{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:Z2FRAPQDCEDVLYWXOPZTY7GFXY","short_pith_number":"pith:Z2FRAPQD","canonical_record":{"source":{"id":"1908.02977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-08T08:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"8760d379c7cadd2166089878589d135124b9820a9bc0d3e9f61a8951a12fb061","abstract_canon_sha256":"e9cbca750e55fc27489d505e0c9228c098ce52250ffe52e154b2c65eaf9a2397"},"schema_version":"1.0"},"canonical_sha256":"ce8b103e03110755e2d773f33c7cc5be274bc99ac423ef7f25f54dd94033b96e","source":{"kind":"arxiv","id":"1908.02977","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.02977","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"arxiv_version","alias_value":"1908.02977v2","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02977","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_12","alias_value":"Z2FRAPQDCEDV","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_16","alias_value":"Z2FRAPQDCEDVLYWX","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_8","alias_value":"Z2FRAPQD","created_at":"2026-07-04T23:56:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:Z2FRAPQDCEDVLYWXOPZTY7GFXY","target":"record","payload":{"canonical_record":{"source":{"id":"1908.02977","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-08T08:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"8760d379c7cadd2166089878589d135124b9820a9bc0d3e9f61a8951a12fb061","abstract_canon_sha256":"e9cbca750e55fc27489d505e0c9228c098ce52250ffe52e154b2c65eaf9a2397"},"schema_version":"1.0"},"canonical_sha256":"ce8b103e03110755e2d773f33c7cc5be274bc99ac423ef7f25f54dd94033b96e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T23:56:36.188402Z","signature_b64":"nFmlZ5lIYdUhwBXVh3wneeM/vIS2T8RI55vWquC5xK9hQMnqYHpS7nM9hx4E9wZhrk1mPbkJoBVgkVr8Rki0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ce8b103e03110755e2d773f33c7cc5be274bc99ac423ef7f25f54dd94033b96e","last_reissued_at":"2026-07-04T23:56:36.187953Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T23:56:36.187953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.02977","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":"2026-07-04T23:56:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QzrNtH2WVcRHiXiNrBvvtMwtTuy5PI162TlsGuSADlsmbVqPt4MlAlN+iNMJcy8tcE3/IewiKTaznLzMV5S2AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T09:32:57.383155Z"},"content_sha256":"ee028ea88c45562a2bc66a16e3650212b965c5563999efec0b91b91c3994e070","schema_version":"1.0","event_id":"sha256:ee028ea88c45562a2bc66a16e3650212b965c5563999efec0b91b91c3994e070"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:Z2FRAPQDCEDVLYWXOPZTY7GFXY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Two remarks on the interpolation space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohammad Daher","submitted_at":"2019-08-08T08:55:34Z","abstract_excerpt":"Dans ce travail, on montre que $(M(\\mathbb{T}),c_0(\\mathbb{Z}))_\\theta = (L^1,c_0(\\mathbb{Z}))_\\theta$, $0<\\theta <1$. Dans la suite on montre pour le couple d'interpolation $(C_0,C_1)$ trouv\\'e par Garling-Smith qu'il existe un isomorphisme $U_\\theta: (C_0,C_0+C_1)_{\\theta ,p}\\rightarrow (C_1,C_0+C_1)_{\\theta, p}$ (resp. $U_\\theta : (C_0,C_0+C_1)_\\theta \\rightarrow (C_1,C_0+C_1)_\\theta)$ tel que sa restriction \\`a $C_{\\theta, p}$ (resp. \\`a $C_\\theta)$ est un isomorphisme : $C_{\\theta, p} \\rightarrow C_{1-\\theta, p}$ (resp. $C_\\theta \\rightarrow C_{1-\\theta })$.\n  --\n  In this work we show th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02977","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/1908.02977/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":"2026-07-04T23:56:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4wV0LVQInoa+CYSSzQWSHjMkBxCq4WQfPoMw1rSHpB+Hn+VoxfmX2VASjOY8U7oQAnMBZXfK/o4MnMOoLnD2Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T09:32:57.383753Z"},"content_sha256":"45edf6ae7541e383305c11d8036b1137e44574811ce62809e2f89e65d654ccf1","schema_version":"1.0","event_id":"sha256:45edf6ae7541e383305c11d8036b1137e44574811ce62809e2f89e65d654ccf1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/bundle.json","state_url":"https://pith.science/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/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":"2026-08-16T09:32:57Z","links":{"resolver":"https://pith.science/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY","bundle":"https://pith.science/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/bundle.json","state":"https://pith.science/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z2FRAPQDCEDVLYWXOPZTY7GFXY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:Z2FRAPQDCEDVLYWXOPZTY7GFXY","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":"e9cbca750e55fc27489d505e0c9228c098ce52250ffe52e154b2c65eaf9a2397","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-08T08:55:34Z","title_canon_sha256":"8760d379c7cadd2166089878589d135124b9820a9bc0d3e9f61a8951a12fb061"},"schema_version":"1.0","source":{"id":"1908.02977","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.02977","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"arxiv_version","alias_value":"1908.02977v2","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02977","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_12","alias_value":"Z2FRAPQDCEDV","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_16","alias_value":"Z2FRAPQDCEDVLYWX","created_at":"2026-07-04T23:56:36Z"},{"alias_kind":"pith_short_8","alias_value":"Z2FRAPQD","created_at":"2026-07-04T23:56:36Z"}],"graph_snapshots":[{"event_id":"sha256:45edf6ae7541e383305c11d8036b1137e44574811ce62809e2f89e65d654ccf1","target":"graph","created_at":"2026-07-04T23:56:36Z","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/1908.02977/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Dans ce travail, on montre que $(M(\\mathbb{T}),c_0(\\mathbb{Z}))_\\theta = (L^1,c_0(\\mathbb{Z}))_\\theta$, $0<\\theta <1$. Dans la suite on montre pour le couple d'interpolation $(C_0,C_1)$ trouv\\'e par Garling-Smith qu'il existe un isomorphisme $U_\\theta: (C_0,C_0+C_1)_{\\theta ,p}\\rightarrow (C_1,C_0+C_1)_{\\theta, p}$ (resp. $U_\\theta : (C_0,C_0+C_1)_\\theta \\rightarrow (C_1,C_0+C_1)_\\theta)$ tel que sa restriction \\`a $C_{\\theta, p}$ (resp. \\`a $C_\\theta)$ est un isomorphisme : $C_{\\theta, p} \\rightarrow C_{1-\\theta, p}$ (resp. $C_\\theta \\rightarrow C_{1-\\theta })$.\n  --\n  In this work we show th","authors_text":"Mohammad Daher","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-08T08:55:34Z","title":"Two remarks on the interpolation space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02977","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:ee028ea88c45562a2bc66a16e3650212b965c5563999efec0b91b91c3994e070","target":"record","created_at":"2026-07-04T23:56:36Z","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":"e9cbca750e55fc27489d505e0c9228c098ce52250ffe52e154b2c65eaf9a2397","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-08-08T08:55:34Z","title_canon_sha256":"8760d379c7cadd2166089878589d135124b9820a9bc0d3e9f61a8951a12fb061"},"schema_version":"1.0","source":{"id":"1908.02977","kind":"arxiv","version":2}},"canonical_sha256":"ce8b103e03110755e2d773f33c7cc5be274bc99ac423ef7f25f54dd94033b96e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ce8b103e03110755e2d773f33c7cc5be274bc99ac423ef7f25f54dd94033b96e","first_computed_at":"2026-07-04T23:56:36.187953Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:56:36.187953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nFmlZ5lIYdUhwBXVh3wneeM/vIS2T8RI55vWquC5xK9hQMnqYHpS7nM9hx4E9wZhrk1mPbkJoBVgkVr8Rki0BA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:56:36.188402Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.02977","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee028ea88c45562a2bc66a16e3650212b965c5563999efec0b91b91c3994e070","sha256:45edf6ae7541e383305c11d8036b1137e44574811ce62809e2f89e65d654ccf1"],"state_sha256":"0f6642cb6596f8d337ef4a303ea016a6d36f5c6da18b0e74e19535f7e59326b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UlehFHS5cHDN82oAcBw6a9B2rBD5fb6XOSkhjhuOnQpzHlpG9YTVklDTnLot31ezw5X5m/tdq7ToJD71skZaDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T09:32:57.388894Z","bundle_sha256":"054d47ead6421ae79dec1e6caa262e8c579ff4d0feb2faa165ae3b35111a8754"}}