{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:TNTSDPPPQGHQRWEMGSGC2OIJ4Y","short_pith_number":"pith:TNTSDPPP","canonical_record":{"source":{"id":"math/0610764","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-10-25T19:12:57Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"9806cfe5b1961b10c929dc6c7b7b304cb4aeaf926e49c482de139d869b42dca9","abstract_canon_sha256":"8aaeb7da52648232b94c059b94c48a7cf8ca9cb5e39e82b67e3c26a3a164ddef"},"schema_version":"1.0"},"canonical_sha256":"9b6721bdef818f08d88c348c2d3909e60ef6e197263ee064fc64efb58b4430ea","source":{"kind":"arxiv","id":"math/0610764","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610764","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610764v1","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610764","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_12","alias_value":"TNTSDPPPQGHQ","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_16","alias_value":"TNTSDPPPQGHQRWEM","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_8","alias_value":"TNTSDPPP","created_at":"2026-07-04T14:57:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:TNTSDPPPQGHQRWEMGSGC2OIJ4Y","target":"record","payload":{"canonical_record":{"source":{"id":"math/0610764","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-10-25T19:12:57Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"9806cfe5b1961b10c929dc6c7b7b304cb4aeaf926e49c482de139d869b42dca9","abstract_canon_sha256":"8aaeb7da52648232b94c059b94c48a7cf8ca9cb5e39e82b67e3c26a3a164ddef"},"schema_version":"1.0"},"canonical_sha256":"9b6721bdef818f08d88c348c2d3909e60ef6e197263ee064fc64efb58b4430ea","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:57:10.374689Z","signature_b64":"kAuU7BOoPoJBMhQ1qD4R9YJg6sq2kd3ahmdiNu65iaXNC+hQB1DLlYSM9ZhShctFb8qpOKM25fg9Irx0pfXxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b6721bdef818f08d88c348c2d3909e60ef6e197263ee064fc64efb58b4430ea","last_reissued_at":"2026-07-04T14:57:10.374341Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:57:10.374341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0610764","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":"2026-07-04T14:57:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iTUYHdS7IwayjC/JZjKiacdkoy2yjqN8ToLgsznXHoMaWu71K3Y8jq49rEpu+KYgH1c4/Sv8AvrCluLkhHZ5DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T12:54:53.822695Z"},"content_sha256":"e54773a52f562d80a38fca900f971e07551cb6127b6aa6b579d6be7a8ef6225a","schema_version":"1.0","event_id":"sha256:e54773a52f562d80a38fca900f971e07551cb6127b6aa6b579d6be7a8ef6225a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:TNTSDPPPQGHQRWEMGSGC2OIJ4Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Fengbo Hang, Huiqiang Jiang","submitted_at":"2006-10-25T19:12:57Z","abstract_excerpt":"In 1992, P. Pol\\'{a}\\v{c}ik\\cite{P2} showed that one could linearly imbed any vector fields into a scalar semi-linear parabolic equation on $\\Omega$ with Neumann boundary condition provided that there exists a smooth vector field $\\Phi=(\\phi_{1},...,\\phi_{n}) $ on $\\bar{\\Omega}$ such that \\[ \\left\\{\\begin{array} [c]{l}% \\operatorname*{rank}(\\Phi(x) ,\\partial_{1}\\Phi(x) ,...,\\partial_{n}\\Phi(x)) =n\\text{for all}x\\in\\bar{\\Omega}, \\frac{\\partial\\Phi}{\\partial\\nu}=0\\text{on}\\partial\\Omega\\text{.}% \\end{array} \\right. \\] In this short note, we give a classification of all the domains on which one m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610764","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0610764/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-04T14:57:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rkwniPl12iT6s4KwlRZBmRvz6UCnwXCfCsuFANFLEwZ8+aQJrllfzLF9QNNX/8XFR5+nFEEy7+V3U/c8jB0jBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T12:54:53.823332Z"},"content_sha256":"951cd37b835f0f3d39c8298b64e2b3ae7d21d2a5e07b10a362f7a4e7d8e920ab","schema_version":"1.0","event_id":"sha256:951cd37b835f0f3d39c8298b64e2b3ae7d21d2a5e07b10a362f7a4e7d8e920ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/bundle.json","state_url":"https://pith.science/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/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":"2026-07-05T12:54:53Z","links":{"resolver":"https://pith.science/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y","bundle":"https://pith.science/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/bundle.json","state":"https://pith.science/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TNTSDPPPQGHQRWEMGSGC2OIJ4Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:TNTSDPPPQGHQRWEMGSGC2OIJ4Y","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":"8aaeb7da52648232b94c059b94c48a7cf8ca9cb5e39e82b67e3c26a3a164ddef","cross_cats_sorted":["math.DS"],"license":"","primary_cat":"math.AP","submitted_at":"2006-10-25T19:12:57Z","title_canon_sha256":"9806cfe5b1961b10c929dc6c7b7b304cb4aeaf926e49c482de139d869b42dca9"},"schema_version":"1.0","source":{"id":"math/0610764","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610764","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610764v1","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610764","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_12","alias_value":"TNTSDPPPQGHQ","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_16","alias_value":"TNTSDPPPQGHQRWEM","created_at":"2026-07-04T14:57:10Z"},{"alias_kind":"pith_short_8","alias_value":"TNTSDPPP","created_at":"2026-07-04T14:57:10Z"}],"graph_snapshots":[{"event_id":"sha256:951cd37b835f0f3d39c8298b64e2b3ae7d21d2a5e07b10a362f7a4e7d8e920ab","target":"graph","created_at":"2026-07-04T14:57:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0610764/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In 1992, P. Pol\\'{a}\\v{c}ik\\cite{P2} showed that one could linearly imbed any vector fields into a scalar semi-linear parabolic equation on $\\Omega$ with Neumann boundary condition provided that there exists a smooth vector field $\\Phi=(\\phi_{1},...,\\phi_{n}) $ on $\\bar{\\Omega}$ such that \\[ \\left\\{\\begin{array} [c]{l}% \\operatorname*{rank}(\\Phi(x) ,\\partial_{1}\\Phi(x) ,...,\\partial_{n}\\Phi(x)) =n\\text{for all}x\\in\\bar{\\Omega}, \\frac{\\partial\\Phi}{\\partial\\nu}=0\\text{on}\\partial\\Omega\\text{.}% \\end{array} \\right. \\] In this short note, we give a classification of all the domains on which one m","authors_text":"Fengbo Hang, Huiqiang Jiang","cross_cats":["math.DS"],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2006-10-25T19:12:57Z","title":"A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610764","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:e54773a52f562d80a38fca900f971e07551cb6127b6aa6b579d6be7a8ef6225a","target":"record","created_at":"2026-07-04T14:57:10Z","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":"8aaeb7da52648232b94c059b94c48a7cf8ca9cb5e39e82b67e3c26a3a164ddef","cross_cats_sorted":["math.DS"],"license":"","primary_cat":"math.AP","submitted_at":"2006-10-25T19:12:57Z","title_canon_sha256":"9806cfe5b1961b10c929dc6c7b7b304cb4aeaf926e49c482de139d869b42dca9"},"schema_version":"1.0","source":{"id":"math/0610764","kind":"arxiv","version":1}},"canonical_sha256":"9b6721bdef818f08d88c348c2d3909e60ef6e197263ee064fc64efb58b4430ea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b6721bdef818f08d88c348c2d3909e60ef6e197263ee064fc64efb58b4430ea","first_computed_at":"2026-07-04T14:57:10.374341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:57:10.374341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kAuU7BOoPoJBMhQ1qD4R9YJg6sq2kd3ahmdiNu65iaXNC+hQB1DLlYSM9ZhShctFb8qpOKM25fg9Irx0pfXxDQ==","signature_status":"signed_v1","signed_at":"2026-07-04T14:57:10.374689Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610764","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e54773a52f562d80a38fca900f971e07551cb6127b6aa6b579d6be7a8ef6225a","sha256:951cd37b835f0f3d39c8298b64e2b3ae7d21d2a5e07b10a362f7a4e7d8e920ab"],"state_sha256":"ba76b111d12c9acb1f5fec61466d0c672bf55dd7d6986d334028111528c4a822"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KJl1K5wiW5MtEF4M0dIxXhTQIrsj/urEQ2C6ttbMxAbLsS9bweBBi4Ft8zzSM1wd2WsC43PtfcNanW3uLQw4CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T12:54:53.827517Z","bundle_sha256":"164f00e05cde0d465aeb01f5673f0944b8ee0362024ba5bb0c20ed2d43a7bd0d"}}