{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:FHHMQXBUK3P6W4ZZD7IQPEAPQW","short_pith_number":"pith:FHHMQXBU","canonical_record":{"source":{"id":"0907.4951","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-07-28T16:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"2d3ee90eec28ee96682153a1d9ce426b7dedc81dccda65022017cd07d3cee5bb","abstract_canon_sha256":"75e1b731b40ff23adebe79acd93c3d095a9103ea56878fabd31149e058a2636b"},"schema_version":"1.0"},"canonical_sha256":"29cec85c3456dfeb73391fd107900f8588046a6452a4ca3de48b78e96e1bdeea","source":{"kind":"arxiv","id":"0907.4951","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.4951","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"0907.4951v1","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4951","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"FHHMQXBUK3P6","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FHHMQXBUK3P6W4ZZ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FHHMQXBU","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:FHHMQXBUK3P6W4ZZD7IQPEAPQW","target":"record","payload":{"canonical_record":{"source":{"id":"0907.4951","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-07-28T16:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"2d3ee90eec28ee96682153a1d9ce426b7dedc81dccda65022017cd07d3cee5bb","abstract_canon_sha256":"75e1b731b40ff23adebe79acd93c3d095a9103ea56878fabd31149e058a2636b"},"schema_version":"1.0"},"canonical_sha256":"29cec85c3456dfeb73391fd107900f8588046a6452a4ca3de48b78e96e1bdeea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:34.515224Z","signature_b64":"OluiyCx7H0tvlWlLrZ1e3dYuqbqLj8ku4NdtDr9B+v1rlhkweSqzJrIgjHChO0ufLPf/hy/DlM8PolkJlxLWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29cec85c3456dfeb73391fd107900f8588046a6452a4ca3de48b78e96e1bdeea","last_reissued_at":"2026-05-18T02:51:34.514749Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:34.514749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.4951","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":"2026-05-18T02:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Osz3X6E7SgAVjZtjwlwHNWIA7W4aFMK/US2K3bqXTwGAato9Un6D8YO91fXf1FRB76imCxvNazgX4RkFOnlTBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:18:37.952891Z"},"content_sha256":"95bd73380b5f9d8a8a688165eadf3650418ed9fee11a375cdff3aa7575a6d69c","schema_version":"1.0","event_id":"sha256:95bd73380b5f9d8a8a688165eadf3650418ed9fee11a375cdff3aa7575a6d69c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:FHHMQXBUK3P6W4ZZD7IQPEAPQW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homogenization and influence of fragmentation in a biological invasion model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francois Hamel, Lionel Roques, Mohammad El Smaily","submitted_at":"2009-07-28T16:39:56Z","abstract_excerpt":"In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4951","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":"2026-05-18T02:51:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kIFhEYjXu9kemcO4QZXbnftGYDk7A9QrRIuFPmA1Eb6zRmng/z39nXEn79YXbLF6kjwu9DYgkVZGgZGOsjbQAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:18:37.953256Z"},"content_sha256":"861b6102710f825f862c86fd2e55f60cadd605d1121ea8913948eb9fb26e4708","schema_version":"1.0","event_id":"sha256:861b6102710f825f862c86fd2e55f60cadd605d1121ea8913948eb9fb26e4708"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/bundle.json","state_url":"https://pith.science/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/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":"2026-06-07T20:18:37Z","links":{"resolver":"https://pith.science/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW","bundle":"https://pith.science/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/bundle.json","state":"https://pith.science/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FHHMQXBUK3P6W4ZZD7IQPEAPQW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:FHHMQXBUK3P6W4ZZD7IQPEAPQW","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":"75e1b731b40ff23adebe79acd93c3d095a9103ea56878fabd31149e058a2636b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-07-28T16:39:56Z","title_canon_sha256":"2d3ee90eec28ee96682153a1d9ce426b7dedc81dccda65022017cd07d3cee5bb"},"schema_version":"1.0","source":{"id":"0907.4951","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.4951","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"arxiv_version","alias_value":"0907.4951v1","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.4951","created_at":"2026-05-18T02:51:34Z"},{"alias_kind":"pith_short_12","alias_value":"FHHMQXBUK3P6","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FHHMQXBUK3P6W4ZZ","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FHHMQXBU","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:861b6102710f825f862c86fd2e55f60cadd605d1121ea8913948eb9fb26e4708","target":"graph","created_at":"2026-05-18T02:51:34Z","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 this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.","authors_text":"Francois Hamel, Lionel Roques, Mohammad El Smaily","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-07-28T16:39:56Z","title":"Homogenization and influence of fragmentation in a biological invasion model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.4951","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":"sha256:95bd73380b5f9d8a8a688165eadf3650418ed9fee11a375cdff3aa7575a6d69c","target":"record","created_at":"2026-05-18T02:51:34Z","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":"75e1b731b40ff23adebe79acd93c3d095a9103ea56878fabd31149e058a2636b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-07-28T16:39:56Z","title_canon_sha256":"2d3ee90eec28ee96682153a1d9ce426b7dedc81dccda65022017cd07d3cee5bb"},"schema_version":"1.0","source":{"id":"0907.4951","kind":"arxiv","version":1}},"canonical_sha256":"29cec85c3456dfeb73391fd107900f8588046a6452a4ca3de48b78e96e1bdeea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29cec85c3456dfeb73391fd107900f8588046a6452a4ca3de48b78e96e1bdeea","first_computed_at":"2026-05-18T02:51:34.514749Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:34.514749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OluiyCx7H0tvlWlLrZ1e3dYuqbqLj8ku4NdtDr9B+v1rlhkweSqzJrIgjHChO0ufLPf/hy/DlM8PolkJlxLWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:34.515224Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.4951","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95bd73380b5f9d8a8a688165eadf3650418ed9fee11a375cdff3aa7575a6d69c","sha256:861b6102710f825f862c86fd2e55f60cadd605d1121ea8913948eb9fb26e4708"],"state_sha256":"cf8ce09d17a1adebce2d3424155c905a76b95ddbbf6ea19f2e6a82d3de46e6eb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K19VRFDXlGRbxgOcD7sgpsjZZbNqTDHAEdMewAp691VGD+/ac/RXEwLnuwWSdSkZk84YrDn026/69aHd9bIWDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T20:18:37.955348Z","bundle_sha256":"4b3e9f520449eb4203c3b9bab7a78ff0ecfed891cb4925c08014b677fcfe4dda"}}