{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:P6JT7T3ZWXYN5HZU2UAIJMUZYQ","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":"752198738dec3795bcc2f4f3b9b7553055cad75d7f7a8dc1296ab759807fc972","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-10-09T19:07:37Z","title_canon_sha256":"624a572194b3eb787039ceac71e3a2ab9508a09b9d82644dc48ce1fcd11747ab"},"schema_version":"1.0","source":{"id":"1210.2694","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2694","created_at":"2026-05-18T03:43:40Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2694v1","created_at":"2026-05-18T03:43:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2694","created_at":"2026-05-18T03:43:40Z"},{"alias_kind":"pith_short_12","alias_value":"P6JT7T3ZWXYN","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"P6JT7T3ZWXYN5HZU","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"P6JT7T3Z","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:d4afe637999a314ce44a1311c70cef005dc51cc53d63e743a7a1bf4466eb90db","target":"graph","created_at":"2026-05-18T03:43:40Z","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":"Alfeld and Schumaker provide a formula for the dimension of the space of piecewise polynomial functions, called splines, of degree $d$ and smoothness $r$ on a generic triangulation of a planar simplicial complex $\\Delta$, for $d \\geq 3r+1$. Schenck and Stiller conjectured that this formula actually holds for all $d \\geq 2r+1$. Up to this moment there was not known a single example where one could show that the bound $d\\geq 2r +1$ is sharp. However, in 2005, a possible such example was constructed to show that this bound is the best possible (i.e., the Alfeld-Schumaker formula does not hold if ","authors_text":"Jan Minac, Stefan O. Tohaneanu","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-10-09T19:07:37Z","title":"From Spline Approximation to Roth's Equation and Schur Functors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2694","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":"sha256:45c92ee92e519a2735462ee521a478ee40be1c145e9ae2a56ebaffc1523fc91e","target":"record","created_at":"2026-05-18T03:43:40Z","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":"752198738dec3795bcc2f4f3b9b7553055cad75d7f7a8dc1296ab759807fc972","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-10-09T19:07:37Z","title_canon_sha256":"624a572194b3eb787039ceac71e3a2ab9508a09b9d82644dc48ce1fcd11747ab"},"schema_version":"1.0","source":{"id":"1210.2694","kind":"arxiv","version":1}},"canonical_sha256":"7f933fcf79b5f0de9f34d50084b299c418dccf2c862dd688deeb0354a2b2a38e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7f933fcf79b5f0de9f34d50084b299c418dccf2c862dd688deeb0354a2b2a38e","first_computed_at":"2026-05-18T03:43:40.907510Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:40.907510Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3FFGoYXcc8LQUcaboYRzHsb7QtEKbq2U/mCga6YCU+mBDJNr/g870f89lHMiYQNfvMxTEQC0gU/eYzQB9EokDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:40.908009Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.2694","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:45c92ee92e519a2735462ee521a478ee40be1c145e9ae2a56ebaffc1523fc91e","sha256:d4afe637999a314ce44a1311c70cef005dc51cc53d63e743a7a1bf4466eb90db"],"state_sha256":"5df225042654a70e3b20e7b9f09ea65fd50cdc61824f4b3aa844ade6020064e7"}