{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:DJDYATFWYFRV5H6MKG4Y4FU75X","short_pith_number":"pith:DJDYATFW","canonical_record":{"source":{"id":"1205.5043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:19:26Z","cross_cats_sorted":[],"title_canon_sha256":"598843ab8bd88a3c4f76fe5054b9d48abc38ca94eafaab10c813bea02bcda879","abstract_canon_sha256":"38345d53a22c2aea815a95492a27e48dd63579b11c66d1ea27a98548015aea78"},"schema_version":"1.0"},"canonical_sha256":"1a47804cb6c1635e9fcc51b98e169fedd0fec4b1d4ce89063a436216dbe3b695","source":{"kind":"arxiv","id":"1205.5043","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.5043","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"arxiv_version","alias_value":"1205.5043v1","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5043","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"pith_short_12","alias_value":"DJDYATFWYFRV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DJDYATFWYFRV5H6M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DJDYATFW","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:DJDYATFWYFRV5H6MKG4Y4FU75X","target":"record","payload":{"canonical_record":{"source":{"id":"1205.5043","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:19:26Z","cross_cats_sorted":[],"title_canon_sha256":"598843ab8bd88a3c4f76fe5054b9d48abc38ca94eafaab10c813bea02bcda879","abstract_canon_sha256":"38345d53a22c2aea815a95492a27e48dd63579b11c66d1ea27a98548015aea78"},"schema_version":"1.0"},"canonical_sha256":"1a47804cb6c1635e9fcc51b98e169fedd0fec4b1d4ce89063a436216dbe3b695","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:02.365178Z","signature_b64":"rJRQWUkuzP+EhqiKqBFZ4/vnkyNT029VZXT2BMdaAmjrkuw/RSoUnYeu64ZwG9UIRCOicoA1s4BsXTa1j9ASAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a47804cb6c1635e9fcc51b98e169fedd0fec4b1d4ce89063a436216dbe3b695","last_reissued_at":"2026-05-18T03:55:02.364495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:02.364495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.5043","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":"2026-05-18T03:55:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gH9xDAcnoJop22jXzSp2vSgtfyO4maqLJmoxRfClfJDM51uch49iodz9PFsSLPusTdu7I2Zn0hPiLW8hUJ2yCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:41:38.119932Z"},"content_sha256":"84125b38c19c2d1b32edfa3fca47dc83b1280bf989eea8e135c161bd18218b18","schema_version":"1.0","event_id":"sha256:84125b38c19c2d1b32edfa3fca47dc83b1280bf989eea8e135c161bd18218b18"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:DJDYATFWYFRV5H6MKG4Y4FU75X","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic expansions for anisotropic heat kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrique Zuazua, Liviu I. Ignat","submitted_at":"2012-05-22T20:19:26Z","abstract_excerpt":"We obtain the asymptotic expansion of the solutions of some anisotropic heat equations when the initial data belong to polynomially weighted Lp-spaces. We mainly address two model examples. In the first one, the diffusivity is of order two in some variables but higher in the other ones. In the second one we consider the heat equation on the Heisenberg group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5043","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":"2026-05-18T03:55:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k6j8XXh8tVndnJXgft4OXDhM/PCrxlTcs5EEW1qoRUWpwkPHtui76e20OOeQA98IW5Iir2tFkOvKZNF9yu3LDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T16:41:38.120270Z"},"content_sha256":"abb7d1db823fcc495164658bf596389bdafee661ab601c15261414f0c9909a25","schema_version":"1.0","event_id":"sha256:abb7d1db823fcc495164658bf596389bdafee661ab601c15261414f0c9909a25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/bundle.json","state_url":"https://pith.science/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/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":"2026-06-09T16:41:38Z","links":{"resolver":"https://pith.science/pith/DJDYATFWYFRV5H6MKG4Y4FU75X","bundle":"https://pith.science/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/bundle.json","state":"https://pith.science/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DJDYATFWYFRV5H6MKG4Y4FU75X/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:DJDYATFWYFRV5H6MKG4Y4FU75X","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":"38345d53a22c2aea815a95492a27e48dd63579b11c66d1ea27a98548015aea78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:19:26Z","title_canon_sha256":"598843ab8bd88a3c4f76fe5054b9d48abc38ca94eafaab10c813bea02bcda879"},"schema_version":"1.0","source":{"id":"1205.5043","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.5043","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"arxiv_version","alias_value":"1205.5043v1","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5043","created_at":"2026-05-18T03:55:02Z"},{"alias_kind":"pith_short_12","alias_value":"DJDYATFWYFRV","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"DJDYATFWYFRV5H6M","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"DJDYATFW","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:abb7d1db823fcc495164658bf596389bdafee661ab601c15261414f0c9909a25","target":"graph","created_at":"2026-05-18T03:55:02Z","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":"We obtain the asymptotic expansion of the solutions of some anisotropic heat equations when the initial data belong to polynomially weighted Lp-spaces. We mainly address two model examples. In the first one, the diffusivity is of order two in some variables but higher in the other ones. In the second one we consider the heat equation on the Heisenberg group.","authors_text":"Enrique Zuazua, Liviu I. Ignat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:19:26Z","title":"Asymptotic expansions for anisotropic heat kernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5043","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":"sha256:84125b38c19c2d1b32edfa3fca47dc83b1280bf989eea8e135c161bd18218b18","target":"record","created_at":"2026-05-18T03:55:02Z","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":"38345d53a22c2aea815a95492a27e48dd63579b11c66d1ea27a98548015aea78","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-05-22T20:19:26Z","title_canon_sha256":"598843ab8bd88a3c4f76fe5054b9d48abc38ca94eafaab10c813bea02bcda879"},"schema_version":"1.0","source":{"id":"1205.5043","kind":"arxiv","version":1}},"canonical_sha256":"1a47804cb6c1635e9fcc51b98e169fedd0fec4b1d4ce89063a436216dbe3b695","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1a47804cb6c1635e9fcc51b98e169fedd0fec4b1d4ce89063a436216dbe3b695","first_computed_at":"2026-05-18T03:55:02.364495Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:02.364495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rJRQWUkuzP+EhqiKqBFZ4/vnkyNT029VZXT2BMdaAmjrkuw/RSoUnYeu64ZwG9UIRCOicoA1s4BsXTa1j9ASAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:02.365178Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.5043","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84125b38c19c2d1b32edfa3fca47dc83b1280bf989eea8e135c161bd18218b18","sha256:abb7d1db823fcc495164658bf596389bdafee661ab601c15261414f0c9909a25"],"state_sha256":"411ef2f44200ab88f2dc28bf51cfb609f95dd05f290b2b1d8c46707385e3108f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ySDJW5XykPBpMUYYqP2vmmUwPPh5Q5BJ1qwW9fMVxqZGcSajlbRcgD/+wn42YqTTu9sQHmRmbgQkwbK3f9xrDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T16:41:38.122552Z","bundle_sha256":"4fb0027058425af1c3c21fd7253d5f3872ae6ba97640d0fe39e93ed6fa2af8e9"}}