{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:KWPURCDWEFDTNPJSFAS43IT6FD","short_pith_number":"pith:KWPURCDW","schema_version":"1.0","canonical_sha256":"559f488876214736bd322825cda27e28ec81646736092662efef658215a3fc31","source":{"kind":"arxiv","id":"1101.1972","version":2},"attestation_state":"computed","paper":{"title":"The generic differentiability of convex-concave functions: Characterization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Abbas Moameni","submitted_at":"2011-01-10T21:16:24Z","abstract_excerpt":"As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \\times D,$ there exist dense subsets ${\\cal N}$ of $C$ and ${\\cal M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on ${\\cal N} \\times D$ (resp. $C \\times {\\cal M}$) and therefore the function is differentiable on ${\\cal N} \\times {\\cal M}$. This is an interesting property of convex-concave functions and it does not hold for convex-conv"},"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":"1101.1972","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-10T21:16:24Z","cross_cats_sorted":[],"title_canon_sha256":"7357903046e5614a5685c73f6172599253c362b21eae71862b23687ac6b80efd","abstract_canon_sha256":"a0102573cd386e2e8df6bc9af76470c8987cf590cc6d29fbaa4313276605b5f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:25.383300Z","signature_b64":"C/P9LDZsWLToxlbbJe+seU7/dflpOMHniCBQFqiHxCImVbxqY2B5fgpSWDyrS9nFkdVwSj8MTzun/1zpZWlADg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"559f488876214736bd322825cda27e28ec81646736092662efef658215a3fc31","last_reissued_at":"2026-05-18T03:36:25.382858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:25.382858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The generic differentiability of convex-concave functions: Characterization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Abbas Moameni","submitted_at":"2011-01-10T21:16:24Z","abstract_excerpt":"As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \\times D,$ there exist dense subsets ${\\cal N}$ of $C$ and ${\\cal M}$ of $D$ such that the partial derivative with respect to the first variable (resp. second variable) exists on ${\\cal N} \\times D$ (resp. $C \\times {\\cal M}$) and therefore the function is differentiable on ${\\cal N} \\times {\\cal M}$. This is an interesting property of convex-concave functions and it does not hold for convex-conv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1972","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":"1101.1972","created_at":"2026-05-18T03:36:25.382917+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.1972v2","created_at":"2026-05-18T03:36:25.382917+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1972","created_at":"2026-05-18T03:36:25.382917+00:00"},{"alias_kind":"pith_short_12","alias_value":"KWPURCDWEFDT","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"KWPURCDWEFDTNPJS","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"KWPURCDW","created_at":"2026-05-18T12:26:34.985390+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/KWPURCDWEFDTNPJSFAS43IT6FD","json":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD.json","graph_json":"https://pith.science/api/pith-number/KWPURCDWEFDTNPJSFAS43IT6FD/graph.json","events_json":"https://pith.science/api/pith-number/KWPURCDWEFDTNPJSFAS43IT6FD/events.json","paper":"https://pith.science/paper/KWPURCDW"},"agent_actions":{"view_html":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD","download_json":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD.json","view_paper":"https://pith.science/paper/KWPURCDW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.1972&json=true","fetch_graph":"https://pith.science/api/pith-number/KWPURCDWEFDTNPJSFAS43IT6FD/graph.json","fetch_events":"https://pith.science/api/pith-number/KWPURCDWEFDTNPJSFAS43IT6FD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD/action/storage_attestation","attest_author":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD/action/author_attestation","sign_citation":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD/action/citation_signature","submit_replication":"https://pith.science/pith/KWPURCDWEFDTNPJSFAS43IT6FD/action/replication_record"}},"created_at":"2026-05-18T03:36:25.382917+00:00","updated_at":"2026-05-18T03:36:25.382917+00:00"}