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Assuming that $0<Z(t)\\leq \\varphi (t)$, where either $\\varphi (t)=o(t)$ as $t\\rightarrow \\infty $ or $\\varphi (t)=at,\\, a>0$, we study the structure of the process $% \\left\\{ Z(s,t),0\\leq s\\leq t\\right\\} ,$ where $Z(s,t)$ is the number of particles in the process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment."},"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":"1809.05029","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-09-13T16:03:44Z","cross_cats_sorted":[],"title_canon_sha256":"87fc8d62e2592e97b0565b7a29c629a65997ffbeb1f2ccfb7c2de8ddf6186d55","abstract_canon_sha256":"d7ca71244a2193a0b92e19285e71828a9f270aa2e47c44f35b13969bcff5cb7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:44.730070Z","signature_b64":"3SpoivmOemkpjARURbLb6n+HttOkZg9ckWDbZcxWyEU9hUKlQ2eU5vLEwmKt0fMcwpHoVYOB4sUxRQkE5AAMAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cafc2e017bb39118c094251cc81df6a39491abb9ece3f6c9790f4c8e23be7788","last_reissued_at":"2026-05-18T00:05:44.729352Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:44.729352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reduced critical Bellman-Harris branching processes for small populations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Vladimir Vatutin, Wenming Hong, Yao Ji","submitted_at":"2018-09-13T16:03:44Z","abstract_excerpt":"Let $\\left\\{ Z(t), t\\geq 0\\right\\} $ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. 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