{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:KYOY3DFH3JFXQEUW4XJQRODIRO","short_pith_number":"pith:KYOY3DFH","schema_version":"1.0","canonical_sha256":"561d8d8ca7da4b781296e5d308b8688b9d9eef70e6dc32657efe6470550135aa","source":{"kind":"arxiv","id":"0803.0060","version":4},"attestation_state":"computed","paper":{"title":"Q-valued functions revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Camillo De Lellis, Emanuele Nunzio Spadaro","submitted_at":"2008-03-03T17:31:12Z","abstract_excerpt":"In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points."},"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":"0803.0060","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-03-03T17:31:12Z","cross_cats_sorted":[],"title_canon_sha256":"9de8b43145006ba730d79adc21b4b3216cd06431ecf717a0fb587dde7823b0ec","abstract_canon_sha256":"20a92885e44a9833950c786a7bb564bab6e01ede4a8d44d82163989d3f668b73"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:43.909843Z","signature_b64":"QRpTRq3Ju1+diqazkjq1plJ8BuBI/GSTVR/ZB6NncToiSbhpU6lHKovGuQhLZyYXKlCPxSh2PBfHepfK12HtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"561d8d8ca7da4b781296e5d308b8688b9d9eef70e6dc32657efe6470550135aa","last_reissued_at":"2026-05-18T04:26:43.909162Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:43.909162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Q-valued functions revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Camillo De Lellis, Emanuele Nunzio Spadaro","submitted_at":"2008-03-03T17:31:12Z","abstract_excerpt":"In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.0060","kind":"arxiv","version":4},"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":"0803.0060","created_at":"2026-05-18T04:26:43.909271+00:00"},{"alias_kind":"arxiv_version","alias_value":"0803.0060v4","created_at":"2026-05-18T04:26:43.909271+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0803.0060","created_at":"2026-05-18T04:26:43.909271+00:00"},{"alias_kind":"pith_short_12","alias_value":"KYOY3DFH3JFX","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"KYOY3DFH3JFXQEUW","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"KYOY3DFH","created_at":"2026-05-18T12:25:57.157939+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/KYOY3DFH3JFXQEUW4XJQRODIRO","json":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO.json","graph_json":"https://pith.science/api/pith-number/KYOY3DFH3JFXQEUW4XJQRODIRO/graph.json","events_json":"https://pith.science/api/pith-number/KYOY3DFH3JFXQEUW4XJQRODIRO/events.json","paper":"https://pith.science/paper/KYOY3DFH"},"agent_actions":{"view_html":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO","download_json":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO.json","view_paper":"https://pith.science/paper/KYOY3DFH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0803.0060&json=true","fetch_graph":"https://pith.science/api/pith-number/KYOY3DFH3JFXQEUW4XJQRODIRO/graph.json","fetch_events":"https://pith.science/api/pith-number/KYOY3DFH3JFXQEUW4XJQRODIRO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO/action/storage_attestation","attest_author":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO/action/author_attestation","sign_citation":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO/action/citation_signature","submit_replication":"https://pith.science/pith/KYOY3DFH3JFXQEUW4XJQRODIRO/action/replication_record"}},"created_at":"2026-05-18T04:26:43.909271+00:00","updated_at":"2026-05-18T04:26:43.909271+00:00"}