{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:BVOUSUHNJFCIRGTX7JBQEQTWIW","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":"620a8f4d340830a9ccfc9fae98cabdd568b9a03f13eceea2d4db8a2b10925253","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-20T09:00:32Z","title_canon_sha256":"ef885db6bfd4af4e5ce86e2577ce14482d14a6618ea8b55851d73081ce7063d6"},"schema_version":"1.0","source":{"id":"1312.5858","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.5858","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"arxiv_version","alias_value":"1312.5858v1","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5858","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"pith_short_12","alias_value":"BVOUSUHNJFCI","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"BVOUSUHNJFCIRGTX","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"BVOUSUHN","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:1709af52a81021a6497f213a4b7a8f949ef8196a0d36ad1c799afc3a13502085","target":"graph","created_at":"2026-05-18T00:41:08Z","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 define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \\ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various i","authors_text":"Alexandra Convent, Jean Van Schaftingen","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-20T09:00:32Z","title":"Intrinsic weak derivatives and Sobolev spaces between manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5858","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":"sha256:99b3e10e39f2487e7a5ec5ff0e7e1dd327e41d40c0bd88517a98ab3834d266c8","target":"record","created_at":"2026-05-18T00:41:08Z","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":"620a8f4d340830a9ccfc9fae98cabdd568b9a03f13eceea2d4db8a2b10925253","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-12-20T09:00:32Z","title_canon_sha256":"ef885db6bfd4af4e5ce86e2577ce14482d14a6618ea8b55851d73081ce7063d6"},"schema_version":"1.0","source":{"id":"1312.5858","kind":"arxiv","version":1}},"canonical_sha256":"0d5d4950ed4944889a77fa4302427645807ce271461013c78ba5ea6e76ed1538","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d5d4950ed4944889a77fa4302427645807ce271461013c78ba5ea6e76ed1538","first_computed_at":"2026-05-18T00:41:08.650398Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:08.650398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SVo3GeRnzlP53/mSFqsWeihB4BYmiLINkaeTTIr2iDf2maNPu7qwrHpX1H6hduTH16UTOm4XlXEnQqXguevZAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:08.650944Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.5858","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:99b3e10e39f2487e7a5ec5ff0e7e1dd327e41d40c0bd88517a98ab3834d266c8","sha256:1709af52a81021a6497f213a4b7a8f949ef8196a0d36ad1c799afc3a13502085"],"state_sha256":"fb0179bada40a03548e524275ccb7d2d85a02169b311a1aa6b2b8dc2022fcd69"}