{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NWTHDNKJ3BBAAV7LILZL7W5NWR","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":"48ae0462eef076d16968094277df8207107cd677c314a05f4cb086b8824dd806","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-24T20:37:18Z","title_canon_sha256":"a176584731f5a3327c8c9258e3f6a3d34a95938c4cb3a3d3a41c604862e7bc1f"},"schema_version":"1.0","source":{"id":"1211.5704","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5704","created_at":"2026-05-18T02:41:09Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5704v2","created_at":"2026-05-18T02:41:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5704","created_at":"2026-05-18T02:41:09Z"},{"alias_kind":"pith_short_12","alias_value":"NWTHDNKJ3BBA","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NWTHDNKJ3BBAAV7L","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NWTHDNKJ","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:01ba11439fb042e873aa7ffbdd8638abde0b8c1d486b47d8e06343e924f62eea","target":"graph","created_at":"2026-05-18T02:41:09Z","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 consider the groups $\\operatorname{Diff}_{\\mathcal B}(\\mathbb R^n)$, $\\operatorname{Diff}_{H^\\infty}(\\mathbb R^n)$, and $\\operatorname{Diff}_{\\mathcal S}(\\mathbb R^n)$ of smooth diffeomorphisms on $\\mathbb R^n$ which differ from the identity by a function which is in either $\\mathcal B$ (bounded in all derivatives), $H^\\infty = \\bigcap_{k\\ge 0}H^k$, or $\\mathcal S$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.","authors_text":"David Mumford, Peter W. Michor","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-24T20:37:18Z","title":"A zoo of diffeomorphism groups on $\\mathbb R^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5704","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":"sha256:8eec91fc3f76be135a7452f8ab157267d1401674b857d427a369120fbb74e63f","target":"record","created_at":"2026-05-18T02:41:09Z","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":"48ae0462eef076d16968094277df8207107cd677c314a05f4cb086b8824dd806","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-11-24T20:37:18Z","title_canon_sha256":"a176584731f5a3327c8c9258e3f6a3d34a95938c4cb3a3d3a41c604862e7bc1f"},"schema_version":"1.0","source":{"id":"1211.5704","kind":"arxiv","version":2}},"canonical_sha256":"6da671b549d8420057eb42f2bfdbadb46b83bd4ea97a82c4c7b59d03f42a0d45","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6da671b549d8420057eb42f2bfdbadb46b83bd4ea97a82c4c7b59d03f42a0d45","first_computed_at":"2026-05-18T02:41:09.210628Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:09.210628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rdu9FxXIYNh4MIJf/r9nD6khlspnj3GUicCBoyXYOuOitvusybkm/FpkgE6/Dd05en/yv2Dt1DkA6NKD9LugCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:09.211172Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5704","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8eec91fc3f76be135a7452f8ab157267d1401674b857d427a369120fbb74e63f","sha256:01ba11439fb042e873aa7ffbdd8638abde0b8c1d486b47d8e06343e924f62eea"],"state_sha256":"a828e7c3baee27eadb158cc4bfea498f228343d1d76c150bbd07d94771510f8c"}