{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:UGPE65QQD2LXX3JZWH4VEVWFI6","short_pith_number":"pith:UGPE65QQ","canonical_record":{"source":{"id":"1405.2813","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-05-12T15:43:57Z","cross_cats_sorted":[],"title_canon_sha256":"77e8a0867ec1290642d04693fe9b6895fb88126d4c8fe58bcf9d67c02afdf971","abstract_canon_sha256":"ae6600352e998a484fa7f63cf2f663e709684ba0d3548af61695a33314920693"},"schema_version":"1.0"},"canonical_sha256":"a19e4f76101e977bed39b1f95256c5478f8a13b4ff9223a2c4260245135c6385","source":{"kind":"arxiv","id":"1405.2813","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2813","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2813v1","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2813","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"pith_short_12","alias_value":"UGPE65QQD2LX","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UGPE65QQD2LXX3JZ","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UGPE65QQ","created_at":"2026-05-18T12:28:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:UGPE65QQD2LXX3JZWH4VEVWFI6","target":"record","payload":{"canonical_record":{"source":{"id":"1405.2813","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-05-12T15:43:57Z","cross_cats_sorted":[],"title_canon_sha256":"77e8a0867ec1290642d04693fe9b6895fb88126d4c8fe58bcf9d67c02afdf971","abstract_canon_sha256":"ae6600352e998a484fa7f63cf2f663e709684ba0d3548af61695a33314920693"},"schema_version":"1.0"},"canonical_sha256":"a19e4f76101e977bed39b1f95256c5478f8a13b4ff9223a2c4260245135c6385","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:07.444827Z","signature_b64":"2MueqmAc7f0p1jrJyZ6vtCxRXEnf71MWt/UXGiFFGnvkPWgnTW96Hcn0oeWoXs5rwyEQCz1AdQK+7knPGCDrDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a19e4f76101e977bed39b1f95256c5478f8a13b4ff9223a2c4260245135c6385","last_reissued_at":"2026-05-18T02:32:07.444377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:07.444377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.2813","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":"2026-05-18T02:32:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"epOrMwaWUIPa/jKSbD01IjfomZ9QJaaiac2+70wV64ZIjJv1c6GLMcweurmO2+85edqJa8wbyimqroSQSbh5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:00:39.205208Z"},"content_sha256":"9b030a028587ff2f08f56028fef4ddb80a78eff157621323e992ee7f6c3ad221","schema_version":"1.0","event_id":"sha256:9b030a028587ff2f08f56028fef4ddb80a78eff157621323e992ee7f6c3ad221"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:UGPE65QQD2LXX3JZWH4VEVWFI6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Artur M. C. Brito da Cruz, Delfim F. M. Torres, Nadia Benkhettou","submitted_at":"2014-05-12T15:43:57Z","abstract_excerpt":"We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2813","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":"2026-05-18T02:32:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kW8L0k9RDyPTVqGLYk4ZBKWWYTbHr/ROvSJxLf9SylMGYhEkboPvlxcs9JeYQTaiXGEPc7gegy8i6sR+3hVdBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T12:00:39.205561Z"},"content_sha256":"7a7aed0bf993ac1431a43e8e19c29fb711c66ac8b3244bb112c16b0bab8b1eea","schema_version":"1.0","event_id":"sha256:7a7aed0bf993ac1431a43e8e19c29fb711c66ac8b3244bb112c16b0bab8b1eea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/bundle.json","state_url":"https://pith.science/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/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":"2026-06-26T12:00:39Z","links":{"resolver":"https://pith.science/pith/UGPE65QQD2LXX3JZWH4VEVWFI6","bundle":"https://pith.science/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/bundle.json","state":"https://pith.science/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UGPE65QQD2LXX3JZWH4VEVWFI6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:UGPE65QQD2LXX3JZWH4VEVWFI6","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":"ae6600352e998a484fa7f63cf2f663e709684ba0d3548af61695a33314920693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-05-12T15:43:57Z","title_canon_sha256":"77e8a0867ec1290642d04693fe9b6895fb88126d4c8fe58bcf9d67c02afdf971"},"schema_version":"1.0","source":{"id":"1405.2813","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.2813","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"arxiv_version","alias_value":"1405.2813v1","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.2813","created_at":"2026-05-18T02:32:07Z"},{"alias_kind":"pith_short_12","alias_value":"UGPE65QQD2LX","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UGPE65QQD2LXX3JZ","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UGPE65QQ","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:7a7aed0bf993ac1431a43e8e19c29fb711c66ac8b3244bb112c16b0bab8b1eea","target":"graph","created_at":"2026-05-18T02:32:07Z","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 introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.","authors_text":"Artur M. C. Brito da Cruz, Delfim F. M. Torres, Nadia Benkhettou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-05-12T15:43:57Z","title":"A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2813","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":"sha256:9b030a028587ff2f08f56028fef4ddb80a78eff157621323e992ee7f6c3ad221","target":"record","created_at":"2026-05-18T02:32:07Z","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":"ae6600352e998a484fa7f63cf2f663e709684ba0d3548af61695a33314920693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-05-12T15:43:57Z","title_canon_sha256":"77e8a0867ec1290642d04693fe9b6895fb88126d4c8fe58bcf9d67c02afdf971"},"schema_version":"1.0","source":{"id":"1405.2813","kind":"arxiv","version":1}},"canonical_sha256":"a19e4f76101e977bed39b1f95256c5478f8a13b4ff9223a2c4260245135c6385","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a19e4f76101e977bed39b1f95256c5478f8a13b4ff9223a2c4260245135c6385","first_computed_at":"2026-05-18T02:32:07.444377Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:07.444377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2MueqmAc7f0p1jrJyZ6vtCxRXEnf71MWt/UXGiFFGnvkPWgnTW96Hcn0oeWoXs5rwyEQCz1AdQK+7knPGCDrDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:07.444827Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.2813","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9b030a028587ff2f08f56028fef4ddb80a78eff157621323e992ee7f6c3ad221","sha256:7a7aed0bf993ac1431a43e8e19c29fb711c66ac8b3244bb112c16b0bab8b1eea"],"state_sha256":"2944fc22839d12e64836bc9f7324b8ac685f305420e291dcfb7a420f9ef84b31"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M8SC1EJdsTtpZLqlbCG6xN72In/YkIPQwjUkcvW6R8wvGGd5A7U6F3D4QmJpoegV5wlu8bDj9KBwve9VP/1VDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T12:00:39.207416Z","bundle_sha256":"8735df55657b65769356b5673ed18ca710f64fee77c85253e3bc12a0fe4be66b"}}