{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6XHN2LRLKSMXNODQKSY224LSPE","short_pith_number":"pith:6XHN2LRL","canonical_record":{"source":{"id":"1804.09935","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-26T08:29:26Z","cross_cats_sorted":[],"title_canon_sha256":"2b280bb93261a976c7ca9dfdd8cd23add86f6f0e570e9ef7c6bb112a7deee012","abstract_canon_sha256":"cb943896ec63036ae85b3709ca89cea73f712362f88e83d75b31512828391269"},"schema_version":"1.0"},"canonical_sha256":"f5cedd2e2b549976b87054b1ad7172792deb5e1c9cff5d033d7805ab2e1b194d","source":{"kind":"arxiv","id":"1804.09935","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.09935","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"arxiv_version","alias_value":"1804.09935v1","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.09935","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"pith_short_12","alias_value":"6XHN2LRLKSMX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6XHN2LRLKSMXNODQ","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6XHN2LRL","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6XHN2LRLKSMXNODQKSY224LSPE","target":"record","payload":{"canonical_record":{"source":{"id":"1804.09935","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-26T08:29:26Z","cross_cats_sorted":[],"title_canon_sha256":"2b280bb93261a976c7ca9dfdd8cd23add86f6f0e570e9ef7c6bb112a7deee012","abstract_canon_sha256":"cb943896ec63036ae85b3709ca89cea73f712362f88e83d75b31512828391269"},"schema_version":"1.0"},"canonical_sha256":"f5cedd2e2b549976b87054b1ad7172792deb5e1c9cff5d033d7805ab2e1b194d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:25.716656Z","signature_b64":"hUpds1gSDfwRBY+nJj98yn6V2MN7AW6E5IV1mZdQRdUTrqBGI0qXcIeRpsG89GE8n4EEHdpD4Cm6teEmHLybCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5cedd2e2b549976b87054b1ad7172792deb5e1c9cff5d033d7805ab2e1b194d","last_reissued_at":"2026-05-18T00:17:25.715851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:25.715851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.09935","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":"2026-05-18T00:17:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zFECpoSrvxvyAOf4LdEwUg1FAlHcSgmCtbadGe0FxqgLl193w9vL07HO0h9H/w56tNr/2NXmrgcmSVC215hUDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T11:42:36.590018Z"},"content_sha256":"d4626448bddedec7f7a6a32e7929ddd5b51c8044abb6e604f271d6373af93f24","schema_version":"1.0","event_id":"sha256:d4626448bddedec7f7a6a32e7929ddd5b51c8044abb6e604f271d6373af93f24"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6XHN2LRLKSMXNODQKSY224LSPE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Null Controllability of the Incompressible Stokes Equations in a 2-D Channel Using Normal Boundary Control","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Debanjana Mitra, Michael Renardy, Shirshendu Chowdhury","submitted_at":"2018-04-26T08:29:26Z","abstract_excerpt":"In this paper, we consider the Stokes equations in a two-dimen- sional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Muntz-Szasz Theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09935","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":"2026-05-18T00:17:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"auY8kLzvbrZ2sDRuevQXoDjbh7VUD1am0kqIrssbwxtMx+CvyiWbDwgFiykpUzuDGp/qutSAnnOa9iGMbDoqBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T11:42:36.590361Z"},"content_sha256":"0360857a26ed25346dfb919829ca47af1c5b20520521408cf162cb5f48ae7b25","schema_version":"1.0","event_id":"sha256:0360857a26ed25346dfb919829ca47af1c5b20520521408cf162cb5f48ae7b25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6XHN2LRLKSMXNODQKSY224LSPE/bundle.json","state_url":"https://pith.science/pith/6XHN2LRLKSMXNODQKSY224LSPE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6XHN2LRLKSMXNODQKSY224LSPE/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":"2026-07-04T11:42:36Z","links":{"resolver":"https://pith.science/pith/6XHN2LRLKSMXNODQKSY224LSPE","bundle":"https://pith.science/pith/6XHN2LRLKSMXNODQKSY224LSPE/bundle.json","state":"https://pith.science/pith/6XHN2LRLKSMXNODQKSY224LSPE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6XHN2LRLKSMXNODQKSY224LSPE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6XHN2LRLKSMXNODQKSY224LSPE","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":"cb943896ec63036ae85b3709ca89cea73f712362f88e83d75b31512828391269","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-26T08:29:26Z","title_canon_sha256":"2b280bb93261a976c7ca9dfdd8cd23add86f6f0e570e9ef7c6bb112a7deee012"},"schema_version":"1.0","source":{"id":"1804.09935","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.09935","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"arxiv_version","alias_value":"1804.09935v1","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.09935","created_at":"2026-05-18T00:17:25Z"},{"alias_kind":"pith_short_12","alias_value":"6XHN2LRLKSMX","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6XHN2LRLKSMXNODQ","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6XHN2LRL","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:0360857a26ed25346dfb919829ca47af1c5b20520521408cf162cb5f48ae7b25","target":"graph","created_at":"2026-05-18T00:17:25Z","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":"In this paper, we consider the Stokes equations in a two-dimen- sional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Muntz-Szasz Theorem.","authors_text":"Debanjana Mitra, Michael Renardy, Shirshendu Chowdhury","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-26T08:29:26Z","title":"Null Controllability of the Incompressible Stokes Equations in a 2-D Channel Using Normal Boundary Control"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09935","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:d4626448bddedec7f7a6a32e7929ddd5b51c8044abb6e604f271d6373af93f24","target":"record","created_at":"2026-05-18T00:17:25Z","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":"cb943896ec63036ae85b3709ca89cea73f712362f88e83d75b31512828391269","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-26T08:29:26Z","title_canon_sha256":"2b280bb93261a976c7ca9dfdd8cd23add86f6f0e570e9ef7c6bb112a7deee012"},"schema_version":"1.0","source":{"id":"1804.09935","kind":"arxiv","version":1}},"canonical_sha256":"f5cedd2e2b549976b87054b1ad7172792deb5e1c9cff5d033d7805ab2e1b194d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f5cedd2e2b549976b87054b1ad7172792deb5e1c9cff5d033d7805ab2e1b194d","first_computed_at":"2026-05-18T00:17:25.715851Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:25.715851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hUpds1gSDfwRBY+nJj98yn6V2MN7AW6E5IV1mZdQRdUTrqBGI0qXcIeRpsG89GE8n4EEHdpD4Cm6teEmHLybCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:25.716656Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.09935","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d4626448bddedec7f7a6a32e7929ddd5b51c8044abb6e604f271d6373af93f24","sha256:0360857a26ed25346dfb919829ca47af1c5b20520521408cf162cb5f48ae7b25"],"state_sha256":"430c011be911f1c7fe915b95fc6a3f678c5b6b95c92bce1750c83917831de513"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3OkfZpVZhxQ80AJ1SBsBS4qcGbA77Rsw6aXnqke9e+/xhr4nYTs2EK8GvF0zsCkj520biOaxFPrRE3R7YZQbAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T11:42:36.592253Z","bundle_sha256":"197e7f611a1fd6b6b353dbcc6e2556589f2e623e305b8efbc04b2cae015cc84a"}}