{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:E5WTNCLN4U3JLHS3WOSFRIGQVH","short_pith_number":"pith:E5WTNCLN","canonical_record":{"source":{"id":"1804.10760","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-28T08:17:57Z","cross_cats_sorted":[],"title_canon_sha256":"963c792ce194997f1c250c39f0d3c45d20bd7c69db011d87732df5ee05fead5b","abstract_canon_sha256":"4410164799bc91c10e9209596b74f5c1b576a654a22c609065a03d6c4fcb41b9"},"schema_version":"1.0"},"canonical_sha256":"276d36896de536959e5bb3a458a0d0a9f9382533a3b236af1ea4d015555a945c","source":{"kind":"arxiv","id":"1804.10760","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10760","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10760v1","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10760","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"pith_short_12","alias_value":"E5WTNCLN4U3J","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"E5WTNCLN4U3JLHS3","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"E5WTNCLN","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:E5WTNCLN4U3JLHS3WOSFRIGQVH","target":"record","payload":{"canonical_record":{"source":{"id":"1804.10760","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-28T08:17:57Z","cross_cats_sorted":[],"title_canon_sha256":"963c792ce194997f1c250c39f0d3c45d20bd7c69db011d87732df5ee05fead5b","abstract_canon_sha256":"4410164799bc91c10e9209596b74f5c1b576a654a22c609065a03d6c4fcb41b9"},"schema_version":"1.0"},"canonical_sha256":"276d36896de536959e5bb3a458a0d0a9f9382533a3b236af1ea4d015555a945c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:15.651528Z","signature_b64":"IjcHgbeL9vIDcvJhFhVVTNy13AtMpBF9jWidcaUX/y3pr9wcrJhsklk5ouYm8ZaaKhBu2yO8H3sxz4C0JCKfBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"276d36896de536959e5bb3a458a0d0a9f9382533a3b236af1ea4d015555a945c","last_reissued_at":"2026-05-18T00:17:15.650863Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:15.650863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.10760","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":"2026-05-18T00:17:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lB28P0lQ3qwv/nxvDzN6jLUv5gpmZ0AGklxCS5zMJlg0SsxsKDqkxMuFK+fT3shULDq5IElhSE/KE0Smy1dqDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:06:59.140173Z"},"content_sha256":"89250d682a06a9539f26e4a3ee8d2494ca73f8e20de5e3c1ae6c56a136480cf2","schema_version":"1.0","event_id":"sha256:89250d682a06a9539f26e4a3ee8d2494ca73f8e20de5e3c1ae6c56a136480cf2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:E5WTNCLN4U3JLHS3WOSFRIGQVH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recent developments of Lyapunov-type inequalities for fractional differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bashir Ahmad, Sotiris K. Ntouyas, Theodoros P. Horikis","submitted_at":"2018-04-28T08:17:57Z","abstract_excerpt":"A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm-Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal and impulsive conditions. Furthermore, our study includes Riemann-Liouville, Caputo, Hadamard, Prabhakar, Hilfer and conformable type fractional derivatives. Results for boundary value problems involving fractional $p$-Laplacian, fractional operators with nonsingular Mittag-Leffler kernels, $q$-difference, di"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10760","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":"2026-05-18T00:17:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wcbtKoUuol1DpRSOkWm3QJqs3Qb9wSlCUB4f57zLr9m0FCB12ear7IjNHCO7ggaZTew5R7z6czyLGybcOJlgBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:06:59.140531Z"},"content_sha256":"4097eb661f80cb08b9ebc0aab099366f84272a3bb2ef4555ac7d2b57404e8cd0","schema_version":"1.0","event_id":"sha256:4097eb661f80cb08b9ebc0aab099366f84272a3bb2ef4555ac7d2b57404e8cd0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/bundle.json","state_url":"https://pith.science/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/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":"2026-06-22T23:06:59Z","links":{"resolver":"https://pith.science/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH","bundle":"https://pith.science/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/bundle.json","state":"https://pith.science/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E5WTNCLN4U3JLHS3WOSFRIGQVH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:E5WTNCLN4U3JLHS3WOSFRIGQVH","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":"4410164799bc91c10e9209596b74f5c1b576a654a22c609065a03d6c4fcb41b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-28T08:17:57Z","title_canon_sha256":"963c792ce194997f1c250c39f0d3c45d20bd7c69db011d87732df5ee05fead5b"},"schema_version":"1.0","source":{"id":"1804.10760","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10760","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10760v1","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10760","created_at":"2026-05-18T00:17:15Z"},{"alias_kind":"pith_short_12","alias_value":"E5WTNCLN4U3J","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"E5WTNCLN4U3JLHS3","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"E5WTNCLN","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:4097eb661f80cb08b9ebc0aab099366f84272a3bb2ef4555ac7d2b57404e8cd0","target":"graph","created_at":"2026-05-18T00:17:15Z","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":"A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm-Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal and impulsive conditions. Furthermore, our study includes Riemann-Liouville, Caputo, Hadamard, Prabhakar, Hilfer and conformable type fractional derivatives. Results for boundary value problems involving fractional $p$-Laplacian, fractional operators with nonsingular Mittag-Leffler kernels, $q$-difference, di","authors_text":"Bashir Ahmad, Sotiris K. Ntouyas, Theodoros P. Horikis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-28T08:17:57Z","title":"Recent developments of Lyapunov-type inequalities for fractional differential equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10760","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":"sha256:89250d682a06a9539f26e4a3ee8d2494ca73f8e20de5e3c1ae6c56a136480cf2","target":"record","created_at":"2026-05-18T00:17:15Z","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":"4410164799bc91c10e9209596b74f5c1b576a654a22c609065a03d6c4fcb41b9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-28T08:17:57Z","title_canon_sha256":"963c792ce194997f1c250c39f0d3c45d20bd7c69db011d87732df5ee05fead5b"},"schema_version":"1.0","source":{"id":"1804.10760","kind":"arxiv","version":1}},"canonical_sha256":"276d36896de536959e5bb3a458a0d0a9f9382533a3b236af1ea4d015555a945c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"276d36896de536959e5bb3a458a0d0a9f9382533a3b236af1ea4d015555a945c","first_computed_at":"2026-05-18T00:17:15.650863Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:15.650863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IjcHgbeL9vIDcvJhFhVVTNy13AtMpBF9jWidcaUX/y3pr9wcrJhsklk5ouYm8ZaaKhBu2yO8H3sxz4C0JCKfBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:15.651528Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10760","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89250d682a06a9539f26e4a3ee8d2494ca73f8e20de5e3c1ae6c56a136480cf2","sha256:4097eb661f80cb08b9ebc0aab099366f84272a3bb2ef4555ac7d2b57404e8cd0"],"state_sha256":"8888d7dd64f0a143a394aaf7d30b674f73a903657c36aaaa65becc5af43b15e1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HcI5BrQ2oIxjhsTCsZ2fpbrGIT2g2BA9gxSzWF005H8V+cbv9RkuGe341ktHx9XnH2uXHIDwd13BI6Of2n2LAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:06:59.142448Z","bundle_sha256":"1433169a7545fa46f411f7fd58a38e85c9b4616a9dfb23e2724a5a16e241f646"}}