{'fraction_non_alphanumeric': 0.05963302752293578, 'fraction_numerical': 0.0, 'mean_word_length': 4.504201680672269, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Automatic continuity of n-homomorphisms between FrÉchet algebras

We impose a condition on a commutative regular Fréchet algebra (A, (pm )) to ensure that A/kerpm is a Fréchet Q-algebra. This implies that if θ is an n-homomorphism on certain Fréchet algebras (A, (pm )) into semisimple commutative Fréchet algebras (B,(qm)) such that θ(kerpm) ⫅ kerqm, for large enough m, then θ is continuous. We also show that if A is a Fréchet Q-algebra, B is a semisimple Fréchet algebra, θ: A → B is a dense range n-homomorphism such that θ(A) is factorizable, and the spectral radius vB is continuous on the separating space (θ), then θ is automatically continuous.