{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:YIEJ3E6Q4CK52M4O2L7SHDYKUH","short_pith_number":"pith:YIEJ3E6Q","canonical_record":{"source":{"id":"1801.00983","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-03T12:58:38Z","cross_cats_sorted":[],"title_canon_sha256":"d9bbcc96d1260d97652d8ea468481ec2d7bc8a56c115b85a705e05909a9846a8","abstract_canon_sha256":"cf50a6e53801c75786b598e2dd8e82c8553eeaf1a6d632307156ca1cd69528d1"},"schema_version":"1.0"},"canonical_sha256":"c2089d93d0e095dd338ed2ff238f0aa1d2d138b021a144532c51dba12fbbde50","source":{"kind":"arxiv","id":"1801.00983","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00983","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00983v2","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00983","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_12","alias_value":"YIEJ3E6Q4CK5","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_16","alias_value":"YIEJ3E6Q4CK52M4O","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_8","alias_value":"YIEJ3E6Q","created_at":"2026-07-05T01:52:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:YIEJ3E6Q4CK52M4O2L7SHDYKUH","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00983","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-03T12:58:38Z","cross_cats_sorted":[],"title_canon_sha256":"d9bbcc96d1260d97652d8ea468481ec2d7bc8a56c115b85a705e05909a9846a8","abstract_canon_sha256":"cf50a6e53801c75786b598e2dd8e82c8553eeaf1a6d632307156ca1cd69528d1"},"schema_version":"1.0"},"canonical_sha256":"c2089d93d0e095dd338ed2ff238f0aa1d2d138b021a144532c51dba12fbbde50","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:52:13.965679Z","signature_b64":"d7SSqLIc9EpmyHuUYzn26FSTaTsxI7Qi556g8q+nFgeLHBOI57tDU0eNxm4NeitF8jRz7FBI07SjpL0OQjuKCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2089d93d0e095dd338ed2ff238f0aa1d2d138b021a144532c51dba12fbbde50","last_reissued_at":"2026-07-05T01:52:13.965189Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:52:13.965189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00983","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":"2026-07-05T01:52:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37I7Ri+L9rghF1DFgtpzCKCqeEnU5SyfW8XHMWhP09RmnMPbvcSFH2a5uFWEWmkbLzoEWye5ZgxemUxvIvXFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:47:35.206551Z"},"content_sha256":"c06c76cf7290ef732b2b499c8aa64cd09dde9437b27b0c8f1ce95077aea7ddfa","schema_version":"1.0","event_id":"sha256:c06c76cf7290ef732b2b499c8aa64cd09dde9437b27b0c8f1ce95077aea7ddfa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:YIEJ3E6Q4CK52M4O2L7SHDYKUH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stabilisation of wave equations on the torus with rough dampings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicolas Burq (LM-Orsay), Patrick G\\'erard (LM-Orsay)","submitted_at":"2018-01-03T12:58:38Z","abstract_excerpt":"For the damped wave equation on a compact manifold with {\\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where  $a(x) = \\sum\\_{j=1}^N a\\_j 1\\_{x\\in R\\_j}$ ($R\\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\\infty$ dampings. We show that this condition is always necessary for uniform stabilisation ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00983","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1801.00983/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:52:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wY27xxpVU3nuY8bnTp2WUXbV1rsa7xMbLGQBCS0zEORkStz0XhkMTP+buI66YLuDgZMU2Pb/w+4aE4njJnBtCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:47:35.207121Z"},"content_sha256":"ea6a7bc5e3eff31913a1db8030320171cb522da78aa2baa56acc033f9566b523","schema_version":"1.0","event_id":"sha256:ea6a7bc5e3eff31913a1db8030320171cb522da78aa2baa56acc033f9566b523"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/bundle.json","state_url":"https://pith.science/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/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":"2026-08-16T05:47:35Z","links":{"resolver":"https://pith.science/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH","bundle":"https://pith.science/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/bundle.json","state":"https://pith.science/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YIEJ3E6Q4CK52M4O2L7SHDYKUH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:YIEJ3E6Q4CK52M4O2L7SHDYKUH","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":"cf50a6e53801c75786b598e2dd8e82c8553eeaf1a6d632307156ca1cd69528d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-03T12:58:38Z","title_canon_sha256":"d9bbcc96d1260d97652d8ea468481ec2d7bc8a56c115b85a705e05909a9846a8"},"schema_version":"1.0","source":{"id":"1801.00983","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00983","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00983v2","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00983","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_12","alias_value":"YIEJ3E6Q4CK5","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_16","alias_value":"YIEJ3E6Q4CK52M4O","created_at":"2026-07-05T01:52:13Z"},{"alias_kind":"pith_short_8","alias_value":"YIEJ3E6Q","created_at":"2026-07-05T01:52:13Z"}],"graph_snapshots":[{"event_id":"sha256:ea6a7bc5e3eff31913a1db8030320171cb522da78aa2baa56acc033f9566b523","target":"graph","created_at":"2026-07-05T01:52:13Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1801.00983/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For the damped wave equation on a compact manifold with {\\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where  $a(x) = \\sum\\_{j=1}^N a\\_j 1\\_{x\\in R\\_j}$ ($R\\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (","authors_text":"Nicolas Burq (LM-Orsay), Patrick G\\'erard (LM-Orsay)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-03T12:58:38Z","title":"Stabilisation of wave equations on the torus with rough dampings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00983","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:c06c76cf7290ef732b2b499c8aa64cd09dde9437b27b0c8f1ce95077aea7ddfa","target":"record","created_at":"2026-07-05T01:52:13Z","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":"cf50a6e53801c75786b598e2dd8e82c8553eeaf1a6d632307156ca1cd69528d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-03T12:58:38Z","title_canon_sha256":"d9bbcc96d1260d97652d8ea468481ec2d7bc8a56c115b85a705e05909a9846a8"},"schema_version":"1.0","source":{"id":"1801.00983","kind":"arxiv","version":2}},"canonical_sha256":"c2089d93d0e095dd338ed2ff238f0aa1d2d138b021a144532c51dba12fbbde50","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2089d93d0e095dd338ed2ff238f0aa1d2d138b021a144532c51dba12fbbde50","first_computed_at":"2026-07-05T01:52:13.965189Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:52:13.965189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d7SSqLIc9EpmyHuUYzn26FSTaTsxI7Qi556g8q+nFgeLHBOI57tDU0eNxm4NeitF8jRz7FBI07SjpL0OQjuKCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:52:13.965679Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00983","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c06c76cf7290ef732b2b499c8aa64cd09dde9437b27b0c8f1ce95077aea7ddfa","sha256:ea6a7bc5e3eff31913a1db8030320171cb522da78aa2baa56acc033f9566b523"],"state_sha256":"0214f1aae48fb42d2d884538eb3fcc15cabe20d77f5bc218d81dbdace77d1bb4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5a3TemVKMYhiLOKTj1dL4ymgxWXV1o8pIzY0M0bLwlEgMDzA7psAz/u1D6sNglyi3qhKV5Mbju4FMRWJgvZGDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:47:35.212131Z","bundle_sha256":"6f9ec7b35615ed7c4b6d3ea639e7ee5d71718431060a55456de0a41df0f9bf5b"}}