{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:37BY3JBDSZ3H64Z3LD6BOANDTN","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":"3b8889573cafb51f833972435ecc5d8db7719a1c399a36ae7a5cfca43e6a9ef0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-12-04T16:43:12Z","title_canon_sha256":"6a39ab9a5c98c98515f5be1186ae10f32bf43cacadad3033ac09b77783cbd350"},"schema_version":"1.0","source":{"id":"1312.1239","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.1239","created_at":"2026-05-18T03:05:34Z"},{"alias_kind":"arxiv_version","alias_value":"1312.1239v1","created_at":"2026-05-18T03:05:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.1239","created_at":"2026-05-18T03:05:34Z"},{"alias_kind":"pith_short_12","alias_value":"37BY3JBDSZ3H","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"37BY3JBDSZ3H64Z3","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"37BY3JBD","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:0f8d179c92c06cbfd990549de82509a5f350c2ec028f4895ff6ecd00a1544010","target":"graph","created_at":"2026-05-18T03:05:34Z","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 present expressions for the Drazin inverse of the generalized Schur complement $A-CD^{d}B$ in terms of the Drazin inverses of $A$ and the generalized Schur complement $D-BA^{d}C$ under less and weaker restrictions, which generalize several results in the literature and the formula of Sherman-Morrison-Woodbury type.","authors_text":"Daochang Zhang, Xiankun Du","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-12-04T16:43:12Z","title":"Representations for the Drazin inverse of the generalized Schur complement"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1239","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":"sha256:8f331c1f7714e8e6b7255b81fad305c719cabd644c19079724206dd4d124d828","target":"record","created_at":"2026-05-18T03:05:34Z","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":"3b8889573cafb51f833972435ecc5d8db7719a1c399a36ae7a5cfca43e6a9ef0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-12-04T16:43:12Z","title_canon_sha256":"6a39ab9a5c98c98515f5be1186ae10f32bf43cacadad3033ac09b77783cbd350"},"schema_version":"1.0","source":{"id":"1312.1239","kind":"arxiv","version":1}},"canonical_sha256":"dfc38da42396767f733b58fc1701a39b60f17083b42bd72be6d746f8cb9e9e09","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfc38da42396767f733b58fc1701a39b60f17083b42bd72be6d746f8cb9e9e09","first_computed_at":"2026-05-18T03:05:34.577376Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:34.577376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xnCtOYSsDNLFXFIvF6LCmAtjXFaMDmw2gzAKzfGdyAFHkgVH2sFSEX88YyH28vfjRXC6U5tcySAxvOYMK0w2Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:34.577906Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.1239","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f331c1f7714e8e6b7255b81fad305c719cabd644c19079724206dd4d124d828","sha256:0f8d179c92c06cbfd990549de82509a5f350c2ec028f4895ff6ecd00a1544010"],"state_sha256":"808c15e488442a55492d15824873b4bbfae4d1e60a962e834247e62172f7d0fd"}