{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:MJ3BH3QAMRGZ5WVCFGRPMAP2A4","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":"8d555eeec23e20be88c2463cfe4e1fd45e5ce98babbf176228b9ea010af30adb","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2007-07-07T19:40:13Z","title_canon_sha256":"c111d9c4df0b82eb4e3ea23da86c0dbea7b7ca22d6f857f76568b641328166df"},"schema_version":"1.0","source":{"id":"0707.1104","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0707.1104","created_at":"2026-05-18T02:55:18Z"},{"alias_kind":"arxiv_version","alias_value":"0707.1104v1","created_at":"2026-05-18T02:55:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.1104","created_at":"2026-05-18T02:55:18Z"},{"alias_kind":"pith_short_12","alias_value":"MJ3BH3QAMRGZ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"MJ3BH3QAMRGZ5WVC","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"MJ3BH3QA","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:9b318b2c7be62ff413ca72394367f496542d19976c4062ef3cd3a133da4562d6","target":"graph","created_at":"2026-05-18T02:55:18Z","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 develop a theory of Hilbert $\\widetilde{\\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\\widetilde{\\C}$-linear functionals and $\\widetilde{\\C}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.","authors_text":"Claudia Garetto, Hans Vernaeve","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2007-07-07T19:40:13Z","title":"Hilbert $\\widetilde{\\C}$-modules: structural properties and applications to variational problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.1104","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":"sha256:2dbe06fceee30cbf5d97608ead47dfad538abe236af0cbb914cb9a16ee2f7bbc","target":"record","created_at":"2026-05-18T02:55:18Z","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":"8d555eeec23e20be88c2463cfe4e1fd45e5ce98babbf176228b9ea010af30adb","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2007-07-07T19:40:13Z","title_canon_sha256":"c111d9c4df0b82eb4e3ea23da86c0dbea7b7ca22d6f857f76568b641328166df"},"schema_version":"1.0","source":{"id":"0707.1104","kind":"arxiv","version":1}},"canonical_sha256":"627613ee00644d9edaa229a2f601fa0719d9c1217738455b0877619f7a5416ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"627613ee00644d9edaa229a2f601fa0719d9c1217738455b0877619f7a5416ab","first_computed_at":"2026-05-18T02:55:18.950620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:18.950620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xqTeirXBOBoc0qU2wp7QHJfBySb2ulryh59yL6ELX6Ikeld8qsVer2orPDWURv1YZPErwYqFVSJjE1i05RNlBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:18.951026Z","signed_message":"canonical_sha256_bytes"},"source_id":"0707.1104","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2dbe06fceee30cbf5d97608ead47dfad538abe236af0cbb914cb9a16ee2f7bbc","sha256:9b318b2c7be62ff413ca72394367f496542d19976c4062ef3cd3a133da4562d6"],"state_sha256":"9a761e21b1fa11824ec7132105b7160fc55527152e63a8bf99a3702cb8a04773"}