{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:4HRAKD6ENL3IYGF5HTCQYWBPKW","short_pith_number":"pith:4HRAKD6E","schema_version":"1.0","canonical_sha256":"e1e2050fc46af68c18bd3cc50c582f55bf8b9dd1014f7a4f0f5a498367c791f8","source":{"kind":"arxiv","id":"1706.05266","version":4},"attestation_state":"computed","paper":{"title":"On some universal Morse-Sard type Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adele Ferone, Alba Roviello, Mikhail V. Korobkov","submitted_at":"2017-06-15T12:39:11Z","abstract_excerpt":"The classical Morse--Sard theorem claims that for a mapping $v:\\mathbb R^n\\to\\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\\ge n-m$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \\{ x \\in \\mathbb R^n : \\, {\\rm rank}\\,\\nabla v(x)\\le m \\}$. Further Dubovitski\\u{\\i} in 1957 and independently Federer and Dubovitski\\u{\\i} in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the $C^k$ category.\n  Here we "},"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":"1706.05266","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-06-15T12:39:11Z","cross_cats_sorted":[],"title_canon_sha256":"c86234dd39b3dc70036a50e55017a2f75ff88c9767cd500d0377f070e7e15ba8","abstract_canon_sha256":"8a82fea760e8fdaaf845b79fd574640df60e6c3231e566abb9a84453675a3cbb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:54.156486Z","signature_b64":"UFLGNQeC6e4zkdksqoT0KzgiheTIB5fUNmSY0c+5qesSNpUf93s9TGOSjfKEUj19D7KyHdvODjYJDQvoifalCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1e2050fc46af68c18bd3cc50c582f55bf8b9dd1014f7a4f0f5a498367c791f8","last_reissued_at":"2026-05-17T23:43:54.155953Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:54.155953Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some universal Morse-Sard type Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adele Ferone, Alba Roviello, Mikhail V. Korobkov","submitted_at":"2017-06-15T12:39:11Z","abstract_excerpt":"The classical Morse--Sard theorem claims that for a mapping $v:\\mathbb R^n\\to\\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\\ge n-m$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \\{ x \\in \\mathbb R^n : \\, {\\rm rank}\\,\\nabla v(x)\\le m \\}$. Further Dubovitski\\u{\\i} in 1957 and independently Federer and Dubovitski\\u{\\i} in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the $C^k$ category.\n  Here we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05266","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":"1706.05266","created_at":"2026-05-17T23:43:54.156029+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.05266v4","created_at":"2026-05-17T23:43:54.156029+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05266","created_at":"2026-05-17T23:43:54.156029+00:00"},{"alias_kind":"pith_short_12","alias_value":"4HRAKD6ENL3I","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"4HRAKD6ENL3IYGF5","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"4HRAKD6E","created_at":"2026-05-18T12:30:58.224056+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/4HRAKD6ENL3IYGF5HTCQYWBPKW","json":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW.json","graph_json":"https://pith.science/api/pith-number/4HRAKD6ENL3IYGF5HTCQYWBPKW/graph.json","events_json":"https://pith.science/api/pith-number/4HRAKD6ENL3IYGF5HTCQYWBPKW/events.json","paper":"https://pith.science/paper/4HRAKD6E"},"agent_actions":{"view_html":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW","download_json":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW.json","view_paper":"https://pith.science/paper/4HRAKD6E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.05266&json=true","fetch_graph":"https://pith.science/api/pith-number/4HRAKD6ENL3IYGF5HTCQYWBPKW/graph.json","fetch_events":"https://pith.science/api/pith-number/4HRAKD6ENL3IYGF5HTCQYWBPKW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW/action/storage_attestation","attest_author":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW/action/author_attestation","sign_citation":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW/action/citation_signature","submit_replication":"https://pith.science/pith/4HRAKD6ENL3IYGF5HTCQYWBPKW/action/replication_record"}},"created_at":"2026-05-17T23:43:54.156029+00:00","updated_at":"2026-05-17T23:43:54.156029+00:00"}