{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:K4GQXO27HG5IZEWYFT3UTOM3OG","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":"5c348ceb74b4ccc602671e83e79b4c338c443d64ea0595dd5dbf0044313f3345","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-02-18T16:48:32Z","title_canon_sha256":"a067020f85185b3b9bce1b77210dc19b4dfe05197c508ba85459f4fba24e6c64"},"schema_version":"1.0","source":{"id":"1602.05872","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05872","created_at":"2026-05-18T01:11:57Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05872v4","created_at":"2026-05-18T01:11:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05872","created_at":"2026-05-18T01:11:57Z"},{"alias_kind":"pith_short_12","alias_value":"K4GQXO27HG5I","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"K4GQXO27HG5IZEWY","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"K4GQXO27","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:b6f315c2ec209b2312e74503a695fc858f880e28a0bc84cd59e1fba69b0975bd","target":"graph","created_at":"2026-05-18T01:11:57Z","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 establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n'$ functions $g_1,...,g_{n'}$ in $\\mathcal{C}^\\infty(M)$ so that every element of $\\mathcal{C}^k(M)$ can be approximated by rational combinations of $f_1,...,f_n$ and polynomial combinations of $g_1,...,g_{n'}$. As an application, we show that the optimal value of $n$ and $n'$ for all manifolds of dimension $m$ is [3m/2], when $k\\geq 1$ and $m\\geq 2$.","authors_text":"Purvi Gupta, Rasul Shafikov","cross_cats":["math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-02-18T16:48:32Z","title":"Rational and Polynomial Density on Compact Real Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05872","kind":"arxiv","version":4},"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":"sha256:8f6d415212990d921c1f1ecc505aeef70dbb3d6d9d389d29554c04844139c301","target":"record","created_at":"2026-05-18T01:11:57Z","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":"5c348ceb74b4ccc602671e83e79b4c338c443d64ea0595dd5dbf0044313f3345","cross_cats_sorted":["math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-02-18T16:48:32Z","title_canon_sha256":"a067020f85185b3b9bce1b77210dc19b4dfe05197c508ba85459f4fba24e6c64"},"schema_version":"1.0","source":{"id":"1602.05872","kind":"arxiv","version":4}},"canonical_sha256":"570d0bbb5f39ba8c92d82cf749b99b718d6dea3c1cc9f8936b8964a158b4a27c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"570d0bbb5f39ba8c92d82cf749b99b718d6dea3c1cc9f8936b8964a158b4a27c","first_computed_at":"2026-05-18T01:11:57.654640Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:57.654640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5F6gQ1I14lzt/OnwljVCCGp6/5bazrppdacPagylbAjuQm0gzAPsEu6i26WwuHCUpYyvvNtW0n9eM/SatHsfDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:57.655014Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.05872","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f6d415212990d921c1f1ecc505aeef70dbb3d6d9d389d29554c04844139c301","sha256:b6f315c2ec209b2312e74503a695fc858f880e28a0bc84cd59e1fba69b0975bd"],"state_sha256":"124185acd3cdcb3f9769192c6ddf3883e708fdf0000f47d72557264119868936"}