{'fraction_non_alphanumeric': 0.046948356807511735, 'fraction_numerical': 0.004694835680751174, 'mean_word_length': 4.714285714285714, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, '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}}
A duality theorem of Chapman type in spaces of orthogonal representations of a compact Lie group

Fundamental theorem. Let the space of a renresentation ν of a compact Lie group G, denoted by E", have a non-trivial fixed point, and let Ε — (£") 2 * + Γ , r > 2, under the coordinatewise action, have two invariant compact subsets that are k v-pseudosmooth in E. For these to be equivariantly shapeequivalent, it is necessary and sufficient that their complements be equivariantly homeomorphic, or (which is equivalent under these conditions) equivariantly diffeomorphic. The dimension n(2k + r) of E, generally speaking, cannot be reduced.