{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:QAP5OG2F2BRBQTQGLBISBPPFYE","short_pith_number":"pith:QAP5OG2F","canonical_record":{"source":{"id":"1003.3529","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-03-18T09:05:12Z","cross_cats_sorted":["math-ph","math.DG","math.MP"],"title_canon_sha256":"e652191cfecb140e86884360f58ff66f0edb3633038d1b5078d20873dde9376c","abstract_canon_sha256":"51a51fb4445a6a53041a3e0b819f16d7a276b319407f31fd35d5b4a117719f08"},"schema_version":"1.0"},"canonical_sha256":"801fd71b45d062184e06585120bde5c125aa6bb9e6d3be9c1747ef2d1f251cff","source":{"kind":"arxiv","id":"1003.3529","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3529","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3529v3","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3529","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"pith_short_12","alias_value":"QAP5OG2F2BRB","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"QAP5OG2F2BRBQTQG","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"QAP5OG2F","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:QAP5OG2F2BRBQTQGLBISBPPFYE","target":"record","payload":{"canonical_record":{"source":{"id":"1003.3529","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-03-18T09:05:12Z","cross_cats_sorted":["math-ph","math.DG","math.MP"],"title_canon_sha256":"e652191cfecb140e86884360f58ff66f0edb3633038d1b5078d20873dde9376c","abstract_canon_sha256":"51a51fb4445a6a53041a3e0b819f16d7a276b319407f31fd35d5b4a117719f08"},"schema_version":"1.0"},"canonical_sha256":"801fd71b45d062184e06585120bde5c125aa6bb9e6d3be9c1747ef2d1f251cff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:04.009094Z","signature_b64":"uw/ya8u2bz8yg73HohAgwHKENZdiEtvRN1v1RxxE8RGOzajtBTy7EkVJo59xyy0PAYakHQg6GGZXadZI6UD4DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"801fd71b45d062184e06585120bde5c125aa6bb9e6d3be9c1747ef2d1f251cff","last_reissued_at":"2026-05-18T04:08:04.008351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:04.008351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.3529","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"agZNEeBEY3KhhkFDSfTSgPwK63ClfzLkRzy4Ux1F6gpajmr+iFAGC2INX4yD+3n8I0Fd42Rl1B8MZOE5nJooAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:40:35.043864Z"},"content_sha256":"976dc7a44c96f4b05bf07f15cbba476f63508753d6f64e109432159e8e7d980c","schema_version":"1.0","event_id":"sha256:976dc7a44c96f4b05bf07f15cbba476f63508753d6f64e109432159e8e7d980c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:QAP5OG2F2BRBQTQGLBISBPPFYE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lie families: theory and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DG","math.MP"],"primary_cat":"math.CA","authors_text":"Janusz Grabowski, Javier de Lucas, Jose F. Carinena","submitted_at":"2010-03-18T09:05:12Z","abstract_excerpt":"We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3529","kind":"arxiv","version":3},"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":"2026-05-18T04:08:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vINwRyfpZP2fc9kutqouVuKTR6WsTLi0qmJBFYd/9lLFzENpT59GFZZM3Vuk6lK1JIHZHZMQAUCIk4fkmD/7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T19:40:35.044631Z"},"content_sha256":"c43b84d7cb5f36488526e90e4337fa9f6669d558b46bf628574070730cb6b1de","schema_version":"1.0","event_id":"sha256:c43b84d7cb5f36488526e90e4337fa9f6669d558b46bf628574070730cb6b1de"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/bundle.json","state_url":"https://pith.science/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/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":"2026-06-05T19:40:35Z","links":{"resolver":"https://pith.science/pith/QAP5OG2F2BRBQTQGLBISBPPFYE","bundle":"https://pith.science/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/bundle.json","state":"https://pith.science/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QAP5OG2F2BRBQTQGLBISBPPFYE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:QAP5OG2F2BRBQTQGLBISBPPFYE","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":"51a51fb4445a6a53041a3e0b819f16d7a276b319407f31fd35d5b4a117719f08","cross_cats_sorted":["math-ph","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-03-18T09:05:12Z","title_canon_sha256":"e652191cfecb140e86884360f58ff66f0edb3633038d1b5078d20873dde9376c"},"schema_version":"1.0","source":{"id":"1003.3529","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3529","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3529v3","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3529","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"pith_short_12","alias_value":"QAP5OG2F2BRB","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"QAP5OG2F2BRBQTQG","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"QAP5OG2F","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:c43b84d7cb5f36488526e90e4337fa9f6669d558b46bf628574070730cb6b1de","target":"graph","created_at":"2026-05-18T04:08:04Z","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 analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study relations of these families, called Lie families, with the theory of Lie and quasi-Lie systems and apply our theory to provide common time-dependent superposition rules for certain Lie families.","authors_text":"Janusz Grabowski, Javier de Lucas, Jose F. Carinena","cross_cats":["math-ph","math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-03-18T09:05:12Z","title":"Lie families: theory and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3529","kind":"arxiv","version":3},"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":"sha256:976dc7a44c96f4b05bf07f15cbba476f63508753d6f64e109432159e8e7d980c","target":"record","created_at":"2026-05-18T04:08:04Z","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":"51a51fb4445a6a53041a3e0b819f16d7a276b319407f31fd35d5b4a117719f08","cross_cats_sorted":["math-ph","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-03-18T09:05:12Z","title_canon_sha256":"e652191cfecb140e86884360f58ff66f0edb3633038d1b5078d20873dde9376c"},"schema_version":"1.0","source":{"id":"1003.3529","kind":"arxiv","version":3}},"canonical_sha256":"801fd71b45d062184e06585120bde5c125aa6bb9e6d3be9c1747ef2d1f251cff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"801fd71b45d062184e06585120bde5c125aa6bb9e6d3be9c1747ef2d1f251cff","first_computed_at":"2026-05-18T04:08:04.008351Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:04.008351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uw/ya8u2bz8yg73HohAgwHKENZdiEtvRN1v1RxxE8RGOzajtBTy7EkVJo59xyy0PAYakHQg6GGZXadZI6UD4DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:04.009094Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.3529","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:976dc7a44c96f4b05bf07f15cbba476f63508753d6f64e109432159e8e7d980c","sha256:c43b84d7cb5f36488526e90e4337fa9f6669d558b46bf628574070730cb6b1de"],"state_sha256":"7c3f47a344b9bdc00a230be8e81745dfa2d43eff64a1cd165875c11ee3d9c4c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D+kkigJxyGecHeObtc/M7Ce0/kTquoF1js0VqxR38ZUQu2sGE56I//0nq9gbt/rARl/KDqtnYQDOOoMAB7nwAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T19:40:35.048412Z","bundle_sha256":"890b78b10d3778c0e476aee40719102ebdf83012d68da8204851f44424bb6629"}}