{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:LCNNKSV6ERRVNOXYRX4NWZWHGF","short_pith_number":"pith:LCNNKSV6","schema_version":"1.0","canonical_sha256":"589ad54abe246356baf88df8db66c731490b9b24905f5aa5f294084fb311230f","source":{"kind":"arxiv","id":"1902.05003","version":1},"attestation_state":"computed","paper":{"title":"An inequality related to M\\\"{o}bius transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Teerapong Suksumran, Themistocles M. Rassias","submitted_at":"2019-02-13T16:47:24Z","abstract_excerpt":"The open unit ball $\\mathbb{B} = \\{\\mathbf{v}\\in\\mathbb{R}^n\\colon\\|\\mathbf{v}\\|<1\\}$ is endowed with M\\\"{o}bius addition $\\oplus_M$ defined by $$\\mathbf{u}\\oplus_M\\mathbf{v} = \\dfrac{(1 + 2\\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{v}\\|^2)\\mathbf{u} + (1 - \\|\\mathbf{u}^2)\\mathbf{v}}{1 + \\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{u}\\|^2\\|\\mathbf{v}\\|^2\\|}$$ for all $\\mathbf{u},\\mathbf{v}\\in \\mathbf{B}$. In this article, we prove the inequality $$ \\dfrac{\\|\\mathbf{u}\\|-\\|\\mathbf{v}\\|}{1+\\|\\mathbf{u}\\|\\|\\mathbf{v}\\|}\\leq \\|\\mathbf{u}\\oplus_M \\mathbf{v}\\| \\leq \\dfrac{\\|\\mathbf{u}\\|+\\|\\m"},"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":"1902.05003","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2019-02-13T16:47:24Z","cross_cats_sorted":[],"title_canon_sha256":"7c8639c1605204cf7575e9a4444100bde52e6467c267812646411c2da8d4eb11","abstract_canon_sha256":"97145a14696535e8ed5bcae78fc8bacab41d782352f3a34657c605bb4d8928b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:03.661232Z","signature_b64":"x5B5LuSCffE7EX0KDdZyEC1/VJ2kAotxefGu1AqQq+dj+gHZdCcwMpSHWCSqfKz1mLeosj1PpPhFuhZ3MxOiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"589ad54abe246356baf88df8db66c731490b9b24905f5aa5f294084fb311230f","last_reissued_at":"2026-05-17T23:54:03.660771Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:03.660771Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An inequality related to M\\\"{o}bius transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Teerapong Suksumran, Themistocles M. Rassias","submitted_at":"2019-02-13T16:47:24Z","abstract_excerpt":"The open unit ball $\\mathbb{B} = \\{\\mathbf{v}\\in\\mathbb{R}^n\\colon\\|\\mathbf{v}\\|<1\\}$ is endowed with M\\\"{o}bius addition $\\oplus_M$ defined by $$\\mathbf{u}\\oplus_M\\mathbf{v} = \\dfrac{(1 + 2\\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{v}\\|^2)\\mathbf{u} + (1 - \\|\\mathbf{u}^2)\\mathbf{v}}{1 + \\langle\\mathbf{u},\\mathbf{v}\\rangle + \\|\\mathbf{u}\\|^2\\|\\mathbf{v}\\|^2\\|}$$ for all $\\mathbf{u},\\mathbf{v}\\in \\mathbf{B}$. In this article, we prove the inequality $$ \\dfrac{\\|\\mathbf{u}\\|-\\|\\mathbf{v}\\|}{1+\\|\\mathbf{u}\\|\\|\\mathbf{v}\\|}\\leq \\|\\mathbf{u}\\oplus_M \\mathbf{v}\\| \\leq \\dfrac{\\|\\mathbf{u}\\|+\\|\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.05003","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":"1902.05003","created_at":"2026-05-17T23:54:03.660837+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.05003v1","created_at":"2026-05-17T23:54:03.660837+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.05003","created_at":"2026-05-17T23:54:03.660837+00:00"},{"alias_kind":"pith_short_12","alias_value":"LCNNKSV6ERRV","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_16","alias_value":"LCNNKSV6ERRVNOXY","created_at":"2026-05-18T12:33:21.387695+00:00"},{"alias_kind":"pith_short_8","alias_value":"LCNNKSV6","created_at":"2026-05-18T12:33:21.387695+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.04895","citing_title":"HyperKG: Hyperbolic Knowledge Graph Embeddings for Knowledge Base Completion","ref_index":40,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF","json":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF.json","graph_json":"https://pith.science/api/pith-number/LCNNKSV6ERRVNOXYRX4NWZWHGF/graph.json","events_json":"https://pith.science/api/pith-number/LCNNKSV6ERRVNOXYRX4NWZWHGF/events.json","paper":"https://pith.science/paper/LCNNKSV6"},"agent_actions":{"view_html":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF","download_json":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF.json","view_paper":"https://pith.science/paper/LCNNKSV6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.05003&json=true","fetch_graph":"https://pith.science/api/pith-number/LCNNKSV6ERRVNOXYRX4NWZWHGF/graph.json","fetch_events":"https://pith.science/api/pith-number/LCNNKSV6ERRVNOXYRX4NWZWHGF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF/action/storage_attestation","attest_author":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF/action/author_attestation","sign_citation":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF/action/citation_signature","submit_replication":"https://pith.science/pith/LCNNKSV6ERRVNOXYRX4NWZWHGF/action/replication_record"}},"created_at":"2026-05-17T23:54:03.660837+00:00","updated_at":"2026-05-17T23:54:03.660837+00:00"}