{"paper":{"title":"On a differential test of homeomorphism, found by N.V. Efimov","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CA","math.DS"],"primary_cat":"math.DG","authors_text":"Victor Alexandrov","submitted_at":"2010-10-18T15:57:01Z","abstract_excerpt":"In the year 1968 N.V. Efimov has proven the following remarkable theorem:\n  \\textit{Let $f:\\mathbb R^2\\to\\mathbb R^2\\in C^1$ be such that $\\det f'(x)<0$ for all $x\\in\\mathbb R^2$ and let there exist a function $a=a(x)>0$ and constants $C_1\\geqslant 0$, $C_2\\geqslant 0$ such that the inequalities $|1/a(x)-1/a(y)|\\leqslant C_1 |x-y|+C_2$ and $|\\det f'(x)|\\geqslant a(x)|{\\rm curl\\,}f(x)|+a^2(x)$ hold true for all $x, y\\in\\mathbb R^2$. Then $f(\\mathbb R^2)$ is a convex domain and $f$ maps $\\mathbb R^2$ onto $f(\\mathbb R^2)$ homeomorhically.}\n  Here ${\\rm curl\\,}f(x)$ stands for the curl of $f$ at "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3637","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"}