{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:NXNT3AZ6IWSXUERBEUZTVNRTHS","short_pith_number":"pith:NXNT3AZ6","canonical_record":{"source":{"id":"1308.0531","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-02T15:21:13Z","cross_cats_sorted":[],"title_canon_sha256":"0d81020dc43ed164c2ca73d63a86f822cba50c7beb5cb40fecf62ad1605e0603","abstract_canon_sha256":"3df9ec951966bd7a9a6a222d80119ae5eb7faf16e1d126ee8f7731b98f133d52"},"schema_version":"1.0"},"canonical_sha256":"6ddb3d833e45a57a122125333ab6333c9c7c075d31446b9586b8d7d95f1bd2c0","source":{"kind":"arxiv","id":"1308.0531","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.0531","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"arxiv_version","alias_value":"1308.0531v2","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0531","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"pith_short_12","alias_value":"NXNT3AZ6IWSX","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"NXNT3AZ6IWSXUERB","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"NXNT3AZ6","created_at":"2026-05-18T12:27:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:NXNT3AZ6IWSXUERBEUZTVNRTHS","target":"record","payload":{"canonical_record":{"source":{"id":"1308.0531","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-02T15:21:13Z","cross_cats_sorted":[],"title_canon_sha256":"0d81020dc43ed164c2ca73d63a86f822cba50c7beb5cb40fecf62ad1605e0603","abstract_canon_sha256":"3df9ec951966bd7a9a6a222d80119ae5eb7faf16e1d126ee8f7731b98f133d52"},"schema_version":"1.0"},"canonical_sha256":"6ddb3d833e45a57a122125333ab6333c9c7c075d31446b9586b8d7d95f1bd2c0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:29.853727Z","signature_b64":"3urfkCDmjThHvDsaXdE0YrOEN2juP05dvKMcUA2DunR1q6Y7u74+dKegIiuPtHEdFhqrqLRBos3S0AyCuJ6hCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ddb3d833e45a57a122125333ab6333c9c7c075d31446b9586b8d7d95f1bd2c0","last_reissued_at":"2026-05-18T02:38:29.853218Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:29.853218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.0531","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:38:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QJq8jB/jyw6amGM6VNn+Ss4P+Gr4Asxseo9oQYm2jgyS1UJJqTYUnrVQJBc4mlvZQxgcfG1gkOQdQK3QmR6FAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:28:34.456645Z"},"content_sha256":"50be72c53daca041e431b41808a1a14e4dfc3db4059fb77e179bb9ac402bee79","schema_version":"1.0","event_id":"sha256:50be72c53daca041e431b41808a1a14e4dfc3db4059fb77e179bb9ac402bee79"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:NXNT3AZ6IWSXUERBEUZTVNRTHS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Liang Kong, Wenxian Shen","submitted_at":"2013-08-02T15:21:13Z","abstract_excerpt":"The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$ u_t(t,x)=(\\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\\quad x\\in\\mathcal{H}, $$ where $\\mathcal{H}=\\RR^N$ or $\\ZZ^N$, $\\mathcal{A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\\mathcal{H}=\\RR^N$ and is a discrete dispersal operator in the case $\\mathcal{H}=\\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\\|x\\|\\to\\infty$ for some time and space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0531","kind":"arxiv","version":2},"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":"2026-05-18T02:38:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QPc/ghl+nABGpSFvoWIY9rEhEehP5tNMaTbF8BD70YjM8MQRVi1Cl3Q5XcGOK2yl9oTjKcPh/DBBvq4dTtYYAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:28:34.457000Z"},"content_sha256":"d384ea8e70cac9d75cdb3aa2473065fb81c55ae985a648a9c9b67c6d8b9730f9","schema_version":"1.0","event_id":"sha256:d384ea8e70cac9d75cdb3aa2473065fb81c55ae985a648a9c9b67c6d8b9730f9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/bundle.json","state_url":"https://pith.science/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/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":"2026-06-01T22:28:34Z","links":{"resolver":"https://pith.science/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS","bundle":"https://pith.science/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/bundle.json","state":"https://pith.science/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NXNT3AZ6IWSXUERBEUZTVNRTHS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:NXNT3AZ6IWSXUERBEUZTVNRTHS","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":"3df9ec951966bd7a9a6a222d80119ae5eb7faf16e1d126ee8f7731b98f133d52","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-02T15:21:13Z","title_canon_sha256":"0d81020dc43ed164c2ca73d63a86f822cba50c7beb5cb40fecf62ad1605e0603"},"schema_version":"1.0","source":{"id":"1308.0531","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.0531","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"arxiv_version","alias_value":"1308.0531v2","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.0531","created_at":"2026-05-18T02:38:29Z"},{"alias_kind":"pith_short_12","alias_value":"NXNT3AZ6IWSX","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_16","alias_value":"NXNT3AZ6IWSXUERB","created_at":"2026-05-18T12:27:52Z"},{"alias_kind":"pith_short_8","alias_value":"NXNT3AZ6","created_at":"2026-05-18T12:27:52Z"}],"graph_snapshots":[{"event_id":"sha256:d384ea8e70cac9d75cdb3aa2473065fb81c55ae985a648a9c9b67c6d8b9730f9","target":"graph","created_at":"2026-05-18T02:38:29Z","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 devoted to the study of semilinear dispersal evolution equations of the form $$ u_t(t,x)=(\\mathcal{A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\\quad x\\in\\mathcal{H}, $$ where $\\mathcal{H}=\\RR^N$ or $\\ZZ^N$, $\\mathcal{A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\\mathcal{H}=\\RR^N$ and is a discrete dispersal operator in the case $\\mathcal{H}=\\ZZ^N$, and $f$ is periodic in $t$, asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to 0 as $\\|x\\|\\to\\infty$ for some time and space periodic function $f_0(t,x,u)$), and is of KPP type in $u$. It","authors_text":"Liang Kong, Wenxian Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-02T15:21:13Z","title":"Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.0531","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":"sha256:50be72c53daca041e431b41808a1a14e4dfc3db4059fb77e179bb9ac402bee79","target":"record","created_at":"2026-05-18T02:38:29Z","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":"3df9ec951966bd7a9a6a222d80119ae5eb7faf16e1d126ee8f7731b98f133d52","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-02T15:21:13Z","title_canon_sha256":"0d81020dc43ed164c2ca73d63a86f822cba50c7beb5cb40fecf62ad1605e0603"},"schema_version":"1.0","source":{"id":"1308.0531","kind":"arxiv","version":2}},"canonical_sha256":"6ddb3d833e45a57a122125333ab6333c9c7c075d31446b9586b8d7d95f1bd2c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ddb3d833e45a57a122125333ab6333c9c7c075d31446b9586b8d7d95f1bd2c0","first_computed_at":"2026-05-18T02:38:29.853218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:29.853218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3urfkCDmjThHvDsaXdE0YrOEN2juP05dvKMcUA2DunR1q6Y7u74+dKegIiuPtHEdFhqrqLRBos3S0AyCuJ6hCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:29.853727Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.0531","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50be72c53daca041e431b41808a1a14e4dfc3db4059fb77e179bb9ac402bee79","sha256:d384ea8e70cac9d75cdb3aa2473065fb81c55ae985a648a9c9b67c6d8b9730f9"],"state_sha256":"aaa2bddb4dfc6a27ec8299481443311b175d568dd9ba5a0b709ebb24de349d0e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"02LCcfhlP5qrtH/tNbMBMLP6HIEmHFTHktgcCJHiETGz7hKPJVCyJM3O1BZIOSlqJNtre+/Dzj4pZO5jsJz1Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T22:28:34.458914Z","bundle_sha256":"62437a06b9aaa25ca0294065a2c1991a68b0901614a8a1a88bf351166ee14411"}}