{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TXAUZF5R7L6SOMDZM3YUQ3VU6E","short_pith_number":"pith:TXAUZF5R","canonical_record":{"source":{"id":"1603.08716","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-03-29T10:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"df42c0f514c035b44aea93bb9aaa72b4f7ae83c0ad9e0a2022042a698c969664","abstract_canon_sha256":"2f932b9eb355d727392065e3845ec8daa35e787880b6e2ac29718de43991e9ed"},"schema_version":"1.0"},"canonical_sha256":"9dc14c97b1fafd27307966f1486eb4f1249cfdb36f4c469d59bbc277c3212d66","source":{"kind":"arxiv","id":"1603.08716","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08716","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08716v1","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08716","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"pith_short_12","alias_value":"TXAUZF5R7L6S","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"TXAUZF5R7L6SOMDZ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"TXAUZF5R","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TXAUZF5R7L6SOMDZM3YUQ3VU6E","target":"record","payload":{"canonical_record":{"source":{"id":"1603.08716","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-03-29T10:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"df42c0f514c035b44aea93bb9aaa72b4f7ae83c0ad9e0a2022042a698c969664","abstract_canon_sha256":"2f932b9eb355d727392065e3845ec8daa35e787880b6e2ac29718de43991e9ed"},"schema_version":"1.0"},"canonical_sha256":"9dc14c97b1fafd27307966f1486eb4f1249cfdb36f4c469d59bbc277c3212d66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:05.514564Z","signature_b64":"CX98uyk1Ffg9LzerH8UdJvFHeejfCFgG1CDUH5IFwX81y4TO19efoMsgMEscTuqhN7eJxxTaAkzMoBKcnfO7AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9dc14c97b1fafd27307966f1486eb4f1249cfdb36f4c469d59bbc277c3212d66","last_reissued_at":"2026-05-18T01:18:05.514025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:05.514025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.08716","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":"2026-05-18T01:18:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BRRvBfj8JqdDole6Hg6ouc0IUCWh7UH+mtb0cdD0t9aW2Zcp9jRNXPCA/h66nBMJxWmj/yNyOBgq59nGljbpCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:26:51.668595Z"},"content_sha256":"a384efec43c242aaef40673df47f4fc0bc9edd408563fee3a627811446c48aae","schema_version":"1.0","event_id":"sha256:a384efec43c242aaef40673df47f4fc0bc9edd408563fee3a627811446c48aae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TXAUZF5R7L6SOMDZM3YUQ3VU6E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Barrier Functionals for the Analysis of Complex Systems: An Optimization-Based Approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Antonis Papachristodoulou, Giorgio Valmorbida, Mohamadreza Ahmadi","submitted_at":"2016-03-29T10:42:58Z","abstract_excerpt":"We propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential equations (PDEs). The proposed method is based on barrier functionals, which are functionals of the states of the studied systems. The proposed method does not require the approximation of solutions nor the stability of trajectories. In the case of SDEs, the formulation relies on a generalized version of the Feynman-Kac formula and results in moments bounds for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08716","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":"2026-05-18T01:18:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gARCrGdXpVqXaWclT1DFrnJk25yBodi25+Nx6Kw8JNrBR7vMgueFu4tE7Eb8bmgAiAFXjh6mnV1qH7f3wSuuBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T12:26:51.668951Z"},"content_sha256":"ca26d4541755a7033f74632d0272fa3d338c36194d724b397fe6d8d4fa738286","schema_version":"1.0","event_id":"sha256:ca26d4541755a7033f74632d0272fa3d338c36194d724b397fe6d8d4fa738286"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/bundle.json","state_url":"https://pith.science/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/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":"2026-05-27T12:26:51Z","links":{"resolver":"https://pith.science/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E","bundle":"https://pith.science/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/bundle.json","state":"https://pith.science/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TXAUZF5R7L6SOMDZM3YUQ3VU6E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TXAUZF5R7L6SOMDZM3YUQ3VU6E","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":"2f932b9eb355d727392065e3845ec8daa35e787880b6e2ac29718de43991e9ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-03-29T10:42:58Z","title_canon_sha256":"df42c0f514c035b44aea93bb9aaa72b4f7ae83c0ad9e0a2022042a698c969664"},"schema_version":"1.0","source":{"id":"1603.08716","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08716","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08716v1","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08716","created_at":"2026-05-18T01:18:05Z"},{"alias_kind":"pith_short_12","alias_value":"TXAUZF5R7L6S","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"TXAUZF5R7L6SOMDZ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"TXAUZF5R","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:ca26d4541755a7033f74632d0272fa3d338c36194d724b397fe6d8d4fa738286","target":"graph","created_at":"2026-05-18T01:18:05Z","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 propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential equations (PDEs). The proposed method is based on barrier functionals, which are functionals of the states of the studied systems. The proposed method does not require the approximation of solutions nor the stability of trajectories. In the case of SDEs, the formulation relies on a generalized version of the Feynman-Kac formula and results in moments bounds for","authors_text":"Antonis Papachristodoulou, Giorgio Valmorbida, Mohamadreza Ahmadi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-03-29T10:42:58Z","title":"Barrier Functionals for the Analysis of Complex Systems: An Optimization-Based Approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08716","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:a384efec43c242aaef40673df47f4fc0bc9edd408563fee3a627811446c48aae","target":"record","created_at":"2026-05-18T01:18:05Z","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":"2f932b9eb355d727392065e3845ec8daa35e787880b6e2ac29718de43991e9ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-03-29T10:42:58Z","title_canon_sha256":"df42c0f514c035b44aea93bb9aaa72b4f7ae83c0ad9e0a2022042a698c969664"},"schema_version":"1.0","source":{"id":"1603.08716","kind":"arxiv","version":1}},"canonical_sha256":"9dc14c97b1fafd27307966f1486eb4f1249cfdb36f4c469d59bbc277c3212d66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9dc14c97b1fafd27307966f1486eb4f1249cfdb36f4c469d59bbc277c3212d66","first_computed_at":"2026-05-18T01:18:05.514025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:05.514025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CX98uyk1Ffg9LzerH8UdJvFHeejfCFgG1CDUH5IFwX81y4TO19efoMsgMEscTuqhN7eJxxTaAkzMoBKcnfO7AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:05.514564Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.08716","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a384efec43c242aaef40673df47f4fc0bc9edd408563fee3a627811446c48aae","sha256:ca26d4541755a7033f74632d0272fa3d338c36194d724b397fe6d8d4fa738286"],"state_sha256":"c64583b9424549071801ea0ae81795c1479d712357daa673d598d0400d78dc50"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fDapPDPkDj4iv5BqMlBQxZRe/+QkJz2ZkOvr+KGyOgzrW8NMvnrmtZmxRXvASPC6bCSprxRsRbzIKuHTKDUWCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T12:26:51.671198Z","bundle_sha256":"3caf283a86cb84a8e000ff6e1b44da4be37bb07aa53495352accfbb665826fa6"}}