{"paper":{"title":"$C^1$ mappings in $\\mathbb{R}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Pawe{\\l} Goldstein, Piotr Haj{\\l}asz","submitted_at":"2018-04-23T08:47:25Z","abstract_excerpt":"We find a counterexample to a conjecture of Ga{\\l}\\k{e}ski by constructing for some positive integers $m<n$ a mapping $f\\in C^1(\\mathbb{R}^n,\\mathbb{R}^n)$ satisfying $\\mathrm{rank}\\, Df\\leq m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_\\varepsilon$ satisfying the same rank constraint $\\mathrm{rank}\\, Df_\\varepsilon\\leq m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08289","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"}