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In this paper, we prove that if there exists some constant $M>1$ (resp. some homeomorphism $\\phi$) such that for all $x\\in D$, $f: B(x,d_D(x))\\to f(B(x,d_D(x)))$ is $M$-QH (resp. $\\phi$-FQC), then $f$ is $M_1$-QH with $M_1=M_1(M)$ (resp. $\\phi_1$-FQC with $\\phi_1=\\phi_1(\\phi)$). We apply our results to establish, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism to be FQC."},"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":"1110.6269","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-10-28T07:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"2a1fee1db45a2bd1ed31ed6a4df12e82fec14d559c9140b8a65abed40c128b43","abstract_canon_sha256":"a7f9621b8b06320c52013a3ba9dece04705f1d438e9aa27771a2f3c81197801b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:27.016275Z","signature_b64":"qfmhOQL5lDkV3ACJcGSwKihwvIUjmhsHlkUZkxUki6+D97nLwtj9PS/De33/0ncvTEi/j9jfa6Q5f+MhXW1NDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2849bc7c987be1d29f67c28e5abd1bd44cddf973a3e2a3c4f740514d1d7c79d1","last_reissued_at":"2026-05-18T04:02:27.015704Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:27.015704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local properties of quasihyperbolic and freely quasiconformal mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Matti Vuorinen, Xiantao Wang, Yaxiang Li","submitted_at":"2011-10-28T07:29:16Z","abstract_excerpt":"Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\\subset E$ and $D'\\subset E'$ are domains, and that $f: D\\to D'$ is a homeomorphism. In this paper, we prove that if there exists some constant $M>1$ (resp. some homeomorphism $\\phi$) such that for all $x\\in D$, $f: B(x,d_D(x))\\to f(B(x,d_D(x)))$ is $M$-QH (resp. $\\phi$-FQC), then $f$ is $M_1$-QH with $M_1=M_1(M)$ (resp. $\\phi_1$-FQC with $\\phi_1=\\phi_1(\\phi)$). We apply our results to establish, in terms of the $j_D$ metric, a sufficient condition for a homeomorphism to be FQC."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6269","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.6269","created_at":"2026-05-18T04:02:27.015794+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.6269v2","created_at":"2026-05-18T04:02:27.015794+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6269","created_at":"2026-05-18T04:02:27.015794+00:00"},{"alias_kind":"pith_short_12","alias_value":"FBE3Y7EYPPQ5","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FBE3Y7EYPPQ5FH3H","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FBE3Y7EY","created_at":"2026-05-18T12:26:28.662955+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/FBE3Y7EYPPQ5FH3HYKHFVPI32R","json":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R.json","graph_json":"https://pith.science/api/pith-number/FBE3Y7EYPPQ5FH3HYKHFVPI32R/graph.json","events_json":"https://pith.science/api/pith-number/FBE3Y7EYPPQ5FH3HYKHFVPI32R/events.json","paper":"https://pith.science/paper/FBE3Y7EY"},"agent_actions":{"view_html":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R","download_json":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R.json","view_paper":"https://pith.science/paper/FBE3Y7EY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.6269&json=true","fetch_graph":"https://pith.science/api/pith-number/FBE3Y7EYPPQ5FH3HYKHFVPI32R/graph.json","fetch_events":"https://pith.science/api/pith-number/FBE3Y7EYPPQ5FH3HYKHFVPI32R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R/action/storage_attestation","attest_author":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R/action/author_attestation","sign_citation":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R/action/citation_signature","submit_replication":"https://pith.science/pith/FBE3Y7EYPPQ5FH3HYKHFVPI32R/action/replication_record"}},"created_at":"2026-05-18T04:02:27.015794+00:00","updated_at":"2026-05-18T04:02:27.015794+00:00"}