{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:OKCJMNMFL25CIONFBK2KYFZH7N","short_pith_number":"pith:OKCJMNMF","canonical_record":{"source":{"id":"1105.3664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-18T15:43:36Z","cross_cats_sorted":["hep-th","math.MP"],"title_canon_sha256":"0aced1a2d05bfacdc3582b98f0362a9a0bc36af4a6b7f80ee07cb78d774715c8","abstract_canon_sha256":"50ba45e16e45d5345546d8191015dc8cc7bc10e0764f7df57742b85e25a79b12"},"schema_version":"1.0"},"canonical_sha256":"72849635855eba2439a50ab4ac1727fb54736957ebc95393a51f73334cfc2f52","source":{"kind":"arxiv","id":"1105.3664","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.3664","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"arxiv_version","alias_value":"1105.3664v2","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.3664","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"pith_short_12","alias_value":"OKCJMNMFL25C","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OKCJMNMFL25CIONF","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OKCJMNMF","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:OKCJMNMFL25CIONFBK2KYFZH7N","target":"record","payload":{"canonical_record":{"source":{"id":"1105.3664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-18T15:43:36Z","cross_cats_sorted":["hep-th","math.MP"],"title_canon_sha256":"0aced1a2d05bfacdc3582b98f0362a9a0bc36af4a6b7f80ee07cb78d774715c8","abstract_canon_sha256":"50ba45e16e45d5345546d8191015dc8cc7bc10e0764f7df57742b85e25a79b12"},"schema_version":"1.0"},"canonical_sha256":"72849635855eba2439a50ab4ac1727fb54736957ebc95393a51f73334cfc2f52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:59.361742Z","signature_b64":"bEeH0xHhI/MaC1MmYpgjBRSjWVhW4bSuzHkwh6ACxIE68ZGA5XZ/8YAPwgCjQNBL4txfsGdDgLHjbVUCU6NYDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72849635855eba2439a50ab4ac1727fb54736957ebc95393a51f73334cfc2f52","last_reissued_at":"2026-05-18T02:21:59.360976Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:59.360976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.3664","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":"2026-05-18T02:21:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6T3nSz7u9BoV4hLQQEwNzV2eHyM/CDZCL8j0PycEFQCcTfPKIzBNaMmGftVuqm0kgypUAUUOdd5x82FE33ggCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T03:22:46.030297Z"},"content_sha256":"bfc105be335c313e430cc23996968e946a2032328ed56fdb6652b54e96142be2","schema_version":"1.0","event_id":"sha256:bfc105be335c313e430cc23996968e946a2032328ed56fdb6652b54e96142be2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:OKCJMNMFL25CIONFBK2KYFZH7N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximate Solutions of Functional Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.MP"],"primary_cat":"math-ph","authors_text":"Cosmas Zachos, Thomas Curtright, Xiang Jin","submitted_at":"2011-05-18T15:43:36Z","abstract_excerpt":"Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by construction of approximate continuous functional iterates for x/(1-x), sin x, and {\\lambda}x(1-x). Simple functional conjugation by these functions, and their inverses, substantially improves the numerical accuracy of formal series approximations for their continuous iterates."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3664","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":""},"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":"2026-05-18T02:21:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kvpD8jyP0kUQGVgDWLeKhjNZ+IRR28Csyrrd7fous8Kz5HUbZN2x4w/maUZ5scnMQYA63n0otdmNGtxFyApTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T03:22:46.030664Z"},"content_sha256":"26c893cc3f6ad0494187e5a9acb873d0bc2d01a0102bbc0f524a9818623aa30a","schema_version":"1.0","event_id":"sha256:26c893cc3f6ad0494187e5a9acb873d0bc2d01a0102bbc0f524a9818623aa30a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OKCJMNMFL25CIONFBK2KYFZH7N/bundle.json","state_url":"https://pith.science/pith/OKCJMNMFL25CIONFBK2KYFZH7N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OKCJMNMFL25CIONFBK2KYFZH7N/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":"2026-07-05T03:22:46Z","links":{"resolver":"https://pith.science/pith/OKCJMNMFL25CIONFBK2KYFZH7N","bundle":"https://pith.science/pith/OKCJMNMFL25CIONFBK2KYFZH7N/bundle.json","state":"https://pith.science/pith/OKCJMNMFL25CIONFBK2KYFZH7N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OKCJMNMFL25CIONFBK2KYFZH7N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:OKCJMNMFL25CIONFBK2KYFZH7N","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":"50ba45e16e45d5345546d8191015dc8cc7bc10e0764f7df57742b85e25a79b12","cross_cats_sorted":["hep-th","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-18T15:43:36Z","title_canon_sha256":"0aced1a2d05bfacdc3582b98f0362a9a0bc36af4a6b7f80ee07cb78d774715c8"},"schema_version":"1.0","source":{"id":"1105.3664","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.3664","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"arxiv_version","alias_value":"1105.3664v2","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.3664","created_at":"2026-05-18T02:21:59Z"},{"alias_kind":"pith_short_12","alias_value":"OKCJMNMFL25C","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OKCJMNMFL25CIONF","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OKCJMNMF","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:26c893cc3f6ad0494187e5a9acb873d0bc2d01a0102bbc0f524a9818623aa30a","target":"graph","created_at":"2026-05-18T02:21:59Z","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":"Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by construction of approximate continuous functional iterates for x/(1-x), sin x, and {\\lambda}x(1-x). Simple functional conjugation by these functions, and their inverses, substantially improves the numerical accuracy of formal series approximations for their continuous iterates.","authors_text":"Cosmas Zachos, Thomas Curtright, Xiang Jin","cross_cats":["hep-th","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-18T15:43:36Z","title":"Approximate Solutions of Functional Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3664","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:bfc105be335c313e430cc23996968e946a2032328ed56fdb6652b54e96142be2","target":"record","created_at":"2026-05-18T02:21:59Z","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":"50ba45e16e45d5345546d8191015dc8cc7bc10e0764f7df57742b85e25a79b12","cross_cats_sorted":["hep-th","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-05-18T15:43:36Z","title_canon_sha256":"0aced1a2d05bfacdc3582b98f0362a9a0bc36af4a6b7f80ee07cb78d774715c8"},"schema_version":"1.0","source":{"id":"1105.3664","kind":"arxiv","version":2}},"canonical_sha256":"72849635855eba2439a50ab4ac1727fb54736957ebc95393a51f73334cfc2f52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72849635855eba2439a50ab4ac1727fb54736957ebc95393a51f73334cfc2f52","first_computed_at":"2026-05-18T02:21:59.360976Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:21:59.360976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bEeH0xHhI/MaC1MmYpgjBRSjWVhW4bSuzHkwh6ACxIE68ZGA5XZ/8YAPwgCjQNBL4txfsGdDgLHjbVUCU6NYDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:21:59.361742Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.3664","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bfc105be335c313e430cc23996968e946a2032328ed56fdb6652b54e96142be2","sha256:26c893cc3f6ad0494187e5a9acb873d0bc2d01a0102bbc0f524a9818623aa30a"],"state_sha256":"430b92a65a34eeb57260259ee85b6bd4ad336ffa6c9839016ebf42cada60acc9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R/Geeqd4q4QP4Xph+Dv5QSBaZaU2tlNXAYtLZGB8k0vlvzbjYgx0UUYQfhOEKYUk3Xoz9YISAaBDDkaGfCblCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T03:22:46.032717Z","bundle_sha256":"8955b5ba7c544e6f1be737dc4e0d0082388d17e520843a1178f83741c3c1ba80"}}