{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:SDTNALWYFOMB2MSY62UKS543DQ","short_pith_number":"pith:SDTNALWY","canonical_record":{"source":{"id":"1901.09669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-18T12:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"c02b3169eacf73a9ce63ae304ee05bba61a377114ade7340171f80d54062d94f","abstract_canon_sha256":"a240b5011c2318574c89c8d9eef5521ec8a4da3cd22f96380064a1ae014bc4bf"},"schema_version":"1.0"},"canonical_sha256":"90e6d02ed82b981d3258f6a8a9779b1c232783bb294ecc690f53752c723b14b8","source":{"kind":"arxiv","id":"1901.09669","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.09669","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"arxiv_version","alias_value":"1901.09669v1","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.09669","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"pith_short_12","alias_value":"SDTNALWYFOMB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SDTNALWYFOMB2MSY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SDTNALWY","created_at":"2026-05-18T12:33:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:SDTNALWYFOMB2MSY62UKS543DQ","target":"record","payload":{"canonical_record":{"source":{"id":"1901.09669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-18T12:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"c02b3169eacf73a9ce63ae304ee05bba61a377114ade7340171f80d54062d94f","abstract_canon_sha256":"a240b5011c2318574c89c8d9eef5521ec8a4da3cd22f96380064a1ae014bc4bf"},"schema_version":"1.0"},"canonical_sha256":"90e6d02ed82b981d3258f6a8a9779b1c232783bb294ecc690f53752c723b14b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:24.152126Z","signature_b64":"n7tekjkSvLZtuu5vqpg5IEWzNSqw00xa+PfSppaPH3IL3hzx+PI19EidiiKudoyev/1RZokGTwaDZIA+6IGqBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"90e6d02ed82b981d3258f6a8a9779b1c232783bb294ecc690f53752c723b14b8","last_reissued_at":"2026-05-17T23:55:24.151555Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:24.151555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.09669","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":"2026-05-17T23:55:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O2XLdbirPx9YOVOzNiCSGDcBtjGrNCj6BC0yqFWNXF1x7Kmj+ifZpOkMvKKQ6yvkKlbLdrPbzza6ahF9P/OtAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T23:04:21.380914Z"},"content_sha256":"67cb57784ec3b8becbcfef16fe1500dd7972733c58b760df58d6aee4598b3a8f","schema_version":"1.0","event_id":"sha256:67cb57784ec3b8becbcfef16fe1500dd7972733c58b760df58d6aee4598b3a8f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:SDTNALWYFOMB2MSY62UKS543DQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local precised approximation in multiscale problems with local defects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Claude Le Bris (MATHERIALS), Marc Josien (MATHERIALS), Xavier Blanc (LJLL)","submitted_at":"2019-01-18T12:11:58Z","abstract_excerpt":"We proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1 < r < +$\\infty$$ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same acc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09669","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":"2026-05-17T23:55:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bDxfdmMnMlb6k9WIWBkOC90aqC8geRef7bT481iMsM2xmLAK0rp4LaOSoScmKDbGEvZsu0HIU93BdMDNtd18AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T23:04:21.381677Z"},"content_sha256":"1c32df292370054903b4d4cc5b5a53720811b27d239f88350bc9fa422de9510b","schema_version":"1.0","event_id":"sha256:1c32df292370054903b4d4cc5b5a53720811b27d239f88350bc9fa422de9510b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SDTNALWYFOMB2MSY62UKS543DQ/bundle.json","state_url":"https://pith.science/pith/SDTNALWYFOMB2MSY62UKS543DQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SDTNALWYFOMB2MSY62UKS543DQ/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":"2026-05-28T23:04:21Z","links":{"resolver":"https://pith.science/pith/SDTNALWYFOMB2MSY62UKS543DQ","bundle":"https://pith.science/pith/SDTNALWYFOMB2MSY62UKS543DQ/bundle.json","state":"https://pith.science/pith/SDTNALWYFOMB2MSY62UKS543DQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SDTNALWYFOMB2MSY62UKS543DQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:SDTNALWYFOMB2MSY62UKS543DQ","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":"a240b5011c2318574c89c8d9eef5521ec8a4da3cd22f96380064a1ae014bc4bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-18T12:11:58Z","title_canon_sha256":"c02b3169eacf73a9ce63ae304ee05bba61a377114ade7340171f80d54062d94f"},"schema_version":"1.0","source":{"id":"1901.09669","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.09669","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"arxiv_version","alias_value":"1901.09669v1","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.09669","created_at":"2026-05-17T23:55:24Z"},{"alias_kind":"pith_short_12","alias_value":"SDTNALWYFOMB","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"SDTNALWYFOMB2MSY","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"SDTNALWY","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:1c32df292370054903b4d4cc5b5a53720811b27d239f88350bc9fa422de9510b","target":"graph","created_at":"2026-05-17T23:55:24Z","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 proceed here with our systematic study, initiated in [3], of multiscale problems with defects, within the context of homogenization theory. The case under consideration here is that of a diffusion equation with a diffusion coefficient of the form of a periodic function perturbed by an $L^r (R^d ) , 1 < r < +$\\infty$$ , function modeling a localized defect. We outline the proof of the following approximation result: the corrector function, the existence of which has been established in [3,4], allows to approximate the solution of the original multiscale equation with essentially the same acc","authors_text":"Claude Le Bris (MATHERIALS), Marc Josien (MATHERIALS), Xavier Blanc (LJLL)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-18T12:11:58Z","title":"Local precised approximation in multiscale problems with local defects"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09669","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":"sha256:67cb57784ec3b8becbcfef16fe1500dd7972733c58b760df58d6aee4598b3a8f","target":"record","created_at":"2026-05-17T23:55:24Z","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":"a240b5011c2318574c89c8d9eef5521ec8a4da3cd22f96380064a1ae014bc4bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-18T12:11:58Z","title_canon_sha256":"c02b3169eacf73a9ce63ae304ee05bba61a377114ade7340171f80d54062d94f"},"schema_version":"1.0","source":{"id":"1901.09669","kind":"arxiv","version":1}},"canonical_sha256":"90e6d02ed82b981d3258f6a8a9779b1c232783bb294ecc690f53752c723b14b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"90e6d02ed82b981d3258f6a8a9779b1c232783bb294ecc690f53752c723b14b8","first_computed_at":"2026-05-17T23:55:24.151555Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:24.151555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"n7tekjkSvLZtuu5vqpg5IEWzNSqw00xa+PfSppaPH3IL3hzx+PI19EidiiKudoyev/1RZokGTwaDZIA+6IGqBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:24.152126Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.09669","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67cb57784ec3b8becbcfef16fe1500dd7972733c58b760df58d6aee4598b3a8f","sha256:1c32df292370054903b4d4cc5b5a53720811b27d239f88350bc9fa422de9510b"],"state_sha256":"65c55cf94c1e0867158b9b49c43a704bb73e09692dd479148f53f77a3e128363"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HLWeVXheY0uLa8mlDQyEyqCz76T73bMu5IY4mUC1vduC+js0bpnmsxNbPSVwhsyDS+cl++eKmZ2RG3GFL2iCDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T23:04:21.385343Z","bundle_sha256":"a5ff3862300075f50b2a10a451110519c85f101bae074883603d0ab840e1fab2"}}