{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:OH4R44ZHCV4OIR6XSZW7ASKSRZ","short_pith_number":"pith:OH4R44ZH","schema_version":"1.0","canonical_sha256":"71f91e73271578e447d7966df049528e4347c4d24a59944ed5b1806932673c80","source":{"kind":"arxiv","id":"1901.01781","version":1},"attestation_state":"computed","paper":{"title":"On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alaa Elshorbagy, Giovanni Bellettini, Maurizio Paolini, Riccardo Scala","submitted_at":"2019-01-07T12:43:37Z","abstract_excerpt":"In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to \"fill the hole\" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces,"},"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":"1901.01781","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-01-07T12:43:37Z","cross_cats_sorted":[],"title_canon_sha256":"e6506050e203c44a9a22e5dce39c2b438b45c0a8a534647fb0a9f77ed43c2719","abstract_canon_sha256":"4ae30e41f19275291beadb8707597a0f3a0c38622aa777c4648e4b8103f66cb2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:51.253410Z","signature_b64":"Ah4kTD8pQC/EIlMAL7bcQ5UTWrj3/Hns3yRkFl21o3iwbuoCmuuYK7FnEBf4Aw8rmYd3N08rWF81hk2T8SWJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71f91e73271578e447d7966df049528e4347c4d24a59944ed5b1806932673c80","last_reissued_at":"2026-05-17T23:56:51.253011Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:51.253011Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the relaxed area of the graph of discontinuous maps from the plane to the plane taking three values with no symmetry assumptions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alaa Elshorbagy, Giovanni Bellettini, Maurizio Paolini, Riccardo Scala","submitted_at":"2019-01-07T12:43:37Z","abstract_excerpt":"In this paper we estimate from above the area of the graph of a singular map $u$ taking a disk to three vectors, the vertices of a triangle, and jumping along three $\\mathcal{C}^2-$ embedded curves that meet transversely at only one point of the disk. We show that the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to \"fill the hole\" in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01781","kind":"arxiv","version":1},"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":"1901.01781","created_at":"2026-05-17T23:56:51.253078+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.01781v1","created_at":"2026-05-17T23:56:51.253078+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01781","created_at":"2026-05-17T23:56:51.253078+00:00"},{"alias_kind":"pith_short_12","alias_value":"OH4R44ZHCV4O","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_16","alias_value":"OH4R44ZHCV4OIR6X","created_at":"2026-05-18T12:33:24.271573+00:00"},{"alias_kind":"pith_short_8","alias_value":"OH4R44ZH","created_at":"2026-05-18T12:33:24.271573+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/OH4R44ZHCV4OIR6XSZW7ASKSRZ","json":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ.json","graph_json":"https://pith.science/api/pith-number/OH4R44ZHCV4OIR6XSZW7ASKSRZ/graph.json","events_json":"https://pith.science/api/pith-number/OH4R44ZHCV4OIR6XSZW7ASKSRZ/events.json","paper":"https://pith.science/paper/OH4R44ZH"},"agent_actions":{"view_html":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ","download_json":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ.json","view_paper":"https://pith.science/paper/OH4R44ZH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.01781&json=true","fetch_graph":"https://pith.science/api/pith-number/OH4R44ZHCV4OIR6XSZW7ASKSRZ/graph.json","fetch_events":"https://pith.science/api/pith-number/OH4R44ZHCV4OIR6XSZW7ASKSRZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ/action/storage_attestation","attest_author":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ/action/author_attestation","sign_citation":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ/action/citation_signature","submit_replication":"https://pith.science/pith/OH4R44ZHCV4OIR6XSZW7ASKSRZ/action/replication_record"}},"created_at":"2026-05-17T23:56:51.253078+00:00","updated_at":"2026-05-17T23:56:51.253078+00:00"}