{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:Y2AZ4A3PSBFHGJOHSDVMNHQ5K7","short_pith_number":"pith:Y2AZ4A3P","canonical_record":{"source":{"id":"1510.04044","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-14T11:20:58Z","cross_cats_sorted":[],"title_canon_sha256":"bcfa8941f7febea796bd6a9c9926d9fb71b5f92e24e2639d5e6703c5003f4316","abstract_canon_sha256":"bdbfdf109df65f8432c2060c4bcb05d2fed4d66cc28077246721cd0c8548f54a"},"schema_version":"1.0"},"canonical_sha256":"c6819e036f904a7325c790eac69e1d57c2527bb98e2d173985719dea6cfacb78","source":{"kind":"arxiv","id":"1510.04044","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.04044","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"arxiv_version","alias_value":"1510.04044v3","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.04044","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_12","alias_value":"Y2AZ4A3PSBFH","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_16","alias_value":"Y2AZ4A3PSBFHGJOH","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_8","alias_value":"Y2AZ4A3P","created_at":"2026-07-05T01:39:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:Y2AZ4A3PSBFHGJOHSDVMNHQ5K7","target":"record","payload":{"canonical_record":{"source":{"id":"1510.04044","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-14T11:20:58Z","cross_cats_sorted":[],"title_canon_sha256":"bcfa8941f7febea796bd6a9c9926d9fb71b5f92e24e2639d5e6703c5003f4316","abstract_canon_sha256":"bdbfdf109df65f8432c2060c4bcb05d2fed4d66cc28077246721cd0c8548f54a"},"schema_version":"1.0"},"canonical_sha256":"c6819e036f904a7325c790eac69e1d57c2527bb98e2d173985719dea6cfacb78","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:39:08.842281Z","signature_b64":"iubRsnfOqiz4v+21O/O8YRDUINiVIs21GnQwANLkRTWBksg0n9FxyDRy35rO9Vwu67hMqznRF5qUFsc1OW36Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6819e036f904a7325c790eac69e1d57c2527bb98e2d173985719dea6cfacb78","last_reissued_at":"2026-07-05T01:39:08.841780Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:39:08.841780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.04044","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:39:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jUKTqx4csWs79CmfVaUBFBafLGAB5K9Cg+iKEemEbToWX7lETK2UHACCyNTH8wvceldnv13iirfgNnEXj1vVBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T22:09:10.753437Z"},"content_sha256":"45892959f3c0c54b5ff8a47c5978b58301982a75de4d03fddbbb25ffd8d24727","schema_version":"1.0","event_id":"sha256:45892959f3c0c54b5ff8a47c5978b58301982a75de4d03fddbbb25ffd8d24727"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:Y2AZ4A3PSBFHGJOHSDVMNHQ5K7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Chuanhou Gao, Zhou Fang","submitted_at":"2015-10-14T11:20:58Z","abstract_excerpt":"In this paper, we develop a method to generate the Lyapunov function for stability analysis for chemical reaction networks. Based on the Chemical Master Equation, we derive the Lyapunov Function partial differential equations (PDEs), whose solution approximates the scaling non-equilibrium potential and serves as the candidate Lyapunov function for the given network. We further prove that for any chemical reaction network the solution (if exists) of the PDEs is dissipative. Moreover, the proposed method of Lyapunov Function PDEs is qualified for analyzing the asymptotic stability of complex bal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04044","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1510.04044/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:39:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mjl9B6JWyvOUxvwIL5ugYV8iIKzodgFNv4le4Zuiwi1VGPbtnf1XSGhzQsEMJCC2aOFUW6t9KoIMKgvM2Xh6DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T22:09:10.753805Z"},"content_sha256":"fa4adf1e45df2e851b86cb8691a771c0127af126560002ea8b02c3b9d021a7b2","schema_version":"1.0","event_id":"sha256:fa4adf1e45df2e851b86cb8691a771c0127af126560002ea8b02c3b9d021a7b2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/bundle.json","state_url":"https://pith.science/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/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":"2026-07-06T22:09:10Z","links":{"resolver":"https://pith.science/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7","bundle":"https://pith.science/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/bundle.json","state":"https://pith.science/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y2AZ4A3PSBFHGJOHSDVMNHQ5K7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Y2AZ4A3PSBFHGJOHSDVMNHQ5K7","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":"bdbfdf109df65f8432c2060c4bcb05d2fed4d66cc28077246721cd0c8548f54a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-14T11:20:58Z","title_canon_sha256":"bcfa8941f7febea796bd6a9c9926d9fb71b5f92e24e2639d5e6703c5003f4316"},"schema_version":"1.0","source":{"id":"1510.04044","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.04044","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"arxiv_version","alias_value":"1510.04044v3","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.04044","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_12","alias_value":"Y2AZ4A3PSBFH","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_16","alias_value":"Y2AZ4A3PSBFHGJOH","created_at":"2026-07-05T01:39:08Z"},{"alias_kind":"pith_short_8","alias_value":"Y2AZ4A3P","created_at":"2026-07-05T01:39:08Z"}],"graph_snapshots":[{"event_id":"sha256:fa4adf1e45df2e851b86cb8691a771c0127af126560002ea8b02c3b9d021a7b2","target":"graph","created_at":"2026-07-05T01:39:08Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1510.04044/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we develop a method to generate the Lyapunov function for stability analysis for chemical reaction networks. Based on the Chemical Master Equation, we derive the Lyapunov Function partial differential equations (PDEs), whose solution approximates the scaling non-equilibrium potential and serves as the candidate Lyapunov function for the given network. We further prove that for any chemical reaction network the solution (if exists) of the PDEs is dissipative. Moreover, the proposed method of Lyapunov Function PDEs is qualified for analyzing the asymptotic stability of complex bal","authors_text":"Chuanhou Gao, Zhou Fang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-14T11:20:58Z","title":"Lyapunov Function Partial Differential Equations for Chemical Reaction Networks: Some Special Cases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04044","kind":"arxiv","version":3},"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":"sha256:45892959f3c0c54b5ff8a47c5978b58301982a75de4d03fddbbb25ffd8d24727","target":"record","created_at":"2026-07-05T01:39:08Z","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":"bdbfdf109df65f8432c2060c4bcb05d2fed4d66cc28077246721cd0c8548f54a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-10-14T11:20:58Z","title_canon_sha256":"bcfa8941f7febea796bd6a9c9926d9fb71b5f92e24e2639d5e6703c5003f4316"},"schema_version":"1.0","source":{"id":"1510.04044","kind":"arxiv","version":3}},"canonical_sha256":"c6819e036f904a7325c790eac69e1d57c2527bb98e2d173985719dea6cfacb78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6819e036f904a7325c790eac69e1d57c2527bb98e2d173985719dea6cfacb78","first_computed_at":"2026-07-05T01:39:08.841780Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:39:08.841780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iubRsnfOqiz4v+21O/O8YRDUINiVIs21GnQwANLkRTWBksg0n9FxyDRy35rO9Vwu67hMqznRF5qUFsc1OW36Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:39:08.842281Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.04044","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:45892959f3c0c54b5ff8a47c5978b58301982a75de4d03fddbbb25ffd8d24727","sha256:fa4adf1e45df2e851b86cb8691a771c0127af126560002ea8b02c3b9d021a7b2"],"state_sha256":"a35c080c02098ead3abb0e7c7e4d6d95765a480cc654bf422036e3e8f476bf55"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A/ZcFVlvZ+SGFXdn1o94dCQ7sLY0Hjsn6vuyxZnCfdI5pXAlJ7TfhoaXyzrx/Xk6fjw2BH60vxyUBjnElhCNBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T22:09:10.755915Z","bundle_sha256":"b3cf297e80ebeaeb3b6e98126f4387e247e695ae102bc95d3b77f4fa57808953"}}