{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NBLYKC2N4NG7S7EKOD5ZGCCE5Y","short_pith_number":"pith:NBLYKC2N","canonical_record":{"source":{"id":"1812.05515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-12-13T16:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"261e54a863f75cd42639c8c3a07fc7892d0c65773245f88bebf2d00dc52a4b6b","abstract_canon_sha256":"269e8ca481e8bcec7e95af1975ad7e842ce1758b0f94a645d1468991c4245b65"},"schema_version":"1.0"},"canonical_sha256":"6857850b4de34df97c8a70fb930844ee2326dd5b9d5ed88cb2946aac510afdeb","source":{"kind":"arxiv","id":"1812.05515","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.05515","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1812.05515v1","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.05515","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"NBLYKC2N4NG7","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NBLYKC2N4NG7S7EK","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NBLYKC2N","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NBLYKC2N4NG7S7EKOD5ZGCCE5Y","target":"record","payload":{"canonical_record":{"source":{"id":"1812.05515","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-12-13T16:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"261e54a863f75cd42639c8c3a07fc7892d0c65773245f88bebf2d00dc52a4b6b","abstract_canon_sha256":"269e8ca481e8bcec7e95af1975ad7e842ce1758b0f94a645d1468991c4245b65"},"schema_version":"1.0"},"canonical_sha256":"6857850b4de34df97c8a70fb930844ee2326dd5b9d5ed88cb2946aac510afdeb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:20.075331Z","signature_b64":"SHzyd+2sUaXTg9uS3ooPFiS1e1q+AirdyxZnOAuIJaZ9PLyqo/lN2GOwTTTLT32eOTY44u1LVymxc4IlcZPbBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6857850b4de34df97c8a70fb930844ee2326dd5b9d5ed88cb2946aac510afdeb","last_reissued_at":"2026-05-17T23:58:20.074946Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:20.074946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.05515","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":"2026-05-17T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HOPBdIShNOagYR/EjYzp0kr70TU5AGg2XkZzVFDneBdUX2ClpODX+ytHaBVcNCgZ19+0eriAznnEbd4HMxyCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:56:56.801044Z"},"content_sha256":"9d6668bc4d0365231fed95b38e58fc9afd041d9cbd41b03849f65774993d7725","schema_version":"1.0","event_id":"sha256:9d6668bc4d0365231fed95b38e58fc9afd041d9cbd41b03849f65774993d7725"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NBLYKC2N4NG7S7EKOD5ZGCCE5Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Population Processes with Immigration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dan Han, Joseph Whitmeyer, Stanislav Molchanov","submitted_at":"2018-12-13T16:45:03Z","abstract_excerpt":"The paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.05515","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":"2026-05-17T23:58:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UqX9GSnV3JrUsirh457QDhwa5aSQiZ5X35PyqfpqeEH2MejbSGQs6u900UH0mPCSOqgPAK5jG8nLk1RJJOCaCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:56:56.801384Z"},"content_sha256":"0f5c9af2276327498df5a7ea1d8d22908857f8266974203126aed0acd8413e2e","schema_version":"1.0","event_id":"sha256:0f5c9af2276327498df5a7ea1d8d22908857f8266974203126aed0acd8413e2e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/bundle.json","state_url":"https://pith.science/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/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":"2026-06-27T02:56:56Z","links":{"resolver":"https://pith.science/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y","bundle":"https://pith.science/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/bundle.json","state":"https://pith.science/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NBLYKC2N4NG7S7EKOD5ZGCCE5Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NBLYKC2N4NG7S7EKOD5ZGCCE5Y","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":"269e8ca481e8bcec7e95af1975ad7e842ce1758b0f94a645d1468991c4245b65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-12-13T16:45:03Z","title_canon_sha256":"261e54a863f75cd42639c8c3a07fc7892d0c65773245f88bebf2d00dc52a4b6b"},"schema_version":"1.0","source":{"id":"1812.05515","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.05515","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"arxiv_version","alias_value":"1812.05515v1","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.05515","created_at":"2026-05-17T23:58:20Z"},{"alias_kind":"pith_short_12","alias_value":"NBLYKC2N4NG7","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NBLYKC2N4NG7S7EK","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NBLYKC2N","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:0f5c9af2276327498df5a7ea1d8d22908857f8266974203126aed0acd8413e2e","target":"graph","created_at":"2026-05-17T23:58:20Z","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 paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions.","authors_text":"Dan Han, Joseph Whitmeyer, Stanislav Molchanov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-12-13T16:45:03Z","title":"Population Processes with Immigration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.05515","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":"sha256:9d6668bc4d0365231fed95b38e58fc9afd041d9cbd41b03849f65774993d7725","target":"record","created_at":"2026-05-17T23:58:20Z","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":"269e8ca481e8bcec7e95af1975ad7e842ce1758b0f94a645d1468991c4245b65","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-12-13T16:45:03Z","title_canon_sha256":"261e54a863f75cd42639c8c3a07fc7892d0c65773245f88bebf2d00dc52a4b6b"},"schema_version":"1.0","source":{"id":"1812.05515","kind":"arxiv","version":1}},"canonical_sha256":"6857850b4de34df97c8a70fb930844ee2326dd5b9d5ed88cb2946aac510afdeb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6857850b4de34df97c8a70fb930844ee2326dd5b9d5ed88cb2946aac510afdeb","first_computed_at":"2026-05-17T23:58:20.074946Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:58:20.074946Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SHzyd+2sUaXTg9uS3ooPFiS1e1q+AirdyxZnOAuIJaZ9PLyqo/lN2GOwTTTLT32eOTY44u1LVymxc4IlcZPbBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:58:20.075331Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.05515","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9d6668bc4d0365231fed95b38e58fc9afd041d9cbd41b03849f65774993d7725","sha256:0f5c9af2276327498df5a7ea1d8d22908857f8266974203126aed0acd8413e2e"],"state_sha256":"df28fd99f42a5b29401299351c002b7b5a8cc8fc1aec5c001d0e80321a48b333"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gsm3dTohvMBvHXBqdX3RdZUlu6bDz6PPq9ycNtknVtoo90xL3ThsHRDnKjsmkYnYk8sFDPy1hZvCZtP+RiEVCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T02:56:56.803126Z","bundle_sha256":"2e3c69ecbc9ab508dc8c404d6844a945e62cd1e35a4997c089f71db053cdc5b2"}}