{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7DBABMXU3HGUUPN4P3CKY4QWHN","short_pith_number":"pith:7DBABMXU","canonical_record":{"source":{"id":"1107.2878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-14T17:45:16Z","cross_cats_sorted":[],"title_canon_sha256":"f28891fbf18d8949ca37a7d19f1f3dafdea29bf75e76d21e5eec52693959ca70","abstract_canon_sha256":"f7ef7ec5ba190aa30c3bf766ea0600fe86404534faee376b118b4ef3af11133b"},"schema_version":"1.0"},"canonical_sha256":"f8c200b2f4d9cd4a3dbc7ec4ac72163b468314e3dd2b565f53a42f5643dcf90a","source":{"kind":"arxiv","id":"1107.2878","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2878","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2878v1","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2878","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"pith_short_12","alias_value":"7DBABMXU3HGU","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7DBABMXU3HGUUPN4","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7DBABMXU","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7DBABMXU3HGUUPN4P3CKY4QWHN","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-14T17:45:16Z","cross_cats_sorted":[],"title_canon_sha256":"f28891fbf18d8949ca37a7d19f1f3dafdea29bf75e76d21e5eec52693959ca70","abstract_canon_sha256":"f7ef7ec5ba190aa30c3bf766ea0600fe86404534faee376b118b4ef3af11133b"},"schema_version":"1.0"},"canonical_sha256":"f8c200b2f4d9cd4a3dbc7ec4ac72163b468314e3dd2b565f53a42f5643dcf90a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:40.879826Z","signature_b64":"0V5sb6yxZ5Q332M+I+4L2/Tonf2t752u62V/kJ3Cec02ElRutXd5ufAeL6pMnDPQThA3CJa7rF7jn8IYd9V/Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8c200b2f4d9cd4a3dbc7ec4ac72163b468314e3dd2b565f53a42f5643dcf90a","last_reissued_at":"2026-05-18T03:29:40.879253Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:40.879253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2878","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":"2026-05-18T03:29:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"37QcA47N0BdlKH5auZc1nxlR5f3tKC37gpge1TyaiOfw17RQIRAX5uPrmawg6nhkmcOSczv0nfokMAFuw5R1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:22:47.730574Z"},"content_sha256":"3c1a6df53a789d0bd955dbfc7eb9f92f2908dfb923bd57af8cf0371cdd369e7e","schema_version":"1.0","event_id":"sha256:3c1a6df53a789d0bd955dbfc7eb9f92f2908dfb923bd57af8cf0371cdd369e7e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7DBABMXU3HGUUPN4P3CKY4QWHN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Randomly Stopped Nonlinear Fractional Birth Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Enzo Orsingher, Federico Polito","submitted_at":"2011-07-14T17:45:16Z","abstract_excerpt":"We present and analyse the nonlinear classical pure birth process $\\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\\mathpzc{N}^\\nu (t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\\alpha$-stable subordinator $\\mathpzc{S}^\\alpha(t)$, $\\alpha \\in (0,1)$, and others. For all of them we derive the state probability distribution $\\hat{p}_k (t)$, $k \\geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for bot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2878","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":"2026-05-18T03:29:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fc8+UVJUzsPXPTszja0+mgUe9WHiVSRD70GeZeozfF/ZY7cLnm/HhEbCEZISeulckdhmLifwPZIZ9Uo8SfbYBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:22:47.731227Z"},"content_sha256":"300edc0e9006e64f40722e509e34661ed28c31f06452f483ba6dba53b0841859","schema_version":"1.0","event_id":"sha256:300edc0e9006e64f40722e509e34661ed28c31f06452f483ba6dba53b0841859"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/bundle.json","state_url":"https://pith.science/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/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":"2026-05-27T23:22:47Z","links":{"resolver":"https://pith.science/pith/7DBABMXU3HGUUPN4P3CKY4QWHN","bundle":"https://pith.science/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/bundle.json","state":"https://pith.science/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7DBABMXU3HGUUPN4P3CKY4QWHN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7DBABMXU3HGUUPN4P3CKY4QWHN","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":"f7ef7ec5ba190aa30c3bf766ea0600fe86404534faee376b118b4ef3af11133b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-14T17:45:16Z","title_canon_sha256":"f28891fbf18d8949ca37a7d19f1f3dafdea29bf75e76d21e5eec52693959ca70"},"schema_version":"1.0","source":{"id":"1107.2878","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2878","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2878v1","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2878","created_at":"2026-05-18T03:29:40Z"},{"alias_kind":"pith_short_12","alias_value":"7DBABMXU3HGU","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7DBABMXU3HGUUPN4","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7DBABMXU","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:300edc0e9006e64f40722e509e34661ed28c31f06452f483ba6dba53b0841859","target":"graph","created_at":"2026-05-18T03:29:40Z","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 present and analyse the nonlinear classical pure birth process $\\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\\mathpzc{N}^\\nu (t)$, $t>0$, subordinated to various random times, namely the first-passage time $T_t$ of the standard Brownian motion $B(t)$, $t>0$, the $\\alpha$-stable subordinator $\\mathpzc{S}^\\alpha(t)$, $\\alpha \\in (0,1)$, and others. For all of them we derive the state probability distribution $\\hat{p}_k (t)$, $k \\geq 1$ and, in some cases, we also present the corresponding governing differential equation. We also highlight interesting interpretations for bot","authors_text":"Enzo Orsingher, Federico Polito","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-14T17:45:16Z","title":"Randomly Stopped Nonlinear Fractional Birth Processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2878","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":"sha256:3c1a6df53a789d0bd955dbfc7eb9f92f2908dfb923bd57af8cf0371cdd369e7e","target":"record","created_at":"2026-05-18T03:29:40Z","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":"f7ef7ec5ba190aa30c3bf766ea0600fe86404534faee376b118b4ef3af11133b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-14T17:45:16Z","title_canon_sha256":"f28891fbf18d8949ca37a7d19f1f3dafdea29bf75e76d21e5eec52693959ca70"},"schema_version":"1.0","source":{"id":"1107.2878","kind":"arxiv","version":1}},"canonical_sha256":"f8c200b2f4d9cd4a3dbc7ec4ac72163b468314e3dd2b565f53a42f5643dcf90a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8c200b2f4d9cd4a3dbc7ec4ac72163b468314e3dd2b565f53a42f5643dcf90a","first_computed_at":"2026-05-18T03:29:40.879253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:40.879253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0V5sb6yxZ5Q332M+I+4L2/Tonf2t752u62V/kJ3Cec02ElRutXd5ufAeL6pMnDPQThA3CJa7rF7jn8IYd9V/Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:40.879826Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2878","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c1a6df53a789d0bd955dbfc7eb9f92f2908dfb923bd57af8cf0371cdd369e7e","sha256:300edc0e9006e64f40722e509e34661ed28c31f06452f483ba6dba53b0841859"],"state_sha256":"eacbb4b3c5966ff777a0b8cf5587031fd7eea8301e8ee0cd02d865354099dad0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VF/AmpHP0g0sJUJffczpg4x1+A6oPw59THqEF0oQkZfY4yOiuEROJaI3s2q/l6GFuTkGUGl5rpySF9lpBsv1Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:22:47.734492Z","bundle_sha256":"28e0b35b6369f0db63fe4f93930bbf587386d57a0a012358e155686fc75f1aa3"}}