{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LVKGUSA76XHSDLDJAEJ5AAMOVW","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":"20b3f276aaf84b6f51d34b08c473d4b83046c6ddda4d5813b37023ca20e75f59","cross_cats_sorted":["hep-th","math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-05-02T16:34:11Z","title_canon_sha256":"4d085bab11b264fc5f123190cd832d7ca5f5ac059a005c9a9e5ca57626f4097d"},"schema_version":"1.0","source":{"id":"1305.0500","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.0500","created_at":"2026-05-18T03:24:41Z"},{"alias_kind":"arxiv_version","alias_value":"1305.0500v2","created_at":"2026-05-18T03:24:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.0500","created_at":"2026-05-18T03:24:41Z"},{"alias_kind":"pith_short_12","alias_value":"LVKGUSA76XHS","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LVKGUSA76XHSDLDJ","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LVKGUSA7","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:aee207774eb863557106f72a96b63324d4fb0fe38e75d7c8a61270c81b37366a","target":"graph","created_at":"2026-05-18T03:24:41Z","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":"I relate some coefficients encountered when computing the functional determinants on spheres to the central differentials of nothing. In doing this I use some historic works, in particular transcribing the elegant symbolic formalism of Jeffery (1861) into central difference form which has computational advantages for Euler numbers, as discovered by Shovelton (1915). I derive sum rules for these, and for the central differentials, the proof of which involves an interesting expression for powers of sech x as multiple derivatives. I present a more general, symbolic treatment of central difference","authors_text":"J.S.Dowker","cross_cats":["hep-th","math-ph","math.CO","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-05-02T16:34:11Z","title":"Central differences, Euler numbers and symbolic methods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0500","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":"sha256:3d6fc48289d5bf43744261a99deb13e26ddd34c99679e5d40d3c7a8631f587db","target":"record","created_at":"2026-05-18T03:24:41Z","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":"20b3f276aaf84b6f51d34b08c473d4b83046c6ddda4d5813b37023ca20e75f59","cross_cats_sorted":["hep-th","math-ph","math.CO","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-05-02T16:34:11Z","title_canon_sha256":"4d085bab11b264fc5f123190cd832d7ca5f5ac059a005c9a9e5ca57626f4097d"},"schema_version":"1.0","source":{"id":"1305.0500","kind":"arxiv","version":2}},"canonical_sha256":"5d546a481ff5cf21ac690113d0018eadb21e265a5b580d4b95a21fde36039a28","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5d546a481ff5cf21ac690113d0018eadb21e265a5b580d4b95a21fde36039a28","first_computed_at":"2026-05-18T03:24:41.147569Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:41.147569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OBSUbxr01bbb50Ju+aOl51Y4TvsBra/mS/vmuetWUjxNMOqCwjFhrZb83kNzJKBfe0Pc2ZMECK3QkrCw4t5YBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:41.148211Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.0500","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3d6fc48289d5bf43744261a99deb13e26ddd34c99679e5d40d3c7a8631f587db","sha256:aee207774eb863557106f72a96b63324d4fb0fe38e75d7c8a61270c81b37366a"],"state_sha256":"a6d49c0ee92ebaf75013c136d6dc91f9c7fd2954e60920c25cdc6df604e66f96"}