{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:RT4FVZHUW6E7GGKPWBHDMQT2LV","short_pith_number":"pith:RT4FVZHU","canonical_record":{"source":{"id":"1808.06563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-17T15:47:08Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"ec23dfee4f7b24a371fbcefabb9943fba2e2bbd6b66191c826a877992d29d805","abstract_canon_sha256":"029926b9979f7b94f537188587a051dcde305287deacf98d5b5068cbad3ee05d"},"schema_version":"1.0"},"canonical_sha256":"8cf85ae4f4b789f3194fb04e36427a5d6c8375f71935e56aaec474990bddf5d9","source":{"kind":"arxiv","id":"1808.06563","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.06563","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"arxiv_version","alias_value":"1808.06563v1","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06563","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"pith_short_12","alias_value":"RT4FVZHUW6E7","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RT4FVZHUW6E7GGKP","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RT4FVZHU","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:RT4FVZHUW6E7GGKPWBHDMQT2LV","target":"record","payload":{"canonical_record":{"source":{"id":"1808.06563","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-17T15:47:08Z","cross_cats_sorted":["math-ph","math.CV","math.MP"],"title_canon_sha256":"ec23dfee4f7b24a371fbcefabb9943fba2e2bbd6b66191c826a877992d29d805","abstract_canon_sha256":"029926b9979f7b94f537188587a051dcde305287deacf98d5b5068cbad3ee05d"},"schema_version":"1.0"},"canonical_sha256":"8cf85ae4f4b789f3194fb04e36427a5d6c8375f71935e56aaec474990bddf5d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:44.437937Z","signature_b64":"36izLgRm7809Qwv7phZZpHIoYV7yZHWkl+ArhWIxq289dLtYaFDy8crbEsh6srhoUBzkNCAlewTPTX1VaVKPDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8cf85ae4f4b789f3194fb04e36427a5d6c8375f71935e56aaec474990bddf5d9","last_reissued_at":"2026-05-18T00:07:44.437206Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:44.437206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1808.06563","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":"2026-05-18T00:07:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XTtdOVdkiNbWrx51tK+e3/5OgCca6Qv2quotmTJVjp6OTDFbgYYE1ysB9DvHtZ56KLguAGWyDa9amIeEUoNUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T12:44:20.715976Z"},"content_sha256":"6e070b1959be4d4b438826bfefffa4df32797380431143162ea6e650e95946c2","schema_version":"1.0","event_id":"sha256:6e070b1959be4d4b438826bfefffa4df32797380431143162ea6e650e95946c2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:RT4FVZHUW6E7GGKPWBHDMQT2LV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Mittag-Leffler function in the thinning theory for renewal processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CV","math.MP"],"primary_cat":"math.PR","authors_text":"Francesco Mainardi, Rudolf Gorenflo","submitted_at":"2018-08-17T15:47:08Z","abstract_excerpt":"The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06563","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":"2026-05-18T00:07:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1NcqKsgJpmh09k5tA2kaY0z8gLOYeC+1vUrqiLTiF2ZSmQY/zNcZjaflEqrFoH72g4AYS0yGPsIvbjS+iV0YBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T12:44:20.716342Z"},"content_sha256":"8bcd5ee02df9fcfb400c239e92e7f14e510bfe2c8c41100958be99820cb29ae8","schema_version":"1.0","event_id":"sha256:8bcd5ee02df9fcfb400c239e92e7f14e510bfe2c8c41100958be99820cb29ae8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/bundle.json","state_url":"https://pith.science/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/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":"2026-06-03T12:44:20Z","links":{"resolver":"https://pith.science/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV","bundle":"https://pith.science/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/bundle.json","state":"https://pith.science/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RT4FVZHUW6E7GGKPWBHDMQT2LV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RT4FVZHUW6E7GGKPWBHDMQT2LV","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":"029926b9979f7b94f537188587a051dcde305287deacf98d5b5068cbad3ee05d","cross_cats_sorted":["math-ph","math.CV","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-17T15:47:08Z","title_canon_sha256":"ec23dfee4f7b24a371fbcefabb9943fba2e2bbd6b66191c826a877992d29d805"},"schema_version":"1.0","source":{"id":"1808.06563","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.06563","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"arxiv_version","alias_value":"1808.06563v1","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06563","created_at":"2026-05-18T00:07:44Z"},{"alias_kind":"pith_short_12","alias_value":"RT4FVZHUW6E7","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RT4FVZHUW6E7GGKP","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RT4FVZHU","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:8bcd5ee02df9fcfb400c239e92e7f14e510bfe2c8c41100958be99820cb29ae8","target":"graph","created_at":"2026-05-18T00:07:44Z","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 main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in the theory of continuous random walk and consequently of fractional diffusion in a plenary lecture by the late Prof Gorenflo at a Seminar on Anomalous Transport held in Bad-Honnef in July 2006, published in a 2008 book. After recalling the basic the","authors_text":"Francesco Mainardi, Rudolf Gorenflo","cross_cats":["math-ph","math.CV","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-17T15:47:08Z","title":"The Mittag-Leffler function in the thinning theory for renewal processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06563","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":"sha256:6e070b1959be4d4b438826bfefffa4df32797380431143162ea6e650e95946c2","target":"record","created_at":"2026-05-18T00:07:44Z","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":"029926b9979f7b94f537188587a051dcde305287deacf98d5b5068cbad3ee05d","cross_cats_sorted":["math-ph","math.CV","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-08-17T15:47:08Z","title_canon_sha256":"ec23dfee4f7b24a371fbcefabb9943fba2e2bbd6b66191c826a877992d29d805"},"schema_version":"1.0","source":{"id":"1808.06563","kind":"arxiv","version":1}},"canonical_sha256":"8cf85ae4f4b789f3194fb04e36427a5d6c8375f71935e56aaec474990bddf5d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8cf85ae4f4b789f3194fb04e36427a5d6c8375f71935e56aaec474990bddf5d9","first_computed_at":"2026-05-18T00:07:44.437206Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:44.437206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"36izLgRm7809Qwv7phZZpHIoYV7yZHWkl+ArhWIxq289dLtYaFDy8crbEsh6srhoUBzkNCAlewTPTX1VaVKPDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:44.437937Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.06563","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e070b1959be4d4b438826bfefffa4df32797380431143162ea6e650e95946c2","sha256:8bcd5ee02df9fcfb400c239e92e7f14e510bfe2c8c41100958be99820cb29ae8"],"state_sha256":"58cf845a47c1a55d79eed8d0a01489318300f8733dcec6269d4cc01a51fb1c4f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DGJEWojmd6XAHMAtVVsoR6NuYBuM24npdmeozV+KJnaa/wLAMUGonW75QWtbbw/K3um/7Mk4q/oSrucTUpPKDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T12:44:20.718260Z","bundle_sha256":"8a471de105d91da94fcf2110251eafab9c4b8e065925428af41c1a0e086370c8"}}