{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:REHMIIR3XC2JSDTLFCWAFNGACK","short_pith_number":"pith:REHMIIR3","canonical_record":{"source":{"id":"1211.4408","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-19T13:31:53Z","cross_cats_sorted":[],"title_canon_sha256":"0de1fe53947e0def4c74b74f6ce94079b0a880a0de2d08c763b44d888f6792ce","abstract_canon_sha256":"3a82dd740d1f2ed9235f0b4d1df4504b2fe66b5b34a9d67c6cc79abdf7bf9296"},"schema_version":"1.0"},"canonical_sha256":"890ec4223bb8b4990e6b28ac02b4c012b7d5b97d1030c089ab07a3a646f562bc","source":{"kind":"arxiv","id":"1211.4408","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.4408","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"1211.4408v2","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4408","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"REHMIIR3XC2J","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"REHMIIR3XC2JSDTL","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"REHMIIR3","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:REHMIIR3XC2JSDTLFCWAFNGACK","target":"record","payload":{"canonical_record":{"source":{"id":"1211.4408","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-19T13:31:53Z","cross_cats_sorted":[],"title_canon_sha256":"0de1fe53947e0def4c74b74f6ce94079b0a880a0de2d08c763b44d888f6792ce","abstract_canon_sha256":"3a82dd740d1f2ed9235f0b4d1df4504b2fe66b5b34a9d67c6cc79abdf7bf9296"},"schema_version":"1.0"},"canonical_sha256":"890ec4223bb8b4990e6b28ac02b4c012b7d5b97d1030c089ab07a3a646f562bc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:40.295023Z","signature_b64":"U3Owxo5laxEx1zFCHoAetnENGGoxiEBE/vOzzrM2Enl28q3MB39rtj+bBE/OIoQvuXfqZGv6Y8MPfZ+bh1V3Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"890ec4223bb8b4990e6b28ac02b4c012b7d5b97d1030c089ab07a3a646f562bc","last_reissued_at":"2026-05-18T03:18:40.294308Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:40.294308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.4408","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":"2026-05-18T03:18:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"izv2qKEiaIwOJYIBlqo31A4GyJT/9flgttTUiwgBgb+OLCeB16TPHbS4llJZC9jxASw9N4SSQYIxlTxKVPokAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:08:02.424529Z"},"content_sha256":"63eb529b672b4e849e8771d2ac4aaa60d53f1bbf1fb30b76b53b7e4b070695fa","schema_version":"1.0","event_id":"sha256:63eb529b672b4e849e8771d2ac4aaa60d53f1bbf1fb30b76b53b7e4b070695fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:REHMIIR3XC2JSDTLFCWAFNGACK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Igor Chueshov","submitted_at":"2012-11-19T13:31:53Z","abstract_excerpt":"We consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove (i) the finiteness of the fractal dimension of the corresponding global attractor, (ii) the existence of finite number of determining modes, and (iii) ergodicity of a related random kick model. All these results provide a new information concerning long tim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4408","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":"2026-05-18T03:18:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6fAK5lOS+gZY33nSc09lsQrzHzmUKWKctlffQnuoZlZokzdrNN1ijLPgS1w6S09ttzOOOhiXDVqvoAvz5cvaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T16:08:02.424876Z"},"content_sha256":"0928f5510bee8f23366f51a2bae01347d335fb8d961dc250ad72fc8050518900","schema_version":"1.0","event_id":"sha256:0928f5510bee8f23366f51a2bae01347d335fb8d961dc250ad72fc8050518900"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/REHMIIR3XC2JSDTLFCWAFNGACK/bundle.json","state_url":"https://pith.science/pith/REHMIIR3XC2JSDTLFCWAFNGACK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/REHMIIR3XC2JSDTLFCWAFNGACK/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":"2026-07-03T16:08:02Z","links":{"resolver":"https://pith.science/pith/REHMIIR3XC2JSDTLFCWAFNGACK","bundle":"https://pith.science/pith/REHMIIR3XC2JSDTLFCWAFNGACK/bundle.json","state":"https://pith.science/pith/REHMIIR3XC2JSDTLFCWAFNGACK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/REHMIIR3XC2JSDTLFCWAFNGACK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:REHMIIR3XC2JSDTLFCWAFNGACK","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":"3a82dd740d1f2ed9235f0b4d1df4504b2fe66b5b34a9d67c6cc79abdf7bf9296","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-19T13:31:53Z","title_canon_sha256":"0de1fe53947e0def4c74b74f6ce94079b0a880a0de2d08c763b44d888f6792ce"},"schema_version":"1.0","source":{"id":"1211.4408","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.4408","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"arxiv_version","alias_value":"1211.4408v2","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4408","created_at":"2026-05-18T03:18:40Z"},{"alias_kind":"pith_short_12","alias_value":"REHMIIR3XC2J","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"REHMIIR3XC2JSDTL","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"REHMIIR3","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:0928f5510bee8f23366f51a2bae01347d335fb8d961dc250ad72fc8050518900","target":"graph","created_at":"2026-05-18T03:18:40Z","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 consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove (i) the finiteness of the fractal dimension of the corresponding global attractor, (ii) the existence of finite number of determining modes, and (iii) ergodicity of a related random kick model. All these results provide a new information concerning long tim","authors_text":"Igor Chueshov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-19T13:31:53Z","title":"A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4408","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":"sha256:63eb529b672b4e849e8771d2ac4aaa60d53f1bbf1fb30b76b53b7e4b070695fa","target":"record","created_at":"2026-05-18T03:18:40Z","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":"3a82dd740d1f2ed9235f0b4d1df4504b2fe66b5b34a9d67c6cc79abdf7bf9296","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-19T13:31:53Z","title_canon_sha256":"0de1fe53947e0def4c74b74f6ce94079b0a880a0de2d08c763b44d888f6792ce"},"schema_version":"1.0","source":{"id":"1211.4408","kind":"arxiv","version":2}},"canonical_sha256":"890ec4223bb8b4990e6b28ac02b4c012b7d5b97d1030c089ab07a3a646f562bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"890ec4223bb8b4990e6b28ac02b4c012b7d5b97d1030c089ab07a3a646f562bc","first_computed_at":"2026-05-18T03:18:40.294308Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:40.294308Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U3Owxo5laxEx1zFCHoAetnENGGoxiEBE/vOzzrM2Enl28q3MB39rtj+bBE/OIoQvuXfqZGv6Y8MPfZ+bh1V3Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:40.295023Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.4408","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:63eb529b672b4e849e8771d2ac4aaa60d53f1bbf1fb30b76b53b7e4b070695fa","sha256:0928f5510bee8f23366f51a2bae01347d335fb8d961dc250ad72fc8050518900"],"state_sha256":"be0f7aefef92eaba5cbf9724d6a2721d2cd240efd09577a62bf3a084f6907dc8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fh6EP9On2CHx0+CMKlfL9MCmdn+4r1KU91BgnmqavAtQrGNlEmktQyoGWTyfsWGwwEQOMO6rjZajUaOo6bETCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T16:08:02.426872Z","bundle_sha256":"3d088c0bec248cfac7f2a9ffdd11daa77b62ffe45a88ebf86153692ee93d7f6f"}}