{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:K5IRLREDTYNI5OGODNZ5FPCVFM","short_pith_number":"pith:K5IRLRED","schema_version":"1.0","canonical_sha256":"575115c4839e1a8eb8ce1b73d2bc552b049d15647f0a474ceb8f64c23b2fc7e8","source":{"kind":"arxiv","id":"1805.07481","version":2},"attestation_state":"computed","paper":{"title":"Apollonian metric, uniformity and Gromov hyperbolicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Matti Vuorinen, Qingshan Zhou, Yaxiang Li","submitted_at":"2018-05-19T00:14:42Z","abstract_excerpt":"The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $\\varphi$-uniformity and $\\delta$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $\\"},"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":"1805.07481","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-05-19T00:14:42Z","cross_cats_sorted":[],"title_canon_sha256":"1422e383370bc2761a95112b0af4ea319e9fa616ee95f1a678c5f413a126fded","abstract_canon_sha256":"23792034876b7e4749b454366f8b54e9618317ecc303da8cad55fd9b8bc9917d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:14.274173Z","signature_b64":"4OnUjzTQ0jCn14UbmaNIAd7iiZAkTPyYS33qHcE/4yBK20vDbjjf4mabl9ltUPqqh9iCYT57WATbQwQvM1XwCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"575115c4839e1a8eb8ce1b73d2bc552b049d15647f0a474ceb8f64c23b2fc7e8","last_reissued_at":"2026-05-17T23:58:14.273744Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:14.273744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Apollonian metric, uniformity and Gromov hyperbolicity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Matti Vuorinen, Qingshan Zhou, Yaxiang Li","submitted_at":"2018-05-19T00:14:42Z","abstract_excerpt":"The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, $\\varphi$-uniformity and $\\delta$-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of $\\mathbb{R}^n$ are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.07481","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":"1805.07481","created_at":"2026-05-17T23:58:14.273809+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.07481v2","created_at":"2026-05-17T23:58:14.273809+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.07481","created_at":"2026-05-17T23:58:14.273809+00:00"},{"alias_kind":"pith_short_12","alias_value":"K5IRLREDTYNI","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"K5IRLREDTYNI5OGO","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"K5IRLRED","created_at":"2026-05-18T12:32:33.847187+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/K5IRLREDTYNI5OGODNZ5FPCVFM","json":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM.json","graph_json":"https://pith.science/api/pith-number/K5IRLREDTYNI5OGODNZ5FPCVFM/graph.json","events_json":"https://pith.science/api/pith-number/K5IRLREDTYNI5OGODNZ5FPCVFM/events.json","paper":"https://pith.science/paper/K5IRLRED"},"agent_actions":{"view_html":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM","download_json":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM.json","view_paper":"https://pith.science/paper/K5IRLRED","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.07481&json=true","fetch_graph":"https://pith.science/api/pith-number/K5IRLREDTYNI5OGODNZ5FPCVFM/graph.json","fetch_events":"https://pith.science/api/pith-number/K5IRLREDTYNI5OGODNZ5FPCVFM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM/action/storage_attestation","attest_author":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM/action/author_attestation","sign_citation":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM/action/citation_signature","submit_replication":"https://pith.science/pith/K5IRLREDTYNI5OGODNZ5FPCVFM/action/replication_record"}},"created_at":"2026-05-17T23:58:14.273809+00:00","updated_at":"2026-05-17T23:58:14.273809+00:00"}