{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:W24UU7UFGVIBSJZILMIEQY73WN","short_pith_number":"pith:W24UU7UF","schema_version":"1.0","canonical_sha256":"b6b94a7e8535501927285b104863fbb37e19d8b0060c037072bcecddbd191aaf","source":{"kind":"arxiv","id":"1510.08992","version":1},"attestation_state":"computed","paper":{"title":"The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Peter Gavin Lawrence Leach, Richard Michael Morris","submitted_at":"2015-10-30T07:40:07Z","abstract_excerpt":"The Ermakov-Pinney Equation, $$\\ddot{x}+\\omega^2 x=\\frac{h^2}{x^3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the $\\omega^2$ and $h^2$. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the $sl(2,\\Re)$ symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.08992","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-10-30T07:40:07Z","cross_cats_sorted":[],"title_canon_sha256":"2448fbb205ecfa737e94365101e4d48468cfd140b95ccc59a8e5fcbe0df919e0","abstract_canon_sha256":"f6aeaa926866d6da99123db2b54d8c92844c7e31e275f722bd9fc37a64da08e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:23.740644Z","signature_b64":"0qpXwPBr526jVNoNz/wIr9l9qb//nKcg1ICjUet/xdveaN9QdedGiPJTS3rRgKfWxmtWCyM2KY0RmfI38TKeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b6b94a7e8535501927285b104863fbb37e19d8b0060c037072bcecddbd191aaf","last_reissued_at":"2026-05-18T01:28:23.740000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:23.740000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Peter Gavin Lawrence Leach, Richard Michael Morris","submitted_at":"2015-10-30T07:40:07Z","abstract_excerpt":"The Ermakov-Pinney Equation, $$\\ddot{x}+\\omega^2 x=\\frac{h^2}{x^3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the $\\omega^2$ and $h^2$. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the $sl(2,\\Re)$ symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08992","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.08992","created_at":"2026-05-18T01:28:23.740084+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.08992v1","created_at":"2026-05-18T01:28:23.740084+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08992","created_at":"2026-05-18T01:28:23.740084+00:00"},{"alias_kind":"pith_short_12","alias_value":"W24UU7UFGVIB","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"W24UU7UFGVIBSJZI","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"W24UU7UF","created_at":"2026-05-18T12:29:47.479230+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2602.06378","citing_title":"Projective Time, Cayley Transformations and the Schwarzian Geometry of the Free Particle--Oscillator Correspondence","ref_index":75,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN","json":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN.json","graph_json":"https://pith.science/api/pith-number/W24UU7UFGVIBSJZILMIEQY73WN/graph.json","events_json":"https://pith.science/api/pith-number/W24UU7UFGVIBSJZILMIEQY73WN/events.json","paper":"https://pith.science/paper/W24UU7UF"},"agent_actions":{"view_html":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN","download_json":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN.json","view_paper":"https://pith.science/paper/W24UU7UF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.08992&json=true","fetch_graph":"https://pith.science/api/pith-number/W24UU7UFGVIBSJZILMIEQY73WN/graph.json","fetch_events":"https://pith.science/api/pith-number/W24UU7UFGVIBSJZILMIEQY73WN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN/action/storage_attestation","attest_author":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN/action/author_attestation","sign_citation":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN/action/citation_signature","submit_replication":"https://pith.science/pith/W24UU7UFGVIBSJZILMIEQY73WN/action/replication_record"}},"created_at":"2026-05-18T01:28:23.740084+00:00","updated_at":"2026-05-18T01:28:23.740084+00:00"}