{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:2F6EN44J44QUZJ4EB74THSMLOK","short_pith_number":"pith:2F6EN44J","canonical_record":{"source":{"id":"1507.08880","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-31T13:56:17Z","cross_cats_sorted":[],"title_canon_sha256":"7a2f393aab43177dd6a42d13c351f81c9e4cfe8c1d25a1f5b61d5024b2c7d8d8","abstract_canon_sha256":"927fe6adac9b6a87a3e8a4c32a10e6cbd665e6fb825a2a999ba262133b6ae668"},"schema_version":"1.0"},"canonical_sha256":"d17c46f389e7214ca7840ff933c98b72b7aa51f9543b52400b0044e08c3c1fc7","source":{"kind":"arxiv","id":"1507.08880","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08880","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08880v2","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08880","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"pith_short_12","alias_value":"2F6EN44J44QU","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"2F6EN44J44QUZJ4E","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"2F6EN44J","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:2F6EN44J44QUZJ4EB74THSMLOK","target":"record","payload":{"canonical_record":{"source":{"id":"1507.08880","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-31T13:56:17Z","cross_cats_sorted":[],"title_canon_sha256":"7a2f393aab43177dd6a42d13c351f81c9e4cfe8c1d25a1f5b61d5024b2c7d8d8","abstract_canon_sha256":"927fe6adac9b6a87a3e8a4c32a10e6cbd665e6fb825a2a999ba262133b6ae668"},"schema_version":"1.0"},"canonical_sha256":"d17c46f389e7214ca7840ff933c98b72b7aa51f9543b52400b0044e08c3c1fc7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:03.158450Z","signature_b64":"tN5IlPVb3ZE/EPveeP1ty77TP2L6NvbVLXRKlSQSJbM0nU6VKcIk/V915xhMQ3ox7vSWqQeZvnbLHofidpTNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d17c46f389e7214ca7840ff933c98b72b7aa51f9543b52400b0044e08c3c1fc7","last_reissued_at":"2026-05-17T23:53:03.157738Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:03.157738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.08880","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":"2026-05-17T23:53:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C/bDyNl8AzHokc9i65ykmVWXWGk277/mSyZYQZhrepmQjWjQChtTPQrDGlUqRwKWF2KMm2Txtej6aOjoyVssCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T13:37:36.971111Z"},"content_sha256":"a0ad378e7734b907e32e61f7f61362cf15e6f96b8a34ee129bb657b86c861f44","schema_version":"1.0","event_id":"sha256:a0ad378e7734b907e32e61f7f61362cf15e6f96b8a34ee129bb657b86c861f44"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:2F6EN44J44QUZJ4EB74THSMLOK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Global Hypoellipticity for First-Order Operators on Closed Smooth Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexandre Kirilov, Fernando de \\'Avila Silva, Todor Gramchev","submitted_at":"2015-07-31T13:56:17Z","abstract_excerpt":"The main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \\in \\mathbb{T} \\times M$, $\\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator on $M$, smoothly depending on the periodic variable $t$. In the case of separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we give necessary and sufficient conditions for the global hypoellipticity of $L$. In particular, we show that, under suitable conditions, the famou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08880","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":"2026-05-17T23:53:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y7rnE1FPYATI7L4bKDd8MlJFtzKrtB1numRLp7HiG6KhXSO36s8eUCjQ4IHFyp0OJ7g38bOn9ql+6nuHZOE5AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T13:37:36.971489Z"},"content_sha256":"a32572a85fc63a33d326bef99f035e28f35c0b534c714ea380c975bc25b3cf6d","schema_version":"1.0","event_id":"sha256:a32572a85fc63a33d326bef99f035e28f35c0b534c714ea380c975bc25b3cf6d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2F6EN44J44QUZJ4EB74THSMLOK/bundle.json","state_url":"https://pith.science/pith/2F6EN44J44QUZJ4EB74THSMLOK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2F6EN44J44QUZJ4EB74THSMLOK/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":"2026-06-23T13:37:36Z","links":{"resolver":"https://pith.science/pith/2F6EN44J44QUZJ4EB74THSMLOK","bundle":"https://pith.science/pith/2F6EN44J44QUZJ4EB74THSMLOK/bundle.json","state":"https://pith.science/pith/2F6EN44J44QUZJ4EB74THSMLOK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2F6EN44J44QUZJ4EB74THSMLOK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:2F6EN44J44QUZJ4EB74THSMLOK","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":"927fe6adac9b6a87a3e8a4c32a10e6cbd665e6fb825a2a999ba262133b6ae668","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-31T13:56:17Z","title_canon_sha256":"7a2f393aab43177dd6a42d13c351f81c9e4cfe8c1d25a1f5b61d5024b2c7d8d8"},"schema_version":"1.0","source":{"id":"1507.08880","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08880","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08880v2","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08880","created_at":"2026-05-17T23:53:03Z"},{"alias_kind":"pith_short_12","alias_value":"2F6EN44J44QU","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"2F6EN44J44QUZJ4E","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"2F6EN44J","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:a32572a85fc63a33d326bef99f035e28f35c0b534c714ea380c975bc25b3cf6d","target":"graph","created_at":"2026-05-17T23:53:03Z","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 main goal of this paper is to address global hypoellipticity issues for the following class of operators: $L = D_t + C(t,x,D_x)$, where $(t,x) \\in \\mathbb{T} \\times M$, $\\mathbb{T}$ is the one-dimensional torus, $M$ is a closed manifold and $C(t,x,D_x)$ is a first order pseudo-differential operator on $M$, smoothly depending on the periodic variable $t$. In the case of separation of variables, namely, $C(t,x,D_x) = a(t)p(x,D_x)+ib(t)q(x,D_x)$, we give necessary and sufficient conditions for the global hypoellipticity of $L$. In particular, we show that, under suitable conditions, the famou","authors_text":"Alexandre Kirilov, Fernando de \\'Avila Silva, Todor Gramchev","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-31T13:56:17Z","title":"Global Hypoellipticity for First-Order Operators on Closed Smooth Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08880","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:a0ad378e7734b907e32e61f7f61362cf15e6f96b8a34ee129bb657b86c861f44","target":"record","created_at":"2026-05-17T23:53:03Z","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":"927fe6adac9b6a87a3e8a4c32a10e6cbd665e6fb825a2a999ba262133b6ae668","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-31T13:56:17Z","title_canon_sha256":"7a2f393aab43177dd6a42d13c351f81c9e4cfe8c1d25a1f5b61d5024b2c7d8d8"},"schema_version":"1.0","source":{"id":"1507.08880","kind":"arxiv","version":2}},"canonical_sha256":"d17c46f389e7214ca7840ff933c98b72b7aa51f9543b52400b0044e08c3c1fc7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d17c46f389e7214ca7840ff933c98b72b7aa51f9543b52400b0044e08c3c1fc7","first_computed_at":"2026-05-17T23:53:03.157738Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:03.157738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tN5IlPVb3ZE/EPveeP1ty77TP2L6NvbVLXRKlSQSJbM0nU6VKcIk/V915xhMQ3ox7vSWqQeZvnbLHofidpTNBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:03.158450Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08880","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0ad378e7734b907e32e61f7f61362cf15e6f96b8a34ee129bb657b86c861f44","sha256:a32572a85fc63a33d326bef99f035e28f35c0b534c714ea380c975bc25b3cf6d"],"state_sha256":"aaebe9524201bc37550aedb92a7c9907e88da775872124996bdf3d3a8abaa96b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"io7b37eRw7WKU/qck2/HUCTz23WGNAq72SorzIELslbURepHFhaoS1MxC5CO2Z9yqOv5jA4DYiS4RjuYICXFAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T13:37:36.973426Z","bundle_sha256":"1ad7687de719e5c5f2ee7d54c915b93d656d1c5f94ce9d7d08919be587bdf3d0"}}