{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:NJCSRFR6PIB4KGYQRVV7UK56KK","short_pith_number":"pith:NJCSRFR6","schema_version":"1.0","canonical_sha256":"6a4528963e7a03c51b108d6bfa2bbe52888ebdd646c60fd8f015d88b9e164b0e","source":{"kind":"arxiv","id":"1612.06086","version":2},"attestation_state":"computed","paper":{"title":"$L^{2}$-Discretization Error Bounds for Maps into Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Hanne Hardering","submitted_at":"2016-12-19T09:19:55Z","abstract_excerpt":"We study the approximation of functions that map a Euclidean domain $\\Omega\\subset \\mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\\subset W^{1,2}(\\Omega,M)$. The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations $S_{h}\\subset H$. We provide a set of conditions on $S_{h}$ such that we can prove a priori $W^{1,2}$- and $L^{2}$-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.06086","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-12-19T09:19:55Z","cross_cats_sorted":[],"title_canon_sha256":"0d31c49cddf06b7383be6072abc25c33e4a3c64c42b105de4cf79736331f3f30","abstract_canon_sha256":"51f55e2b23d4142715617ff23df3ccb66686e3b376cabc0b0e53f2bd040dbe09"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:05.677590Z","signature_b64":"6VEy8rCBwL5b3XTZLbcj8DsPUeahzCCSIAeOKdZBw32FBG0Cj5ObcJUeD7hHBEwuXZ+3XWgNVb7fEMLkmJjWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a4528963e7a03c51b108d6bfa2bbe52888ebdd646c60fd8f015d88b9e164b0e","last_reissued_at":"2026-05-18T00:15:05.677080Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:05.677080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L^{2}$-Discretization Error Bounds for Maps into Riemannian Manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Hanne Hardering","submitted_at":"2016-12-19T09:19:55Z","abstract_excerpt":"We study the approximation of functions that map a Euclidean domain $\\Omega\\subset \\mathbb{R}^{d}$ into an $n$-dimensional Riemannian manifold $(M,g)$ minimizing an elliptic, semilinear energy in a function set $H\\subset W^{1,2}(\\Omega,M)$. The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations $S_{h}\\subset H$. We provide a set of conditions on $S_{h}$ such that we can prove a priori $W^{1,2}$- and $L^{2}$-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06086","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.06086","created_at":"2026-05-18T00:15:05.677155+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.06086v2","created_at":"2026-05-18T00:15:05.677155+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06086","created_at":"2026-05-18T00:15:05.677155+00:00"},{"alias_kind":"pith_short_12","alias_value":"NJCSRFR6PIB4","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"NJCSRFR6PIB4KGYQ","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"NJCSRFR6","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK","json":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK.json","graph_json":"https://pith.science/api/pith-number/NJCSRFR6PIB4KGYQRVV7UK56KK/graph.json","events_json":"https://pith.science/api/pith-number/NJCSRFR6PIB4KGYQRVV7UK56KK/events.json","paper":"https://pith.science/paper/NJCSRFR6"},"agent_actions":{"view_html":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK","download_json":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK.json","view_paper":"https://pith.science/paper/NJCSRFR6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.06086&json=true","fetch_graph":"https://pith.science/api/pith-number/NJCSRFR6PIB4KGYQRVV7UK56KK/graph.json","fetch_events":"https://pith.science/api/pith-number/NJCSRFR6PIB4KGYQRVV7UK56KK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK/action/storage_attestation","attest_author":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK/action/author_attestation","sign_citation":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK/action/citation_signature","submit_replication":"https://pith.science/pith/NJCSRFR6PIB4KGYQRVV7UK56KK/action/replication_record"}},"created_at":"2026-05-18T00:15:05.677155+00:00","updated_at":"2026-05-18T00:15:05.677155+00:00"}