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Assuming that $0<Z(n)\\leq \\varphi (n)$, where either $\\varphi (n)=an$ for some $a>0$ or $\\varphi (n)=o(n)$ as $n\\rightarrow \\infty $, we study the structure of the process $% \\left\\{ Z(m,n),0\\leq m\\leq n\\right\\} ,$ where $Z(m,n)$ is the number of particles in the process at moment $m\\leq n$ having a positive number of descendants at moment $n$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1801.03217","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-10T02:13:13Z","cross_cats_sorted":[],"title_canon_sha256":"8cbfd6299cd8ea616a0294c1260d76d90ee5711e0836f1350da829a6d159e443","abstract_canon_sha256":"6f396c62b4e32d4fc0f2bca0d14db45228cf790d7f49eb48234bd3b64139f381"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:17.708497Z","signature_b64":"iOdmnmaub97upWN5WcfZPHkphFQohdMO9YTRcEe/uv80evchVnOqQWZ+tQtYc5s1pdOmlvmibEMGDzdd+UMODw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"361821d8b2fb443e22b87afa53bb44c7ad24569aa70edd23ddde66581387fe85","last_reissued_at":"2026-05-18T00:26:17.707944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:17.707944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reduced critical processes for small populations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Minzhi Liu, Vladimir Vatutin","submitted_at":"2018-01-10T02:13:13Z","abstract_excerpt":"Let $\\left\\{ Z(n),n\\geq 1\\right\\} $ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. 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