{'fraction_non_alphanumeric': 0.012718600953895072, 'fraction_numerical': 0.001589825119236884, 'mean_word_length': 5.494845360824742, '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}}
Analysis of Algorithms Generalizing B-Spline Subdivision

A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic B-spline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than 1 on the second largest eigenvalue of any smoothly converging subdivision.