{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:WX67LNE3RSUBPXMDT5JP5PAUVK","short_pith_number":"pith:WX67LNE3","canonical_record":{"source":{"id":"1809.00721","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-03T21:19:24Z","cross_cats_sorted":[],"title_canon_sha256":"c140c4cb8810a54dbbdf39e8445550a912078f49c67776027e7075a53d4b5c8a","abstract_canon_sha256":"f2c2b514d7800dabd647b50741a590e189abf4d388211f7162703c870a56fba7"},"schema_version":"1.0"},"canonical_sha256":"b5fdf5b49b8ca817dd839f52febc14aaa6474ae8fec60f092e8a28bded1d60ef","source":{"kind":"arxiv","id":"1809.00721","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.00721","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"arxiv_version","alias_value":"1809.00721v1","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00721","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"pith_short_12","alias_value":"WX67LNE3RSUB","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WX67LNE3RSUBPXMD","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WX67LNE3","created_at":"2026-05-18T12:33:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:WX67LNE3RSUBPXMDT5JP5PAUVK","target":"record","payload":{"canonical_record":{"source":{"id":"1809.00721","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-03T21:19:24Z","cross_cats_sorted":[],"title_canon_sha256":"c140c4cb8810a54dbbdf39e8445550a912078f49c67776027e7075a53d4b5c8a","abstract_canon_sha256":"f2c2b514d7800dabd647b50741a590e189abf4d388211f7162703c870a56fba7"},"schema_version":"1.0"},"canonical_sha256":"b5fdf5b49b8ca817dd839f52febc14aaa6474ae8fec60f092e8a28bded1d60ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:32.655594Z","signature_b64":"7gxrCepdcmZDRDIVBiy1e7nL44VWIxGYnxB1Xk5Nob/JwD3d7KyXeQyin7tCCmxojXa0+Q1twUQ2vR0R1f8rDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5fdf5b49b8ca817dd839f52febc14aaa6474ae8fec60f092e8a28bded1d60ef","last_reissued_at":"2026-05-18T00:06:32.655177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:32.655177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.00721","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":"2026-05-18T00:06:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"airLASYbDaE19Bq6sEtBEkBriOB1xh1jcxk2sTwxe4YVebyjgU3XRmismPdtyagjbhOS9nFPFx67n5Ytjd2DCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:58:21.953867Z"},"content_sha256":"d4386c046fa9f5f57cac43f58de5f212e9d8a90f7ac0718841f738606d1051e5","schema_version":"1.0","event_id":"sha256:d4386c046fa9f5f57cac43f58de5f212e9d8a90f7ac0718841f738606d1051e5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:WX67LNE3RSUBPXMDT5JP5PAUVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazuo Yamazaki","submitted_at":"2018-09-03T21:19:24Z","abstract_excerpt":"Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of \\cite{EM01, R04} on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-li"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00721","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":"2026-05-18T00:06:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z1pnaYkpLVDV8HIvaTUZzhXAENNsbBl18wLw80dgBLpU5cKBWbY/DBmjCTG5Jsy3v8+F8489aXX1039cpJs4CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T20:58:21.954549Z"},"content_sha256":"36892ff816f2f03097c97249a66397b1fe2ba7484d67173047301ed775a05e37","schema_version":"1.0","event_id":"sha256:36892ff816f2f03097c97249a66397b1fe2ba7484d67173047301ed775a05e37"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/bundle.json","state_url":"https://pith.science/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/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":"2026-06-06T20:58:21Z","links":{"resolver":"https://pith.science/pith/WX67LNE3RSUBPXMDT5JP5PAUVK","bundle":"https://pith.science/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/bundle.json","state":"https://pith.science/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WX67LNE3RSUBPXMDT5JP5PAUVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:WX67LNE3RSUBPXMDT5JP5PAUVK","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":"f2c2b514d7800dabd647b50741a590e189abf4d388211f7162703c870a56fba7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-03T21:19:24Z","title_canon_sha256":"c140c4cb8810a54dbbdf39e8445550a912078f49c67776027e7075a53d4b5c8a"},"schema_version":"1.0","source":{"id":"1809.00721","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.00721","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"arxiv_version","alias_value":"1809.00721v1","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00721","created_at":"2026-05-18T00:06:32Z"},{"alias_kind":"pith_short_12","alias_value":"WX67LNE3RSUB","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_16","alias_value":"WX67LNE3RSUBPXMD","created_at":"2026-05-18T12:33:01Z"},{"alias_kind":"pith_short_8","alias_value":"WX67LNE3","created_at":"2026-05-18T12:33:01Z"}],"graph_snapshots":[{"event_id":"sha256:36892ff816f2f03097c97249a66397b1fe2ba7484d67173047301ed775a05e37","target":"graph","created_at":"2026-05-18T00:06:32Z","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":"Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of \\cite{EM01, R04} on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-li","authors_text":"Kazuo Yamazaki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-03T21:19:24Z","title":"Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00721","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:d4386c046fa9f5f57cac43f58de5f212e9d8a90f7ac0718841f738606d1051e5","target":"record","created_at":"2026-05-18T00:06:32Z","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":"f2c2b514d7800dabd647b50741a590e189abf4d388211f7162703c870a56fba7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-09-03T21:19:24Z","title_canon_sha256":"c140c4cb8810a54dbbdf39e8445550a912078f49c67776027e7075a53d4b5c8a"},"schema_version":"1.0","source":{"id":"1809.00721","kind":"arxiv","version":1}},"canonical_sha256":"b5fdf5b49b8ca817dd839f52febc14aaa6474ae8fec60f092e8a28bded1d60ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5fdf5b49b8ca817dd839f52febc14aaa6474ae8fec60f092e8a28bded1d60ef","first_computed_at":"2026-05-18T00:06:32.655177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:32.655177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7gxrCepdcmZDRDIVBiy1e7nL44VWIxGYnxB1Xk5Nob/JwD3d7KyXeQyin7tCCmxojXa0+Q1twUQ2vR0R1f8rDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:32.655594Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.00721","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4386c046fa9f5f57cac43f58de5f212e9d8a90f7ac0718841f738606d1051e5","sha256:36892ff816f2f03097c97249a66397b1fe2ba7484d67173047301ed775a05e37"],"state_sha256":"a76cbb732f4a029fed8577ace59358ffab5ede6cb48700fc232479a27dc35596"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uH/LJIW8xafZZa7Vck4Xed0tuPHXN89+TiO8yMsTRocblwkky6yBtZx884MSxj2lkwcqXIlVCKkaDmOjiWa+AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T20:58:21.958432Z","bundle_sha256":"98b6b3470459cace5ac23c51f3cfd5ebfee4250d85a57a9355c82c25941b0894"}}