{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:CNPR6DKNWQWPVYIGVZWEGQDU7P","short_pith_number":"pith:CNPR6DKN","canonical_record":{"source":{"id":"0812.1367","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-12-07T17:04:06Z","cross_cats_sorted":[],"title_canon_sha256":"2cfe09e652b8bb3e2093a97f848df863cd4a4a3a537ee2f30f077d607687141e","abstract_canon_sha256":"891dfc8d498bb15e821021272e8e965132b098e12b162e3d958a7b0fa675786d"},"schema_version":"1.0"},"canonical_sha256":"135f1f0d4db42cfae106ae6c434074fbce4b0d7e17056aed51f4cbff7e5d0764","source":{"kind":"arxiv","id":"0812.1367","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.1367","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"arxiv_version","alias_value":"0812.1367v1","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1367","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"pith_short_12","alias_value":"CNPR6DKNWQWP","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"CNPR6DKNWQWPVYIG","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"CNPR6DKN","created_at":"2026-05-18T12:25:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:CNPR6DKNWQWPVYIGVZWEGQDU7P","target":"record","payload":{"canonical_record":{"source":{"id":"0812.1367","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-12-07T17:04:06Z","cross_cats_sorted":[],"title_canon_sha256":"2cfe09e652b8bb3e2093a97f848df863cd4a4a3a537ee2f30f077d607687141e","abstract_canon_sha256":"891dfc8d498bb15e821021272e8e965132b098e12b162e3d958a7b0fa675786d"},"schema_version":"1.0"},"canonical_sha256":"135f1f0d4db42cfae106ae6c434074fbce4b0d7e17056aed51f4cbff7e5d0764","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:43.154539Z","signature_b64":"nRbacheJYAfpkF50T72xpqui2mIkctLMUZnwKQ6BVg1fDLMMu+l8Zgu7GoR/55Q2TVjMr63TkRDP/lJJtv3iBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"135f1f0d4db42cfae106ae6c434074fbce4b0d7e17056aed51f4cbff7e5d0764","last_reissued_at":"2026-05-17T23:50:43.154134Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:43.154134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0812.1367","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:50:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0io0GSn76I4fQvf8jQC72hhyRIUIMlgwi+vguj5TqT2Oj4/ytIy8Z0RcDEGe75Zbq2GwWAKDAB7GL+4ZSsBMAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T22:52:37.710642Z"},"content_sha256":"5c14f5c29cff61a1bf136cc5fa56a026e676cadb8bac81ed13072c29878fa201","schema_version":"1.0","event_id":"sha256:5c14f5c29cff61a1bf136cc5fa56a026e676cadb8bac81ed13072c29878fa201"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:CNPR6DKNWQWPVYIGVZWEGQDU7P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hierarchical size-structured populations: The linearized semigroup approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jozsef Z. Farkas, Thomas C. Hagen","submitted_at":"2008-12-07T17:04:06Z","abstract_excerpt":"In the present paper we analyze the linear stability of a hierarchical size-structured population model where the vital rates (mortality, fertility and growth rate) depend both on size and a general functional of the population density (\"environment\"). We derive regularity properties of the governing linear semigroup, implying that linear stability is governed by a dominant real eigenvalue of the semigroup generator, which arises as a zero of an associated characteristic function. In the special case where neither the growth rate nor the mortality depend on the environment, we explicitly calcu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1367","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:50:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rvgmiuDJOh2f8AU0nHWSomAH/r/jdpoDEXOriiKfWaDFQzzqjPgLfIUCW8mDkRvyFd/n0G+bPRGwo2yf4CkSBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T22:52:37.710985Z"},"content_sha256":"21f2cfdc90bc4a69139707bde1eeb2c06111cabddf46ef906a86128cca52b9de","schema_version":"1.0","event_id":"sha256:21f2cfdc90bc4a69139707bde1eeb2c06111cabddf46ef906a86128cca52b9de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/bundle.json","state_url":"https://pith.science/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T22:52:37Z","links":{"resolver":"https://pith.science/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P","bundle":"https://pith.science/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/bundle.json","state":"https://pith.science/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CNPR6DKNWQWPVYIGVZWEGQDU7P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:CNPR6DKNWQWPVYIGVZWEGQDU7P","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":"891dfc8d498bb15e821021272e8e965132b098e12b162e3d958a7b0fa675786d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-12-07T17:04:06Z","title_canon_sha256":"2cfe09e652b8bb3e2093a97f848df863cd4a4a3a537ee2f30f077d607687141e"},"schema_version":"1.0","source":{"id":"0812.1367","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.1367","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"arxiv_version","alias_value":"0812.1367v1","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.1367","created_at":"2026-05-17T23:50:43Z"},{"alias_kind":"pith_short_12","alias_value":"CNPR6DKNWQWP","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"CNPR6DKNWQWPVYIG","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"CNPR6DKN","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:21f2cfdc90bc4a69139707bde1eeb2c06111cabddf46ef906a86128cca52b9de","target":"graph","created_at":"2026-05-17T23:50:43Z","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":"In the present paper we analyze the linear stability of a hierarchical size-structured population model where the vital rates (mortality, fertility and growth rate) depend both on size and a general functional of the population density (\"environment\"). We derive regularity properties of the governing linear semigroup, implying that linear stability is governed by a dominant real eigenvalue of the semigroup generator, which arises as a zero of an associated characteristic function. In the special case where neither the growth rate nor the mortality depend on the environment, we explicitly calcu","authors_text":"Jozsef Z. Farkas, Thomas C. Hagen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-12-07T17:04:06Z","title":"Hierarchical size-structured populations: The linearized semigroup approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.1367","kind":"arxiv","version":1},"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":"sha256:5c14f5c29cff61a1bf136cc5fa56a026e676cadb8bac81ed13072c29878fa201","target":"record","created_at":"2026-05-17T23:50:43Z","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":"891dfc8d498bb15e821021272e8e965132b098e12b162e3d958a7b0fa675786d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-12-07T17:04:06Z","title_canon_sha256":"2cfe09e652b8bb3e2093a97f848df863cd4a4a3a537ee2f30f077d607687141e"},"schema_version":"1.0","source":{"id":"0812.1367","kind":"arxiv","version":1}},"canonical_sha256":"135f1f0d4db42cfae106ae6c434074fbce4b0d7e17056aed51f4cbff7e5d0764","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"135f1f0d4db42cfae106ae6c434074fbce4b0d7e17056aed51f4cbff7e5d0764","first_computed_at":"2026-05-17T23:50:43.154134Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:43.154134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nRbacheJYAfpkF50T72xpqui2mIkctLMUZnwKQ6BVg1fDLMMu+l8Zgu7GoR/55Q2TVjMr63TkRDP/lJJtv3iBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:43.154539Z","signed_message":"canonical_sha256_bytes"},"source_id":"0812.1367","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5c14f5c29cff61a1bf136cc5fa56a026e676cadb8bac81ed13072c29878fa201","sha256:21f2cfdc90bc4a69139707bde1eeb2c06111cabddf46ef906a86128cca52b9de"],"state_sha256":"2dca7cf7bba07c41f8bf0125ca4a39e46926e73d73c6023dc022ab12d3b76dcf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i2fmJBlUjSKE2QKR8X8wWsDWtmxiefJuChJHoRPcS1rIlhI/Wo/MvWNCU2oKo2A1OXwXV+D9wHv9pD9ebKd2Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T22:52:37.712898Z","bundle_sha256":"83eb543913b5402971cfbf4020af633e77a427f6ee1321845c7d8dfdde26a54b"}}