{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6FCXN7CRZ2YA3D2BVI4X5M2R5S","short_pith_number":"pith:6FCXN7CR","canonical_record":{"source":{"id":"1309.4899","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-19T09:07:02Z","cross_cats_sorted":["math.NA","math.OC"],"title_canon_sha256":"2f2c307a9a446cedef3805ac707fc6726c806a2c1165da10d0baa1f749e55bd8","abstract_canon_sha256":"1c1dee5808b863e82a89a8ef6741d651d63549def19df79849ca2343db31c585"},"schema_version":"1.0"},"canonical_sha256":"f14576fc51ceb00d8f41aa397eb351ec9a59c8d1021baff633700c6f95034bef","source":{"kind":"arxiv","id":"1309.4899","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4899","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4899v1","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4899","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"pith_short_12","alias_value":"6FCXN7CRZ2YA","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6FCXN7CRZ2YA3D2B","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6FCXN7CR","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6FCXN7CRZ2YA3D2BVI4X5M2R5S","target":"record","payload":{"canonical_record":{"source":{"id":"1309.4899","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-19T09:07:02Z","cross_cats_sorted":["math.NA","math.OC"],"title_canon_sha256":"2f2c307a9a446cedef3805ac707fc6726c806a2c1165da10d0baa1f749e55bd8","abstract_canon_sha256":"1c1dee5808b863e82a89a8ef6741d651d63549def19df79849ca2343db31c585"},"schema_version":"1.0"},"canonical_sha256":"f14576fc51ceb00d8f41aa397eb351ec9a59c8d1021baff633700c6f95034bef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:46.682953Z","signature_b64":"AknpNkGT/RXtYAOik0Sw1nr6126drRPIlQn8LTgXCOdBmboCbzUOIaYHMzg5qUCMO74d41iHasTTdif+mAZlCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f14576fc51ceb00d8f41aa397eb351ec9a59c8d1021baff633700c6f95034bef","last_reissued_at":"2026-05-18T03:08:46.682462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:46.682462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.4899","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":"2026-05-18T03:08:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"me312qgbmT+mWf/bNchp11wM78G9Ra5xn3Kmspwd1PSid8Lge3umGPEm7UE7eE/NXTXxmIaaAVXDDHxwwg8vDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:41:53.127406Z"},"content_sha256":"0120133bde50448d066fa73dbf6c2827c827df901400b66d3878a1ab9b08b63f","schema_version":"1.0","event_id":"sha256:0120133bde50448d066fa73dbf6c2827c827df901400b66d3878a1ab9b08b63f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6FCXN7CRZ2YA3D2BVI4X5M2R5S","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An expansion formula with higher-order derivatives for fractional operators of variable order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA","math.OC"],"primary_cat":"math.CA","authors_text":"Delfim F. M. Torres, Ricardo Almeida","submitted_at":"2013-09-19T09:07:02Z","abstract_excerpt":"We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations and problems of the calculus of variations that depend on fractional derivatives of Marchaud type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4899","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":"2026-05-18T03:08:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Se5KIe0TMenW2QQgoayHD2qSMSZx8/BvdsfShFyA2wEqxH6GJW3m6Egw//y5cM653cOI6WauZ3Z3TyrQ17ToBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:41:53.127761Z"},"content_sha256":"280bf8af50bf27847830047b8ce9e0f951ba4610aff4f0ecdad54dc9908e37fa","schema_version":"1.0","event_id":"sha256:280bf8af50bf27847830047b8ce9e0f951ba4610aff4f0ecdad54dc9908e37fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/bundle.json","state_url":"https://pith.science/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/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":"2026-06-03T04:41:53Z","links":{"resolver":"https://pith.science/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S","bundle":"https://pith.science/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/bundle.json","state":"https://pith.science/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6FCXN7CRZ2YA3D2BVI4X5M2R5S/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6FCXN7CRZ2YA3D2BVI4X5M2R5S","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":"1c1dee5808b863e82a89a8ef6741d651d63549def19df79849ca2343db31c585","cross_cats_sorted":["math.NA","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-19T09:07:02Z","title_canon_sha256":"2f2c307a9a446cedef3805ac707fc6726c806a2c1165da10d0baa1f749e55bd8"},"schema_version":"1.0","source":{"id":"1309.4899","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4899","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4899v1","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4899","created_at":"2026-05-18T03:08:46Z"},{"alias_kind":"pith_short_12","alias_value":"6FCXN7CRZ2YA","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6FCXN7CRZ2YA3D2B","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6FCXN7CR","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:280bf8af50bf27847830047b8ce9e0f951ba4610aff4f0ecdad54dc9908e37fa","target":"graph","created_at":"2026-05-18T03:08:46Z","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":"We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.","authors_text":"Delfim F. M. Torres, Ricardo Almeida","cross_cats":["math.NA","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-19T09:07:02Z","title":"An expansion formula with higher-order derivatives for fractional operators of variable order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4899","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":"sha256:0120133bde50448d066fa73dbf6c2827c827df901400b66d3878a1ab9b08b63f","target":"record","created_at":"2026-05-18T03:08:46Z","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":"1c1dee5808b863e82a89a8ef6741d651d63549def19df79849ca2343db31c585","cross_cats_sorted":["math.NA","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-09-19T09:07:02Z","title_canon_sha256":"2f2c307a9a446cedef3805ac707fc6726c806a2c1165da10d0baa1f749e55bd8"},"schema_version":"1.0","source":{"id":"1309.4899","kind":"arxiv","version":1}},"canonical_sha256":"f14576fc51ceb00d8f41aa397eb351ec9a59c8d1021baff633700c6f95034bef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f14576fc51ceb00d8f41aa397eb351ec9a59c8d1021baff633700c6f95034bef","first_computed_at":"2026-05-18T03:08:46.682462Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:08:46.682462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AknpNkGT/RXtYAOik0Sw1nr6126drRPIlQn8LTgXCOdBmboCbzUOIaYHMzg5qUCMO74d41iHasTTdif+mAZlCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:08:46.682953Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4899","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0120133bde50448d066fa73dbf6c2827c827df901400b66d3878a1ab9b08b63f","sha256:280bf8af50bf27847830047b8ce9e0f951ba4610aff4f0ecdad54dc9908e37fa"],"state_sha256":"f21a448bf984579a3b143fc8da8e509948d5cfeb230564c9adad9e24e0cf4d2b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZZh1df5j4mU7NjX4RYBMiX9t5HhU4qgcv+8ZylXAltJxuNKVEVUf/bKN9bHzgf4eBQ/653SxXq3uyJm63jLwCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:41:53.129747Z","bundle_sha256":"119ceda352bf64f85af37743df44210ad9fa92fa32a73cd54efdf37ba858cd36"}}