{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:KWF3FOXVJ7EIK6N7DC7GOGS2FF","short_pith_number":"pith:KWF3FOXV","canonical_record":{"source":{"id":"1209.3533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-17T00:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"72c83fc7dbde7876cd6843e5ffaceeeea244bb616d3c464eaf15c18adc81b2de","abstract_canon_sha256":"ece323e70881e2625b06253662e9b02c69fba9feda2a0203d63b2b154af3e06b"},"schema_version":"1.0"},"canonical_sha256":"558bb2baf54fc88579bf18be671a5a29696b1f297473f2e76a855c5dc5e41a96","source":{"kind":"arxiv","id":"1209.3533","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3533","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3533v1","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3533","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"pith_short_12","alias_value":"KWF3FOXVJ7EI","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KWF3FOXVJ7EIK6N7","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KWF3FOXV","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:KWF3FOXVJ7EIK6N7DC7GOGS2FF","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3533","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-17T00:50:17Z","cross_cats_sorted":[],"title_canon_sha256":"72c83fc7dbde7876cd6843e5ffaceeeea244bb616d3c464eaf15c18adc81b2de","abstract_canon_sha256":"ece323e70881e2625b06253662e9b02c69fba9feda2a0203d63b2b154af3e06b"},"schema_version":"1.0"},"canonical_sha256":"558bb2baf54fc88579bf18be671a5a29696b1f297473f2e76a855c5dc5e41a96","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:24.249488Z","signature_b64":"jGJKvr/ychDBQnlp85xpbRT4w+hYaB1CIc5s7LSbsw4+t7JB2XSr4qQPxDH+Q8UdUEKJHwGCMAZ40y30AEKHCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"558bb2baf54fc88579bf18be671a5a29696b1f297473f2e76a855c5dc5e41a96","last_reissued_at":"2026-05-18T02:57:24.248970Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:24.248970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3533","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":"2026-05-18T02:57:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5Vruhyb4fDaxNn5mJbl0c//JKzTkxSu6CO2A0dsr4V6jZf358A7dOhN87Cp/RUpMcZ3MvHd6sloKhGJYpoaVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:44:39.805600Z"},"content_sha256":"8a8320a14784a6ba92557a9b128af38c3af908c18e3e54f6c7253c8a7fd4f25b","schema_version":"1.0","event_id":"sha256:8a8320a14784a6ba92557a9b128af38c3af908c18e3e54f6c7253c8a7fd4f25b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:KWF3FOXVJ7EIK6N7DC7GOGS2FF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized inverses of Markovian kernels in terms of properties of the Markov chain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jeffrey J. Hunter","submitted_at":"2012-09-17T00:50:17Z","abstract_excerpt":"All one-condition generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include the group inverse of I - P, Kemeny and Snell's fundamental matrix of the Markov chain and the Moore-Penrose g-inverse. The elements of some sub-families of the generalized inverses can also be re-expressed involving the second moments of the recurrence time variables. Some applications to Kemeny's"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3533","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":"2026-05-18T02:57:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"39KgHssx2Wryoc3D5L6joDvbV9I/siKVnF1EAsET7x+o2sCjhA14HCW/istriSlRq7RQC7q/ZUn0DTypUKh8Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:44:39.805956Z"},"content_sha256":"5e7ef52de94d7dd518c1215d4b755ad4aaed661b1dea4186e0f27a14a89914e2","schema_version":"1.0","event_id":"sha256:5e7ef52de94d7dd518c1215d4b755ad4aaed661b1dea4186e0f27a14a89914e2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/bundle.json","state_url":"https://pith.science/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/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":"2026-05-30T08:44:39Z","links":{"resolver":"https://pith.science/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF","bundle":"https://pith.science/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/bundle.json","state":"https://pith.science/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KWF3FOXVJ7EIK6N7DC7GOGS2FF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:KWF3FOXVJ7EIK6N7DC7GOGS2FF","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":"ece323e70881e2625b06253662e9b02c69fba9feda2a0203d63b2b154af3e06b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-17T00:50:17Z","title_canon_sha256":"72c83fc7dbde7876cd6843e5ffaceeeea244bb616d3c464eaf15c18adc81b2de"},"schema_version":"1.0","source":{"id":"1209.3533","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3533","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3533v1","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3533","created_at":"2026-05-18T02:57:24Z"},{"alias_kind":"pith_short_12","alias_value":"KWF3FOXVJ7EI","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KWF3FOXVJ7EIK6N7","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KWF3FOXV","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:5e7ef52de94d7dd518c1215d4b755ad4aaed661b1dea4186e0f27a14a89914e2","target":"graph","created_at":"2026-05-18T02:57:24Z","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":"All one-condition generalized inverses of the Markovian kernel I - P, where P is the transition matrix of a finite irreducible Markov chain, can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include the group inverse of I - P, Kemeny and Snell's fundamental matrix of the Markov chain and the Moore-Penrose g-inverse. The elements of some sub-families of the generalized inverses can also be re-expressed involving the second moments of the recurrence time variables. Some applications to Kemeny's","authors_text":"Jeffrey J. Hunter","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-17T00:50:17Z","title":"Generalized inverses of Markovian kernels in terms of properties of the Markov chain"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3533","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":"sha256:8a8320a14784a6ba92557a9b128af38c3af908c18e3e54f6c7253c8a7fd4f25b","target":"record","created_at":"2026-05-18T02:57:24Z","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":"ece323e70881e2625b06253662e9b02c69fba9feda2a0203d63b2b154af3e06b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-09-17T00:50:17Z","title_canon_sha256":"72c83fc7dbde7876cd6843e5ffaceeeea244bb616d3c464eaf15c18adc81b2de"},"schema_version":"1.0","source":{"id":"1209.3533","kind":"arxiv","version":1}},"canonical_sha256":"558bb2baf54fc88579bf18be671a5a29696b1f297473f2e76a855c5dc5e41a96","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"558bb2baf54fc88579bf18be671a5a29696b1f297473f2e76a855c5dc5e41a96","first_computed_at":"2026-05-18T02:57:24.248970Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:24.248970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jGJKvr/ychDBQnlp85xpbRT4w+hYaB1CIc5s7LSbsw4+t7JB2XSr4qQPxDH+Q8UdUEKJHwGCMAZ40y30AEKHCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:24.249488Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3533","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a8320a14784a6ba92557a9b128af38c3af908c18e3e54f6c7253c8a7fd4f25b","sha256:5e7ef52de94d7dd518c1215d4b755ad4aaed661b1dea4186e0f27a14a89914e2"],"state_sha256":"d8f8745b2737b63f8ff793202b6d5171539395f4fe1c378793bb0a6e27989e53"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nONjPVIqqt4alwAWxbVJVzKK48YYTXSZXdA7BJdanvJ465aVKj0Q2tGmdykbtenCR0IsHs46ZlHTedb8B+8ABA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T08:44:39.808007Z","bundle_sha256":"4aff4d2ffd8276c8e897487b3d6b88969f59a0d1c4bf672248d43c34bfaec15d"}}