{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:5YXSPPIDQL2V2FDUGNNCHLDD3N","short_pith_number":"pith:5YXSPPID","canonical_record":{"source":{"id":"1805.00720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-05-02T10:39:07Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"20b9c0c3e167b76ce452e39002cfa026199a4e1ccc246fbefd402d5162157a2c","abstract_canon_sha256":"6839bcac99ed09643f1acb564fd7af5a74957f3b92c0cfb207695f61e867f5cb"},"schema_version":"1.0"},"canonical_sha256":"ee2f27bd0382f55d1474335a23ac63db7c0caede03718b51b8352ef9b9c57bfb","source":{"kind":"arxiv","id":"1805.00720","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00720","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00720v2","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00720","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"pith_short_12","alias_value":"5YXSPPIDQL2V","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"5YXSPPIDQL2V2FDU","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"5YXSPPID","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:5YXSPPIDQL2V2FDUGNNCHLDD3N","target":"record","payload":{"canonical_record":{"source":{"id":"1805.00720","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-05-02T10:39:07Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"20b9c0c3e167b76ce452e39002cfa026199a4e1ccc246fbefd402d5162157a2c","abstract_canon_sha256":"6839bcac99ed09643f1acb564fd7af5a74957f3b92c0cfb207695f61e867f5cb"},"schema_version":"1.0"},"canonical_sha256":"ee2f27bd0382f55d1474335a23ac63db7c0caede03718b51b8352ef9b9c57bfb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:04.143602Z","signature_b64":"ZAkDoEN+nKtttbiQKrPoF6FBYzCKlfLg/rL114UvkTfxI5a4WYueoSpu4EiSQhggZZ1S7oG/62Fy3+9H233UDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee2f27bd0382f55d1474335a23ac63db7c0caede03718b51b8352ef9b9c57bfb","last_reissued_at":"2026-05-18T00:13:04.142829Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:04.142829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.00720","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:13:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"byAxWnj5/2lQ/64OjGffOWTGfKWU93qIhzQmth31jpcUE6RTZ7I/3opxG1jCcEmTXqoqh2DmWknwOu45y2EdBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:26:07.967297Z"},"content_sha256":"acca009a6f21d5834caf53619b584dbcb490768e1b6f712a7bbb91e63248d0e6","schema_version":"1.0","event_id":"sha256:acca009a6f21d5834caf53619b584dbcb490768e1b6f712a7bbb91e63248d0e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:5YXSPPIDQL2V2FDUGNNCHLDD3N","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Variable-Order Fractional Calculus of Variations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.OC","authors_text":"Delfim F. M. Torres, Dina Tavares, Ricardo Almeida","submitted_at":"2018-05-02T10:39:07Z","abstract_excerpt":"This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00720","kind":"arxiv","version":2},"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":"2026-05-18T00:13:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uUvERsnE4Q/1IuotTtiny2C5xCUA7oleCvs1H7ABUIE/Aq4V2V4xckbQK2ieHaHTznjcMQ3GSbhW08kjMwVaAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:26:07.967668Z"},"content_sha256":"95942e1db047bda9568e43a137e3fa8210de7fa4804012c9df511bb99efeb5a4","schema_version":"1.0","event_id":"sha256:95942e1db047bda9568e43a137e3fa8210de7fa4804012c9df511bb99efeb5a4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/bundle.json","state_url":"https://pith.science/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/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":"2026-06-03T05:26:07Z","links":{"resolver":"https://pith.science/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N","bundle":"https://pith.science/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/bundle.json","state":"https://pith.science/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5YXSPPIDQL2V2FDUGNNCHLDD3N/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:5YXSPPIDQL2V2FDUGNNCHLDD3N","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":"6839bcac99ed09643f1acb564fd7af5a74957f3b92c0cfb207695f61e867f5cb","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-05-02T10:39:07Z","title_canon_sha256":"20b9c0c3e167b76ce452e39002cfa026199a4e1ccc246fbefd402d5162157a2c"},"schema_version":"1.0","source":{"id":"1805.00720","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00720","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00720v2","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00720","created_at":"2026-05-18T00:13:04Z"},{"alias_kind":"pith_short_12","alias_value":"5YXSPPIDQL2V","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"5YXSPPIDQL2V2FDU","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"5YXSPPID","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:95942e1db047bda9568e43a137e3fa8210de7fa4804012c9df511bb99efeb5a4","target":"graph","created_at":"2026-05-18T00:13:04Z","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":"This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivativ","authors_text":"Delfim F. M. Torres, Dina Tavares, Ricardo Almeida","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-05-02T10:39:07Z","title":"The Variable-Order Fractional Calculus of Variations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00720","kind":"arxiv","version":2},"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:acca009a6f21d5834caf53619b584dbcb490768e1b6f712a7bbb91e63248d0e6","target":"record","created_at":"2026-05-18T00:13:04Z","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":"6839bcac99ed09643f1acb564fd7af5a74957f3b92c0cfb207695f61e867f5cb","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-05-02T10:39:07Z","title_canon_sha256":"20b9c0c3e167b76ce452e39002cfa026199a4e1ccc246fbefd402d5162157a2c"},"schema_version":"1.0","source":{"id":"1805.00720","kind":"arxiv","version":2}},"canonical_sha256":"ee2f27bd0382f55d1474335a23ac63db7c0caede03718b51b8352ef9b9c57bfb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee2f27bd0382f55d1474335a23ac63db7c0caede03718b51b8352ef9b9c57bfb","first_computed_at":"2026-05-18T00:13:04.142829Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:04.142829Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZAkDoEN+nKtttbiQKrPoF6FBYzCKlfLg/rL114UvkTfxI5a4WYueoSpu4EiSQhggZZ1S7oG/62Fy3+9H233UDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:04.143602Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.00720","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:acca009a6f21d5834caf53619b584dbcb490768e1b6f712a7bbb91e63248d0e6","sha256:95942e1db047bda9568e43a137e3fa8210de7fa4804012c9df511bb99efeb5a4"],"state_sha256":"2ea85c82658035aae00e6c5992446c6377459dc461afe5d9d9e03e01bf0203f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x44hRUG2BiFxxlvSeQAdBEY5Z8vbPc5YG4rhTUf4UsKzLP1KkAL3t3FtnvJhdS3kEhcruw+knV1LRcA5GmopBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:26:07.969664Z","bundle_sha256":"26218556ed96e40f1fa450c0a026bd445ce738dfd177f3e0c2545063b2df890f"}}