{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:YOSKYIT755MNRM6NOG3E7HCXUQ","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":"cbd5139c5f6f963cb6776fde07d542631e7d443ba96fa7ca0606fec8d79bd22b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-20T18:40:26Z","title_canon_sha256":"5fa706ce1f6ac03118558adca3b3be46f46e6470c5e6d086524856fe60a74297"},"schema_version":"1.0","source":{"id":"1405.5183","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.5183","created_at":"2026-05-18T02:51:25Z"},{"alias_kind":"arxiv_version","alias_value":"1405.5183v1","created_at":"2026-05-18T02:51:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.5183","created_at":"2026-05-18T02:51:25Z"},{"alias_kind":"pith_short_12","alias_value":"YOSKYIT755MN","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"YOSKYIT755MNRM6N","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"YOSKYIT7","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:9738b3014d257af2c1f19b579670df1b59b69eeced7984133d0d3f680314d3d3","target":"graph","created_at":"2026-05-18T02:51:25Z","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 consider an $\\alpha$-relaxed projection $P_A^\\alpha:H\\to H$ given by $P_A^\\alpha(x)=\\alpha P_A(x)+(1-\\alpha)x$ where $\\alpha\\in[0,1]$ and $P_A$ is the projection onto a non-empty, convex and closed subset $A$ of the real Hilbert space $H$. We characterise all the sets $F\\subset[0,1]$ such that for some non-empty, convex and closed subsets $A_1,A_2,\\dots,A_k\\subset H$ the composition $P_{A_k}^\\alpha P_{A_{k-1}}^\\alpha\\dots P_{A_1}^\\alpha$ has a fixed point iff $\\alpha\\in F$. It proves, that if $\\dim H\\geq 3$ and $k\\geq3$ then the class of the derscribed above sets $F$ of coefficients $\\alpha","authors_text":"Adam Paszkiewicz, Andrzej Komisarski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-20T18:40:26Z","title":"On the stability of the existence of fixed points for the projection-iterative methods with relaxation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5183","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":"sha256:46c3926352606b45ef593641848e79abf02062ace1a9c9cd1959c88fbc8912f9","target":"record","created_at":"2026-05-18T02:51:25Z","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":"cbd5139c5f6f963cb6776fde07d542631e7d443ba96fa7ca0606fec8d79bd22b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-20T18:40:26Z","title_canon_sha256":"5fa706ce1f6ac03118558adca3b3be46f46e6470c5e6d086524856fe60a74297"},"schema_version":"1.0","source":{"id":"1405.5183","kind":"arxiv","version":1}},"canonical_sha256":"c3a4ac227fef58d8b3cd71b64f9c57a42e68af5e6f0769c4de3b4c3b097d7370","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3a4ac227fef58d8b3cd71b64f9c57a42e68af5e6f0769c4de3b4c3b097d7370","first_computed_at":"2026-05-18T02:51:25.897154Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:25.897154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2+wwzvTgNpmnYjPuovG4qWqBp3PhqbveAjepAhmspCtfV9scZUcCuGuCk6v/QmMysvMZuaMqRPn8f75TWifQBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:25.897687Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.5183","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46c3926352606b45ef593641848e79abf02062ace1a9c9cd1959c88fbc8912f9","sha256:9738b3014d257af2c1f19b579670df1b59b69eeced7984133d0d3f680314d3d3"],"state_sha256":"02d7b0f3e3c1d9f083e991fa9f37f12f567c8b91203df211dec67701eda78378"}