined definition denote derivatives determined differential equations elements equianharmonic euclidean group exists Fermat's finite number follows formula fundamental function given groups of order Hence identity independent infinite integers integral invariant irreducible isomorphic Lemma Math Mathematical matrix modalities multiplier necessary and sufficient obtained operation orthogonal osculating parameter plane polynomial prime proof proved pseudoline pseudosphere quadratic quadratic fields rational reduced relation respect ruled surface satisfy sequence solution square stochastically independent Sturmian system sufficient condition summable suppose Theorem theory Trans transformation triangle Tschirnhaus transformation variables zero