{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ZABJKWQ577K7QNW6TA6L7OTU63","short_pith_number":"pith:ZABJKWQ5","canonical_record":{"source":{"id":"1202.1812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-08T20:48:57Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"79a01613088522b6a12b0418cf77d2eecfc38fe711e46e5d3a0333e0772c4ced","abstract_canon_sha256":"4a4a98c866eba2b52e55f64497461c61f412d6dd267a7aa5e6871ebdc3061e9d"},"schema_version":"1.0"},"canonical_sha256":"c802955a1dffd5f836de983cbfba74f6e4d86da6ec0b32d7255bc70898e96303","source":{"kind":"arxiv","id":"1202.1812","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1812","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1812v1","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1812","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"pith_short_12","alias_value":"ZABJKWQ577K7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZABJKWQ577K7QNW6","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZABJKWQ5","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ZABJKWQ577K7QNW6TA6L7OTU63","target":"record","payload":{"canonical_record":{"source":{"id":"1202.1812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-08T20:48:57Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"79a01613088522b6a12b0418cf77d2eecfc38fe711e46e5d3a0333e0772c4ced","abstract_canon_sha256":"4a4a98c866eba2b52e55f64497461c61f412d6dd267a7aa5e6871ebdc3061e9d"},"schema_version":"1.0"},"canonical_sha256":"c802955a1dffd5f836de983cbfba74f6e4d86da6ec0b32d7255bc70898e96303","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:30.102283Z","signature_b64":"aYnz6fdsTmDItek4+Hj6xzIRUdhbli0M32ls+HMj9jGKBv+NMc4d3Ze9fNJezXQCaiUTMK4aviFJ8pDDxWinDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c802955a1dffd5f836de983cbfba74f6e4d86da6ec0b32d7255bc70898e96303","last_reissued_at":"2026-05-18T02:38:30.101776Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:30.101776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.1812","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":"2026-05-18T02:38:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ilb0Tami+uovz+uAvUEQJuP5qMZ0k/uFdH22++dHUyzt6CCVFYxBVuVrO22PRw8jxeVzkmckShq/QAaAXXLcDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:11:27.304625Z"},"content_sha256":"88bcabc12a62e9860015704b155293ae96bf9421f865b654300639ce0c93db66","schema_version":"1.0","event_id":"sha256:88bcabc12a62e9860015704b155293ae96bf9421f865b654300639ce0c93db66"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ZABJKWQ577K7QNW6TA6L7OTU63","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DS","authors_text":"Liang Kong, Wenxian Shen","submitted_at":"2012-02-08T20:48:57Z","abstract_excerpt":"The current paper is concerned with positive stationary solutions and spatial spreading speeds of KPP type evolution equations with random or nonlocal or discrete dispersal in locally spatially inhomogeneous media. It is shown that such an equation has a unique globally stable positive stationary solution and has a spreading speed in every direction. Moreover, it is shown that the localized spatial inhomogeneity of the medium neither slows down nor speeds up the spatial spreading in all the directions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1812","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":"2026-05-18T02:38:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8PfkSZ7u2nq6rYJHsOT6wJNE2nUqpQNzWRPprbz4o6KBZUb8frCx/T481hJoF0IJf5qXtZT5MZCKQDLGhqSfDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:11:27.305195Z"},"content_sha256":"93d24dd8e00e9c763d4e54154543a9254b27ffd4d5f5fd849de82591395ca172","schema_version":"1.0","event_id":"sha256:93d24dd8e00e9c763d4e54154543a9254b27ffd4d5f5fd849de82591395ca172"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZABJKWQ577K7QNW6TA6L7OTU63/bundle.json","state_url":"https://pith.science/pith/ZABJKWQ577K7QNW6TA6L7OTU63/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZABJKWQ577K7QNW6TA6L7OTU63/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":"2026-05-31T19:11:27Z","links":{"resolver":"https://pith.science/pith/ZABJKWQ577K7QNW6TA6L7OTU63","bundle":"https://pith.science/pith/ZABJKWQ577K7QNW6TA6L7OTU63/bundle.json","state":"https://pith.science/pith/ZABJKWQ577K7QNW6TA6L7OTU63/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZABJKWQ577K7QNW6TA6L7OTU63/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZABJKWQ577K7QNW6TA6L7OTU63","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":"4a4a98c866eba2b52e55f64497461c61f412d6dd267a7aa5e6871ebdc3061e9d","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-08T20:48:57Z","title_canon_sha256":"79a01613088522b6a12b0418cf77d2eecfc38fe711e46e5d3a0333e0772c4ced"},"schema_version":"1.0","source":{"id":"1202.1812","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1812","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1812v1","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1812","created_at":"2026-05-18T02:38:30Z"},{"alias_kind":"pith_short_12","alias_value":"ZABJKWQ577K7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZABJKWQ577K7QNW6","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZABJKWQ5","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:93d24dd8e00e9c763d4e54154543a9254b27ffd4d5f5fd849de82591395ca172","target":"graph","created_at":"2026-05-18T02:38:30Z","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":"The current paper is concerned with positive stationary solutions and spatial spreading speeds of KPP type evolution equations with random or nonlocal or discrete dispersal in locally spatially inhomogeneous media. It is shown that such an equation has a unique globally stable positive stationary solution and has a spreading speed in every direction. Moreover, it is shown that the localized spatial inhomogeneity of the medium neither slows down nor speeds up the spatial spreading in all the directions.","authors_text":"Liang Kong, Wenxian Shen","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-08T20:48:57Z","title":"Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1812","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":"sha256:88bcabc12a62e9860015704b155293ae96bf9421f865b654300639ce0c93db66","target":"record","created_at":"2026-05-18T02:38:30Z","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":"4a4a98c866eba2b52e55f64497461c61f412d6dd267a7aa5e6871ebdc3061e9d","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-02-08T20:48:57Z","title_canon_sha256":"79a01613088522b6a12b0418cf77d2eecfc38fe711e46e5d3a0333e0772c4ced"},"schema_version":"1.0","source":{"id":"1202.1812","kind":"arxiv","version":1}},"canonical_sha256":"c802955a1dffd5f836de983cbfba74f6e4d86da6ec0b32d7255bc70898e96303","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c802955a1dffd5f836de983cbfba74f6e4d86da6ec0b32d7255bc70898e96303","first_computed_at":"2026-05-18T02:38:30.101776Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:30.101776Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aYnz6fdsTmDItek4+Hj6xzIRUdhbli0M32ls+HMj9jGKBv+NMc4d3Ze9fNJezXQCaiUTMK4aviFJ8pDDxWinDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:30.102283Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.1812","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88bcabc12a62e9860015704b155293ae96bf9421f865b654300639ce0c93db66","sha256:93d24dd8e00e9c763d4e54154543a9254b27ffd4d5f5fd849de82591395ca172"],"state_sha256":"2bf4c87346d198d33789ac2cee395ceb2d47f7907f76d1b6c56e85c0e0adb7fd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rgtsdzfH3P910e1z+e7uKyIVan8JODZVWkmsTs1iGHjaBdD/BEwXYNYYhoMj3J7Lda82ZJerj4lCYTz8bohLAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:11:27.309033Z","bundle_sha256":"d642b6ddaee91a02fa095764aa63747e0590ac98c1a704adbd8b0d8e15afc09a"}}