{"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"}