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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 "},"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":"1010.3637","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-10-18T15:57:01Z","cross_cats_sorted":["math.AC","math.CA","math.DS"],"title_canon_sha256":"9b1b1abc99e6d0780b00786a69b6cf29c5d6a665785e93ef7b54eda0f601ec60","abstract_canon_sha256":"2436c995deb16f7d28cdcde12fdf4cf6d93519f116e29edf47e01d96919f5da6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:51.710862Z","signature_b64":"34D2jwN42f8+YtA9RE2jIKKU+U1iENHawepUZ+DkDqW6fJcUO6jh6vkETRl3oFbqOdJi7lbAXo0H1ofBKkaoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72e98a76008accd5daf09b232c1f853f817622366226e305a41a015bbbb80dca","last_reissued_at":"2026-05-18T03:58:51.710337Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:51.710337Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a differential test of homeomorphism, found by N.V. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.3637","created_at":"2026-05-18T03:58:51.710417+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.3637v1","created_at":"2026-05-18T03:58:51.710417+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.3637","created_at":"2026-05-18T03:58:51.710417+00:00"},{"alias_kind":"pith_short_12","alias_value":"OLUYU5QARLGN","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"OLUYU5QARLGNLWXQ","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"OLUYU5QA","created_at":"2026-05-18T12:26:12.377268+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/OLUYU5QARLGNLWXQTMRSYH4FH6","json":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6.json","graph_json":"https://pith.science/api/pith-number/OLUYU5QARLGNLWXQTMRSYH4FH6/graph.json","events_json":"https://pith.science/api/pith-number/OLUYU5QARLGNLWXQTMRSYH4FH6/events.json","paper":"https://pith.science/paper/OLUYU5QA"},"agent_actions":{"view_html":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6","download_json":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6.json","view_paper":"https://pith.science/paper/OLUYU5QA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.3637&json=true","fetch_graph":"https://pith.science/api/pith-number/OLUYU5QARLGNLWXQTMRSYH4FH6/graph.json","fetch_events":"https://pith.science/api/pith-number/OLUYU5QARLGNLWXQTMRSYH4FH6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6/action/storage_attestation","attest_author":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6/action/author_attestation","sign_citation":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6/action/citation_signature","submit_replication":"https://pith.science/pith/OLUYU5QARLGNLWXQTMRSYH4FH6/action/replication_record"}},"created_at":"2026-05-18T03:58:51.710417+00:00","updated_at":"2026-05-18T03:58:51.710417+00:00"}