{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:CC475V3QQBCDSUXRH6CPUDMH6C","short_pith_number":"pith:CC475V3Q","canonical_record":{"source":{"id":"1111.3812","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CA","submitted_at":"2011-11-16T14:26:14Z","cross_cats_sorted":[],"title_canon_sha256":"15aa64ff5304597afd2b5447493e6bc332d77c0ceea941d330d4bc70311cd1ce","abstract_canon_sha256":"c40757f67b2f415bfbc33481a9d0b8abc3512d7a2828882272f82219f115ce28"},"schema_version":"1.0"},"canonical_sha256":"10b9fed77080443952f13f84fa0d87f0a8fd11ea555643fbb8d2a2059059a6ad","source":{"kind":"arxiv","id":"1111.3812","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3812","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3812v4","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3812","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"pith_short_12","alias_value":"CC475V3QQBCD","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CC475V3QQBCDSUXR","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CC475V3Q","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:CC475V3QQBCDSUXRH6CPUDMH6C","target":"record","payload":{"canonical_record":{"source":{"id":"1111.3812","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CA","submitted_at":"2011-11-16T14:26:14Z","cross_cats_sorted":[],"title_canon_sha256":"15aa64ff5304597afd2b5447493e6bc332d77c0ceea941d330d4bc70311cd1ce","abstract_canon_sha256":"c40757f67b2f415bfbc33481a9d0b8abc3512d7a2828882272f82219f115ce28"},"schema_version":"1.0"},"canonical_sha256":"10b9fed77080443952f13f84fa0d87f0a8fd11ea555643fbb8d2a2059059a6ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:17.693293Z","signature_b64":"wiiVyHDHjwgufwuLG3wOjoUe9uyA4pSCc5Bky4sDUcpkPEIzG89/Vq2kaSIj0Vhg1QVMPlfRqqrrWKWhLTsdDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10b9fed77080443952f13f84fa0d87f0a8fd11ea555643fbb8d2a2059059a6ad","last_reissued_at":"2026-05-18T03:28:17.692662Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:17.692662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.3812","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:28:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iEPgs4yMewSRaB4/fiPsbe2RoRpMcXhyT5A2lkUxbRQTcYgZ/Su9XvXeqDsqDYhuFtIt4DVIO4Oe2JCAI+LeAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T05:39:04.240069Z"},"content_sha256":"8b90279ad7679a0ef2be4949202af7e01024faceb9f1c22585b0c758aafb22b3","schema_version":"1.0","event_id":"sha256:8b90279ad7679a0ef2be4949202af7e01024faceb9f1c22585b0c758aafb22b3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:CC475V3QQBCDSUXRH6CPUDMH6C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On exterior moduli of quadrilaterals and special functions","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Matti Vuorinen, Xiaohui Zhang","submitted_at":"2011-11-16T14:26:14Z","abstract_excerpt":"In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the properties of monotonicity and convexity of this function. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3812","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:28:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7tEmoH5nQNSCLgbYifTlAF/1GSpJQoA14BdtOJ9Wv2ZX6Rb10f1Fj3tR7UCqnea6RUpNLNStyq8b4mkHJsuHCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T05:39:04.240429Z"},"content_sha256":"7eb0e5afa3941f8329a33ad6fee6af80b120e51e577b579311612927411f3ae3","schema_version":"1.0","event_id":"sha256:7eb0e5afa3941f8329a33ad6fee6af80b120e51e577b579311612927411f3ae3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CC475V3QQBCDSUXRH6CPUDMH6C/bundle.json","state_url":"https://pith.science/pith/CC475V3QQBCDSUXRH6CPUDMH6C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CC475V3QQBCDSUXRH6CPUDMH6C/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T05:39:04Z","links":{"resolver":"https://pith.science/pith/CC475V3QQBCDSUXRH6CPUDMH6C","bundle":"https://pith.science/pith/CC475V3QQBCDSUXRH6CPUDMH6C/bundle.json","state":"https://pith.science/pith/CC475V3QQBCDSUXRH6CPUDMH6C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CC475V3QQBCDSUXRH6CPUDMH6C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CC475V3QQBCDSUXRH6CPUDMH6C","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c40757f67b2f415bfbc33481a9d0b8abc3512d7a2828882272f82219f115ce28","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CA","submitted_at":"2011-11-16T14:26:14Z","title_canon_sha256":"15aa64ff5304597afd2b5447493e6bc332d77c0ceea941d330d4bc70311cd1ce"},"schema_version":"1.0","source":{"id":"1111.3812","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.3812","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"arxiv_version","alias_value":"1111.3812v4","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.3812","created_at":"2026-05-18T03:28:17Z"},{"alias_kind":"pith_short_12","alias_value":"CC475V3QQBCD","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CC475V3QQBCDSUXR","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CC475V3Q","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:7eb0e5afa3941f8329a33ad6fee6af80b120e51e577b579311612927411f3ae3","target":"graph","created_at":"2026-05-18T03:28:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the properties of monotonicity and convexity of this function. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.","authors_text":"Matti Vuorinen, Xiaohui Zhang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CA","submitted_at":"2011-11-16T14:26:14Z","title":"On exterior moduli of quadrilaterals and special functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3812","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8b90279ad7679a0ef2be4949202af7e01024faceb9f1c22585b0c758aafb22b3","target":"record","created_at":"2026-05-18T03:28:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c40757f67b2f415bfbc33481a9d0b8abc3512d7a2828882272f82219f115ce28","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CA","submitted_at":"2011-11-16T14:26:14Z","title_canon_sha256":"15aa64ff5304597afd2b5447493e6bc332d77c0ceea941d330d4bc70311cd1ce"},"schema_version":"1.0","source":{"id":"1111.3812","kind":"arxiv","version":4}},"canonical_sha256":"10b9fed77080443952f13f84fa0d87f0a8fd11ea555643fbb8d2a2059059a6ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"10b9fed77080443952f13f84fa0d87f0a8fd11ea555643fbb8d2a2059059a6ad","first_computed_at":"2026-05-18T03:28:17.692662Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:17.692662Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wiiVyHDHjwgufwuLG3wOjoUe9uyA4pSCc5Bky4sDUcpkPEIzG89/Vq2kaSIj0Vhg1QVMPlfRqqrrWKWhLTsdDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:17.693293Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.3812","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b90279ad7679a0ef2be4949202af7e01024faceb9f1c22585b0c758aafb22b3","sha256:7eb0e5afa3941f8329a33ad6fee6af80b120e51e577b579311612927411f3ae3"],"state_sha256":"93d076b1fed039e9a8dd886c9fd4ebce3fcc7704cbe2db3f077f20125bb6ef17"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fvt9RvD5UpprHaeDCqasNUfedNDWpxRuKawG4futrs2vMad1xNNVanOukqhVQnWMISosgDQtjXhI5u9xYEXmAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T05:39:04.242286Z","bundle_sha256":"4320073f9cedf2a4936e17c732fbf5df590012e22e959edff159e65ce645c25a"}}