{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:QJY3QB47Y44ZESPRVOJBU5XIFW","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":"3f6bffb67ee09505732c0179ccb68e27f6411c809b56cb4f8d0f4932635d9f27","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-06-10T14:56:17Z","title_canon_sha256":"2cab033142800f33e5cae404b16f0fad25199695967ff45a249e42f58bf3481c"},"schema_version":"1.0","source":{"id":"1006.2047","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.2047","created_at":"2026-05-18T03:46:17Z"},{"alias_kind":"arxiv_version","alias_value":"1006.2047v1","created_at":"2026-05-18T03:46:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.2047","created_at":"2026-05-18T03:46:17Z"},{"alias_kind":"pith_short_12","alias_value":"QJY3QB47Y44Z","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"QJY3QB47Y44ZESPR","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"QJY3QB47","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:91e3991d35d3d5712b3accd5432bacd7e486f8dd5fb5e7aacbe160c8acdc1e90","target":"graph","created_at":"2026-05-18T03:46:17Z","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":"A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.","authors_text":"Catalin Badea, Sophie Grivaux, Vladimir Muller","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-06-10T14:56:17Z","title":"The rate of convergence in the method of alternating projections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.2047","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":"sha256:2c7fd5abf2e3d52641557f7f93f8aeb0107126d6326dd5f02722dfd3473b3993","target":"record","created_at":"2026-05-18T03:46:17Z","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":"3f6bffb67ee09505732c0179ccb68e27f6411c809b56cb4f8d0f4932635d9f27","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-06-10T14:56:17Z","title_canon_sha256":"2cab033142800f33e5cae404b16f0fad25199695967ff45a249e42f58bf3481c"},"schema_version":"1.0","source":{"id":"1006.2047","kind":"arxiv","version":1}},"canonical_sha256":"8271b8079fc7399249f1ab921a76e82d87d3db428ebe3c57180fcc4da3e6517f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8271b8079fc7399249f1ab921a76e82d87d3db428ebe3c57180fcc4da3e6517f","first_computed_at":"2026-05-18T03:46:17.080385Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:17.080385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DKlXrvgprKv0iJTweKBC+HXt6Q6jUB5g5dMq3Ge4R3+7GBPS4cGZK9DYgPHRKg/S1JgO1jDZeoGMjedAcocSDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:17.081021Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.2047","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c7fd5abf2e3d52641557f7f93f8aeb0107126d6326dd5f02722dfd3473b3993","sha256:91e3991d35d3d5712b3accd5432bacd7e486f8dd5fb5e7aacbe160c8acdc1e90"],"state_sha256":"ec66b20ee639650c5ce7739daf6439d42201d1f4c6c22f1163d112c4da7e08ca"}