{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:3UXMCRIEU2SR4S2E3MHPASNOYR","short_pith_number":"pith:3UXMCRIE","canonical_record":{"source":{"id":"1811.05780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-13T08:03:06Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a9065566693f000c31379ecf27d80e11a0fbf95981632be3daa5fec24d9c287e","abstract_canon_sha256":"fbad8428d72593930bbac43b3e694f22b3a1229ac2bb5b0ff2f14b1aca3ba746"},"schema_version":"1.0"},"canonical_sha256":"dd2ec14504a6a51e4b44db0ef049aec44c30d3bac891be3bfd03996b83875201","source":{"kind":"arxiv","id":"1811.05780","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05780","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05780v1","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05780","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"pith_short_12","alias_value":"3UXMCRIEU2SR","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"3UXMCRIEU2SR4S2E","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"3UXMCRIE","created_at":"2026-05-18T12:32:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:3UXMCRIEU2SR4S2E3MHPASNOYR","target":"record","payload":{"canonical_record":{"source":{"id":"1811.05780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-13T08:03:06Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a9065566693f000c31379ecf27d80e11a0fbf95981632be3daa5fec24d9c287e","abstract_canon_sha256":"fbad8428d72593930bbac43b3e694f22b3a1229ac2bb5b0ff2f14b1aca3ba746"},"schema_version":"1.0"},"canonical_sha256":"dd2ec14504a6a51e4b44db0ef049aec44c30d3bac891be3bfd03996b83875201","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:42.109148Z","signature_b64":"tyjnLiDAaylzzPwAd+MbCzNAb4MMtDo0uHrp9crGCAhuhAq2839xleJ42fTddyi+9PIs6UBIfwB2uVXhGXKKDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd2ec14504a6a51e4b44db0ef049aec44c30d3bac891be3bfd03996b83875201","last_reissued_at":"2026-05-18T00:00:42.108686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:42.108686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.05780","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":"2026-05-18T00:00:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D1n5hNWopaa1ECFWrGV1N0R9usU5BpjUXqnEgSzBGCng4IXHS2hZhWNBEUdigkFCEcg+myINrrx6vsLj8IynAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:43:15.435918Z"},"content_sha256":"40e5dd778fc20658470fb83c4bb04af368c01a15520744b65f6340552728f982","schema_version":"1.0","event_id":"sha256:40e5dd778fc20658470fb83c4bb04af368c01a15520744b65f6340552728f982"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:3UXMCRIEU2SR4S2E3MHPASNOYR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Observability and Control Property for a Singular Heat Equation with Variable Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Shumin Li, Xue Qin","submitted_at":"2018-11-13T08:03:06Z","abstract_excerpt":"The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential:\\,$\\partial_tu(x,t)-{\\rm div}(p(x)\\nabla u(x,t))-({\\mu}/{|x|^2})u(x,t)=f(x,t).$ Here $\\mu$ is a real constant . It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when $0\\leq{\\mu\\leq p_1(n-2)^2/4}$, and in this paper, we mainly consider the case $0\\leq\\mu<({ p_1^2}/{ p_2})(n-2)^2/4$ , where $ p_1,p_2$ are two positive constants which satisfy:\\, $ 0< p_1\\leq p(x)\\leq p_2 , \\forall"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05780","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":"2026-05-18T00:00:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mNKZRlPuOjtZxCpXuGwamC0njsFKAwUz2XHJZvyJ1lyme5qBV7vsFzHyYEyJCZzy6aSRKdxS2VXZpdwdg7EgBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:43:15.436400Z"},"content_sha256":"a2563f4ad88e1487102938cb231f9c58ddfddff503b122e9a497e03c98b39a70","schema_version":"1.0","event_id":"sha256:a2563f4ad88e1487102938cb231f9c58ddfddff503b122e9a497e03c98b39a70"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/bundle.json","state_url":"https://pith.science/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/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":"2026-06-25T23:43:15Z","links":{"resolver":"https://pith.science/pith/3UXMCRIEU2SR4S2E3MHPASNOYR","bundle":"https://pith.science/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/bundle.json","state":"https://pith.science/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3UXMCRIEU2SR4S2E3MHPASNOYR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3UXMCRIEU2SR4S2E3MHPASNOYR","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":"fbad8428d72593930bbac43b3e694f22b3a1229ac2bb5b0ff2f14b1aca3ba746","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-13T08:03:06Z","title_canon_sha256":"a9065566693f000c31379ecf27d80e11a0fbf95981632be3daa5fec24d9c287e"},"schema_version":"1.0","source":{"id":"1811.05780","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05780","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05780v1","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05780","created_at":"2026-05-18T00:00:42Z"},{"alias_kind":"pith_short_12","alias_value":"3UXMCRIEU2SR","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"3UXMCRIEU2SR4S2E","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"3UXMCRIE","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:a2563f4ad88e1487102938cb231f9c58ddfddff503b122e9a497e03c98b39a70","target":"graph","created_at":"2026-05-18T00:00:42Z","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 goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential:\\,$\\partial_tu(x,t)-{\\rm div}(p(x)\\nabla u(x,t))-({\\mu}/{|x|^2})u(x,t)=f(x,t).$ Here $\\mu$ is a real constant . It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when $0\\leq{\\mu\\leq p_1(n-2)^2/4}$, and in this paper, we mainly consider the case $0\\leq\\mu<({ p_1^2}/{ p_2})(n-2)^2/4$ , where $ p_1,p_2$ are two positive constants which satisfy:\\, $ 0< p_1\\leq p(x)\\leq p_2 , \\forall","authors_text":"Shumin Li, Xue Qin","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-13T08:03:06Z","title":"Observability and Control Property for a Singular Heat Equation with Variable Coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05780","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":"sha256:40e5dd778fc20658470fb83c4bb04af368c01a15520744b65f6340552728f982","target":"record","created_at":"2026-05-18T00:00:42Z","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":"fbad8428d72593930bbac43b3e694f22b3a1229ac2bb5b0ff2f14b1aca3ba746","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-13T08:03:06Z","title_canon_sha256":"a9065566693f000c31379ecf27d80e11a0fbf95981632be3daa5fec24d9c287e"},"schema_version":"1.0","source":{"id":"1811.05780","kind":"arxiv","version":1}},"canonical_sha256":"dd2ec14504a6a51e4b44db0ef049aec44c30d3bac891be3bfd03996b83875201","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd2ec14504a6a51e4b44db0ef049aec44c30d3bac891be3bfd03996b83875201","first_computed_at":"2026-05-18T00:00:42.108686Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:00:42.108686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tyjnLiDAaylzzPwAd+MbCzNAb4MMtDo0uHrp9crGCAhuhAq2839xleJ42fTddyi+9PIs6UBIfwB2uVXhGXKKDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:00:42.109148Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.05780","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40e5dd778fc20658470fb83c4bb04af368c01a15520744b65f6340552728f982","sha256:a2563f4ad88e1487102938cb231f9c58ddfddff503b122e9a497e03c98b39a70"],"state_sha256":"24eb128f2c665d7a9ead88b6c4af85740308b9cb112ce51f9bac05763196cee7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7s8QvEyW7VR1nnlzPv59rabPJeGKld7m+7FxkZJAvfaQ3C01HNOQP5z5RMkqoeAlV4luwVoVzt9BuTT/ccqXBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:43:15.439044Z","bundle_sha256":"0c804488d2b8a9631075d5ce442d4f1647777f302de9484fd458008def1dd98e"}}