{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:FW5RLKDLKAHV2F3G45HM22S6JH","short_pith_number":"pith:FW5RLKDL","canonical_record":{"source":{"id":"1706.05841","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-06-19T09:20:51Z","cross_cats_sorted":[],"title_canon_sha256":"09fcbaee80f68f4d26f0868bbaf13e397c3b3c48243d7433ffe6bd7848f90b3e","abstract_canon_sha256":"45287028b89e26bbfa49df2b8dcbaaffa25cdce3841fe9a59d6e93259bc7cdb5"},"schema_version":"1.0"},"canonical_sha256":"2dbb15a86b500f5d1766e74ecd6a5e49fd85edfc0e6f1471b9575a41401f73e8","source":{"kind":"arxiv","id":"1706.05841","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.05841","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"arxiv_version","alias_value":"1706.05841v1","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05841","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"pith_short_12","alias_value":"FW5RLKDLKAHV","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FW5RLKDLKAHV2F3G","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FW5RLKDL","created_at":"2026-05-18T12:31:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:FW5RLKDLKAHV2F3G45HM22S6JH","target":"record","payload":{"canonical_record":{"source":{"id":"1706.05841","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-06-19T09:20:51Z","cross_cats_sorted":[],"title_canon_sha256":"09fcbaee80f68f4d26f0868bbaf13e397c3b3c48243d7433ffe6bd7848f90b3e","abstract_canon_sha256":"45287028b89e26bbfa49df2b8dcbaaffa25cdce3841fe9a59d6e93259bc7cdb5"},"schema_version":"1.0"},"canonical_sha256":"2dbb15a86b500f5d1766e74ecd6a5e49fd85edfc0e6f1471b9575a41401f73e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:05.795110Z","signature_b64":"aW43bUzzyILfLPaSYpW92Ji84607/JoMn8upVDicHAY1VURHd+EwG4QFMCSPTlkpDwjDwbwHdUHRYySRGjk/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2dbb15a86b500f5d1766e74ecd6a5e49fd85edfc0e6f1471b9575a41401f73e8","last_reissued_at":"2026-05-18T00:09:05.794599Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:05.794599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.05841","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:09:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TN5Bk9TJKQlQMbXsLl7jPpg51vbpkdtwiE3ruab8koWtYPGp0JKsMxrIXjxzhGJRjrP1wd+MIin8HMe2O/CwAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:00:01.881227Z"},"content_sha256":"41b41fd8ef7110c578708167d4ca8cbd625fece05914e003fa368e4ac7d48cda","schema_version":"1.0","event_id":"sha256:41b41fd8ef7110c578708167d4ca8cbd625fece05914e003fa368e4ac7d48cda"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:FW5RLKDLKAHV2F3G45HM22S6JH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some results on $\\varphi$--convex functions and geodesic $\\varphi$-convex functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Absos Ali Shaikh, Akhlad Iqbal, Chandan Kumar Mondal","submitted_at":"2017-06-19T09:20:51Z","abstract_excerpt":"As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $\\varphi$-convex function and deduce some basic properties of $\\varphi$-convex function and geodesic $\\varphi$-convex function. We also introduce the concept of geodesic $\\varphi$-convex set and $\\varphi$-epigraph and investigate a characterization of geodesic $\\varphi$-convex functions in terms of their $\\varphi$-epigraphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05841","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:09:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZQ9O3ZyXM4qWSoKhnmCBlvSBBXFG2zlQUmpgXnSfWtMcdZXhm9W72X3CRQWMXgPtk/n/fK4fYVdDWuNN5cRnAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:00:01.881906Z"},"content_sha256":"b98ed98de7d22fad7a1d6beb3c2b1c6c7ca7770bb24e1248eb1f6f59421b8274","schema_version":"1.0","event_id":"sha256:b98ed98de7d22fad7a1d6beb3c2b1c6c7ca7770bb24e1248eb1f6f59421b8274"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FW5RLKDLKAHV2F3G45HM22S6JH/bundle.json","state_url":"https://pith.science/pith/FW5RLKDLKAHV2F3G45HM22S6JH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FW5RLKDLKAHV2F3G45HM22S6JH/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":"2026-05-25T21:00:01Z","links":{"resolver":"https://pith.science/pith/FW5RLKDLKAHV2F3G45HM22S6JH","bundle":"https://pith.science/pith/FW5RLKDLKAHV2F3G45HM22S6JH/bundle.json","state":"https://pith.science/pith/FW5RLKDLKAHV2F3G45HM22S6JH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FW5RLKDLKAHV2F3G45HM22S6JH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:FW5RLKDLKAHV2F3G45HM22S6JH","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":"45287028b89e26bbfa49df2b8dcbaaffa25cdce3841fe9a59d6e93259bc7cdb5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-06-19T09:20:51Z","title_canon_sha256":"09fcbaee80f68f4d26f0868bbaf13e397c3b3c48243d7433ffe6bd7848f90b3e"},"schema_version":"1.0","source":{"id":"1706.05841","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.05841","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"arxiv_version","alias_value":"1706.05841v1","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.05841","created_at":"2026-05-18T00:09:05Z"},{"alias_kind":"pith_short_12","alias_value":"FW5RLKDLKAHV","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_16","alias_value":"FW5RLKDLKAHV2F3G","created_at":"2026-05-18T12:31:15Z"},{"alias_kind":"pith_short_8","alias_value":"FW5RLKDL","created_at":"2026-05-18T12:31:15Z"}],"graph_snapshots":[{"event_id":"sha256:b98ed98de7d22fad7a1d6beb3c2b1c6c7ca7770bb24e1248eb1f6f59421b8274","target":"graph","created_at":"2026-05-18T00:09:05Z","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":"As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $\\varphi$-convex function and deduce some basic properties of $\\varphi$-convex function and geodesic $\\varphi$-convex function. We also introduce the concept of geodesic $\\varphi$-convex set and $\\varphi$-epigraph and investigate a characterization of geodesic $\\varphi$-convex functions in terms of their $\\varphi$-epigraphs.","authors_text":"Absos Ali Shaikh, Akhlad Iqbal, Chandan Kumar Mondal","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-06-19T09:20:51Z","title":"Some results on $\\varphi$--convex functions and geodesic $\\varphi$-convex functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05841","kind":"arxiv","version":1},"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":"sha256:41b41fd8ef7110c578708167d4ca8cbd625fece05914e003fa368e4ac7d48cda","target":"record","created_at":"2026-05-18T00:09:05Z","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":"45287028b89e26bbfa49df2b8dcbaaffa25cdce3841fe9a59d6e93259bc7cdb5","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2017-06-19T09:20:51Z","title_canon_sha256":"09fcbaee80f68f4d26f0868bbaf13e397c3b3c48243d7433ffe6bd7848f90b3e"},"schema_version":"1.0","source":{"id":"1706.05841","kind":"arxiv","version":1}},"canonical_sha256":"2dbb15a86b500f5d1766e74ecd6a5e49fd85edfc0e6f1471b9575a41401f73e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2dbb15a86b500f5d1766e74ecd6a5e49fd85edfc0e6f1471b9575a41401f73e8","first_computed_at":"2026-05-18T00:09:05.794599Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:05.794599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aW43bUzzyILfLPaSYpW92Ji84607/JoMn8upVDicHAY1VURHd+EwG4QFMCSPTlkpDwjDwbwHdUHRYySRGjk/Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:05.795110Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.05841","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41b41fd8ef7110c578708167d4ca8cbd625fece05914e003fa368e4ac7d48cda","sha256:b98ed98de7d22fad7a1d6beb3c2b1c6c7ca7770bb24e1248eb1f6f59421b8274"],"state_sha256":"2ef6bf29aeaf1f781aee7e28ae94dca3aedfc3caf3fdea590ffcb2b71d9e9b74"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N+EJ8fpXNpTa6z1q3viT+347ORwuM1ZGyZ0dQB1C2fW5HCuVIhVyBXasX/AYI3qqV90bX7gPJW7BDkSAar7XBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T21:00:01.885699Z","bundle_sha256":"883eabacc770d27738062abfba9912573326237ab92501cf53f1978fdbddd00e"}}