{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ARQRXO5RM6UHIOU25WTQAWQRH7","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":"02c1b52b85407b410e237783e1659cb41425d4c51fc3e41920010a3ca8588289","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-18T23:29:11Z","title_canon_sha256":"9ba8ecd8f11eeabca1e686786362371ce4cce391eb25cf86de2187d17b679d1b"},"schema_version":"1.0","source":{"id":"1602.06001","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.06001","created_at":"2026-05-18T01:20:21Z"},{"alias_kind":"arxiv_version","alias_value":"1602.06001v1","created_at":"2026-05-18T01:20:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.06001","created_at":"2026-05-18T01:20:21Z"},{"alias_kind":"pith_short_12","alias_value":"ARQRXO5RM6UH","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"ARQRXO5RM6UHIOU2","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"ARQRXO5R","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:da3acf5f771373439fe3e3cd5a7447b6cb72e011d318ec8a9ca3b817d202bf51","target":"graph","created_at":"2026-05-18T01:20:21Z","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":"A symmetric relation in the probabilistic Green's function for birth-death chains is explored. Two proofs are given, each of which makes use of the known symmetry of the Green's functions in other contexts. The first uses as primary tool the local time of Brownian motion, while the second uses the reciprocity principle from electric network theory. We also show that the the second proof extends easily to cover birth-death chains (a.k.a. state-dependent random walks) on trees.","authors_text":"Greg Markowsky, Jos\\'e Luis Palacios","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-18T23:29:11Z","title":"Symmetry in the Green's function for birth-death chains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.06001","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":"sha256:1a5d5a343fb5abf3e8647f50985b2f30f918bd7c536eb8814fb774c7c70cd670","target":"record","created_at":"2026-05-18T01:20:21Z","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":"02c1b52b85407b410e237783e1659cb41425d4c51fc3e41920010a3ca8588289","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-18T23:29:11Z","title_canon_sha256":"9ba8ecd8f11eeabca1e686786362371ce4cce391eb25cf86de2187d17b679d1b"},"schema_version":"1.0","source":{"id":"1602.06001","kind":"arxiv","version":1}},"canonical_sha256":"04611bbbb167a8743a9aeda7005a113fd1a42489c6c0b77c5a266da9aefb23ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"04611bbbb167a8743a9aeda7005a113fd1a42489c6c0b77c5a266da9aefb23ae","first_computed_at":"2026-05-18T01:20:21.866410Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:21.866410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HG5lqeS8L84vSHIl0ygUK6jQpSxaYWO5cNFOAzB9fzRlSEWtMLN8UuCVKjVB7iJCpheGgfAhfgP5wRsRtD7dDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:21.867023Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.06001","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a5d5a343fb5abf3e8647f50985b2f30f918bd7c536eb8814fb774c7c70cd670","sha256:da3acf5f771373439fe3e3cd5a7447b6cb72e011d318ec8a9ca3b817d202bf51"],"state_sha256":"f072ad069d819e03541ce0403dff14db9b1468ff89f1394b50f5c94228cf6557"}