{'fraction_non_alphanumeric': 0.019975031210986267, 'fraction_numerical': 0.0, 'mean_word_length': 4.897058823529412, '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}}
Excursion set mass functions for hierarchical Gaussian fluctuations

Most schemes for determining the mass function of virialized objects from the statistics of the initial density perturbation field suffer from the «cloud-in-cloud» problem of miscounting the number of low-mass clumps, many of which would have been subsumed into larger objects. We propose a solution based on the theory of the excursion sets of F(r, R f ), the four-dimensional initial density perturbation field smoothed with a continuous hierarchy of filters of radii Rf. We identify the mass fraction of matter in virialized objects with mass greater than M with the fraction of space in which the initial density contrast lies above a critical overdensity when smoothed on some filter of radius greater than or equal to R f (M).