{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:JHWZJ64BK67JHXTKL7KPB4UK7W","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4facf099f6d64c0a19c4865f94d793d317f9a0523bb1804bc8b39624423f6bdc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-20T11:19:29Z","title_canon_sha256":"cc4766bb5f1cbb81bb1a5454759262650b07c13840300108bf6f175ba5c212a9"},"schema_version":"1.0","source":{"id":"1704.06087","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06087","created_at":"2026-05-18T00:00:31Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06087v2","created_at":"2026-05-18T00:00:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06087","created_at":"2026-05-18T00:00:31Z"},{"alias_kind":"pith_short_12","alias_value":"JHWZJ64BK67J","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JHWZJ64BK67JHXTK","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JHWZJ64B","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:bfa52b5442bb32f25bf2652f7b7b54a776d521f9191713f3e26e0a0c5e2a8aa2","target":"graph","created_at":"2026-05-18T00:00:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We give here an explicit formula for the following critical case of the growth-fragmentation equation $$\\frac{\\partial}{\\partial t} u(t, x) + \\frac{\\partial}{\\partial x} (gxu(t, x)) + bu(t, x) = b\\alpha^2 u(t, \\alpha x), \\qquad u(0, x) = u\\_0 (x),$$ for some constants $g > 0$, $b > 0$ and $\\alpha > 1$ - the case $\\alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.","authors_text":"Bruce Van Brunt, Marie Doumic (MAMBA)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-20T11:19:29Z","title":"Explicit Solution and Fine Asymptotics for a Critical Growth-Fragmentation Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06087","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:617464890fabefc819b68b115cc22ce022c29f78a6c0a2e1c2deeb3990a3700a","target":"record","created_at":"2026-05-18T00:00:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4facf099f6d64c0a19c4865f94d793d317f9a0523bb1804bc8b39624423f6bdc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-20T11:19:29Z","title_canon_sha256":"cc4766bb5f1cbb81bb1a5454759262650b07c13840300108bf6f175ba5c212a9"},"schema_version":"1.0","source":{"id":"1704.06087","kind":"arxiv","version":2}},"canonical_sha256":"49ed94fb8157be93de6a5fd4f0f28afd86819f2e62e22604dd0d1a06e3242c7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49ed94fb8157be93de6a5fd4f0f28afd86819f2e62e22604dd0d1a06e3242c7f","first_computed_at":"2026-05-18T00:00:31.123221Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:31.123221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hRXegF1cdrNZNXKNt8wQIfi8uzNWleCeATqzbo66N7lcCYsLBgbPFJ+ZO0L8vhpTxMK2giGSJd7alKt2IvoTDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:31.123769Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.06087","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:617464890fabefc819b68b115cc22ce022c29f78a6c0a2e1c2deeb3990a3700a","sha256:bfa52b5442bb32f25bf2652f7b7b54a776d521f9191713f3e26e0a0c5e2a8aa2"],"state_sha256":"9d49109d29b508e91f69662f6d803055cab967cae4a91ee200b1ec1ec0385e2b"}