{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:7WOLORVESBUGHV3H3DKTFXYREJ","short_pith_number":"pith:7WOLORVE","canonical_record":{"source":{"id":"1507.07798","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-25T14:06:47Z","cross_cats_sorted":[],"title_canon_sha256":"52b24bd3ea877c9aab6dbf83a3b1fbdcfd4eec4e8fdf146ac34899f05298b22c","abstract_canon_sha256":"aa2aded6691a02c34dc4672eb2f01367139b5c14991cb3a506ba0e91a192ca3b"},"schema_version":"1.0"},"canonical_sha256":"fd9cb746a4906863d767d8d532df112264e5f9cf53791b9a2cd2c09c50593735","source":{"kind":"arxiv","id":"1507.07798","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07798","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07798v1","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07798","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"pith_short_12","alias_value":"7WOLORVESBUG","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7WOLORVESBUGHV3H","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7WOLORVE","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:7WOLORVESBUGHV3H3DKTFXYREJ","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07798","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-25T14:06:47Z","cross_cats_sorted":[],"title_canon_sha256":"52b24bd3ea877c9aab6dbf83a3b1fbdcfd4eec4e8fdf146ac34899f05298b22c","abstract_canon_sha256":"aa2aded6691a02c34dc4672eb2f01367139b5c14991cb3a506ba0e91a192ca3b"},"schema_version":"1.0"},"canonical_sha256":"fd9cb746a4906863d767d8d532df112264e5f9cf53791b9a2cd2c09c50593735","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:11.870931Z","signature_b64":"IGy/qzNdz78sLQbXctAM8NWfm5wTG/05jy/0QhAqQ9XFzSBSB77MeF57umxi5B/kf1QoAwy6H8Q4Q/jFqFsbAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd9cb746a4906863d767d8d532df112264e5f9cf53791b9a2cd2c09c50593735","last_reissued_at":"2026-05-18T01:36:11.870305Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:11.870305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07798","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":"2026-05-18T01:36:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LKv1NzzF4fObGwIIRNKjlt3GyDcnJXQG3r5LfQ0daPxDMbj5R80sQ85eFd20ySa2ejRz3uRsiY4yOgYJCco2Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:00:11.949624Z"},"content_sha256":"b47627fe216a5bd75ad59c992a6b89f2ca1533bf2893fd0ef5990a319bbe6f3f","schema_version":"1.0","event_id":"sha256:b47627fe216a5bd75ad59c992a6b89f2ca1533bf2893fd0ef5990a319bbe6f3f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:7WOLORVESBUGHV3H3DKTFXYREJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Invalidity of Fourier Series Expansions of Fractional Order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ahmed I. Zayed, Peter Massopust","submitted_at":"2015-07-25T14:06:47Z","abstract_excerpt":"The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{in\\omega x}$ are replaced by the Mittag-Leffler functions $E_\\alpha \\left (i (n\\omega x)^\\alpha\\right) ,$ over the interval $[0, M_\\alpha/ \\omega]$ where $0< \\omega<\\infty $ and $M_\\alpha$ is the period of the function $E_\\alpha \\left( ix^\\alpha\\right),$ i.e., $E_\\alpha \\left( ix^\\alpha\\right)=E_\\alpha \\left( i(x+M_\\alpha)^\\alpha\\right).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07798","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":"2026-05-18T01:36:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sVCJQBSPGf6K3SoQLe1JsA/4rcbnX/Shb1oKM1IF5dgoHlAt9bolsJcSE6UpgX9iLhjlJFLhy8iTTuDKiyx+Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T13:00:11.949954Z"},"content_sha256":"b2e591760f51c70540bc4f86f5e31198a957473cad59e76fed44a0a9b6ad07e8","schema_version":"1.0","event_id":"sha256:b2e591760f51c70540bc4f86f5e31198a957473cad59e76fed44a0a9b6ad07e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7WOLORVESBUGHV3H3DKTFXYREJ/bundle.json","state_url":"https://pith.science/pith/7WOLORVESBUGHV3H3DKTFXYREJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7WOLORVESBUGHV3H3DKTFXYREJ/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":"2026-06-22T13:00:11Z","links":{"resolver":"https://pith.science/pith/7WOLORVESBUGHV3H3DKTFXYREJ","bundle":"https://pith.science/pith/7WOLORVESBUGHV3H3DKTFXYREJ/bundle.json","state":"https://pith.science/pith/7WOLORVESBUGHV3H3DKTFXYREJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7WOLORVESBUGHV3H3DKTFXYREJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7WOLORVESBUGHV3H3DKTFXYREJ","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":"aa2aded6691a02c34dc4672eb2f01367139b5c14991cb3a506ba0e91a192ca3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-25T14:06:47Z","title_canon_sha256":"52b24bd3ea877c9aab6dbf83a3b1fbdcfd4eec4e8fdf146ac34899f05298b22c"},"schema_version":"1.0","source":{"id":"1507.07798","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07798","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07798v1","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07798","created_at":"2026-05-18T01:36:11Z"},{"alias_kind":"pith_short_12","alias_value":"7WOLORVESBUG","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7WOLORVESBUGHV3H","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7WOLORVE","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:b2e591760f51c70540bc4f86f5e31198a957473cad59e76fed44a0a9b6ad07e8","target":"graph","created_at":"2026-05-18T01:36:11Z","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":"The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions $e^{in\\omega x}$ are replaced by the Mittag-Leffler functions $E_\\alpha \\left (i (n\\omega x)^\\alpha\\right) ,$ over the interval $[0, M_\\alpha/ \\omega]$ where $0< \\omega<\\infty $ and $M_\\alpha$ is the period of the function $E_\\alpha \\left( ix^\\alpha\\right),$ i.e., $E_\\alpha \\left( ix^\\alpha\\right)=E_\\alpha \\left( i(x+M_\\alpha)^\\alpha\\right).$","authors_text":"Ahmed I. Zayed, Peter Massopust","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-25T14:06:47Z","title":"On the Invalidity of Fourier Series Expansions of Fractional Order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07798","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:b47627fe216a5bd75ad59c992a6b89f2ca1533bf2893fd0ef5990a319bbe6f3f","target":"record","created_at":"2026-05-18T01:36:11Z","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":"aa2aded6691a02c34dc4672eb2f01367139b5c14991cb3a506ba0e91a192ca3b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-25T14:06:47Z","title_canon_sha256":"52b24bd3ea877c9aab6dbf83a3b1fbdcfd4eec4e8fdf146ac34899f05298b22c"},"schema_version":"1.0","source":{"id":"1507.07798","kind":"arxiv","version":1}},"canonical_sha256":"fd9cb746a4906863d767d8d532df112264e5f9cf53791b9a2cd2c09c50593735","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd9cb746a4906863d767d8d532df112264e5f9cf53791b9a2cd2c09c50593735","first_computed_at":"2026-05-18T01:36:11.870305Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:11.870305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IGy/qzNdz78sLQbXctAM8NWfm5wTG/05jy/0QhAqQ9XFzSBSB77MeF57umxi5B/kf1QoAwy6H8Q4Q/jFqFsbAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:11.870931Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07798","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b47627fe216a5bd75ad59c992a6b89f2ca1533bf2893fd0ef5990a319bbe6f3f","sha256:b2e591760f51c70540bc4f86f5e31198a957473cad59e76fed44a0a9b6ad07e8"],"state_sha256":"5e0c74549966b4f0e82561481688c90f6dddb3e6aff53c09df205a3d1cd49b87"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xUOjLq80iM2+wJJikdfeJGIw04QS2mN4qFU1YYKuF6UZb5H3LZrk/A9ymEm5GBDhjHAXgLA/G6+JvgR9hpD5AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T13:00:11.951783Z","bundle_sha256":"89bf33e9aedbb85fcc80c1ab1af307da13e4757ea2f1a8b4d5117a9bf2617e23"}}