{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:P3QBTS6PV56FFVZ4CGGRVXCSLO","short_pith_number":"pith:P3QBTS6P","canonical_record":{"source":{"id":"2011.00180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-10-31T03:50:32Z","cross_cats_sorted":[],"title_canon_sha256":"176df4278d494de203a4fcff5abfa6be567f944faf285082867e5e05fdd2bd4a","abstract_canon_sha256":"9fb83e2918f9dd82de7128e314de10071ff8bceccafd87cdcb15cdb949bd54d9"},"schema_version":"1.0"},"canonical_sha256":"7ee019cbcfaf7c52d73c118d1adc525b9a88d384645a776d0422838d805d4854","source":{"kind":"arxiv","id":"2011.00180","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00180","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00180v1","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00180","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_12","alias_value":"P3QBTS6PV56F","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_16","alias_value":"P3QBTS6PV56FFVZ4","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_8","alias_value":"P3QBTS6P","created_at":"2026-07-05T01:47:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:P3QBTS6PV56FFVZ4CGGRVXCSLO","target":"record","payload":{"canonical_record":{"source":{"id":"2011.00180","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-10-31T03:50:32Z","cross_cats_sorted":[],"title_canon_sha256":"176df4278d494de203a4fcff5abfa6be567f944faf285082867e5e05fdd2bd4a","abstract_canon_sha256":"9fb83e2918f9dd82de7128e314de10071ff8bceccafd87cdcb15cdb949bd54d9"},"schema_version":"1.0"},"canonical_sha256":"7ee019cbcfaf7c52d73c118d1adc525b9a88d384645a776d0422838d805d4854","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:47:58.143593Z","signature_b64":"FvYstYpxpo/+5un5L26MqB6LxbT+37Gkk19WLw2pDuWe3gb3M5sOfUhnWUkt3Hjlyak2SEKSHoYxl1iLIWDTBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ee019cbcfaf7c52d73c118d1adc525b9a88d384645a776d0422838d805d4854","last_reissued_at":"2026-07-05T01:47:58.143105Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:47:58.143105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2011.00180","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":"2026-07-05T01:47:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qj8kaQauH+z+1sWsZk4hoyMw06mHD3sNJB8o1DtxX269zsWTPEsxcwLfDHY9MA7FqS7VqKU9QWObkBKxKjliAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:09:28.805598Z"},"content_sha256":"f1f3a477d8241c8b2d048692d55458f4add84ad937631c2827194513ea5d683c","schema_version":"1.0","event_id":"sha256:f1f3a477d8241c8b2d048692d55458f4add84ad937631c2827194513ea5d683c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:P3QBTS6PV56FFVZ4CGGRVXCSLO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Revisit of the Velocity Averaging Lemma: on the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chun-Hsiung Hsia, I-Kun Chen, Jhe-Kuan Su, Ping-Han Chuang","submitted_at":"2020-10-31T03:50:32Z","abstract_excerpt":"In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00180","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2011.00180/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":"2026-07-05T01:47:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CDG0NgPeEfkRHitAbMxdgfX5VYiwwaufLB/uqY3btchiq90z0jYLzMUkowkJnlegd0CMBZ4LPBuzOrknUMpuAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:09:28.806109Z"},"content_sha256":"96c0bb15f1bda03ccc14ffd4c886e2d1302f625ceb8db2644bbf06f0d5dfb725","schema_version":"1.0","event_id":"sha256:96c0bb15f1bda03ccc14ffd4c886e2d1302f625ceb8db2644bbf06f0d5dfb725"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/bundle.json","state_url":"https://pith.science/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/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":"2026-08-08T03:09:28Z","links":{"resolver":"https://pith.science/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO","bundle":"https://pith.science/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/bundle.json","state":"https://pith.science/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P3QBTS6PV56FFVZ4CGGRVXCSLO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:P3QBTS6PV56FFVZ4CGGRVXCSLO","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":"9fb83e2918f9dd82de7128e314de10071ff8bceccafd87cdcb15cdb949bd54d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-10-31T03:50:32Z","title_canon_sha256":"176df4278d494de203a4fcff5abfa6be567f944faf285082867e5e05fdd2bd4a"},"schema_version":"1.0","source":{"id":"2011.00180","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00180","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00180v1","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00180","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_12","alias_value":"P3QBTS6PV56F","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_16","alias_value":"P3QBTS6PV56FFVZ4","created_at":"2026-07-05T01:47:58Z"},{"alias_kind":"pith_short_8","alias_value":"P3QBTS6P","created_at":"2026-07-05T01:47:58Z"}],"graph_snapshots":[{"event_id":"sha256:96c0bb15f1bda03ccc14ffd4c886e2d1302f625ceb8db2644bbf06f0d5dfb725","target":"graph","created_at":"2026-07-05T01:47:58Z","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/2011.00180/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$.","authors_text":"Chun-Hsiung Hsia, I-Kun Chen, Jhe-Kuan Su, Ping-Han Chuang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-10-31T03:50:32Z","title":"A Revisit of the Velocity Averaging Lemma: on the Regularity of Stationary Boltzmann Equation in a Bounded Convex Domain"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00180","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:f1f3a477d8241c8b2d048692d55458f4add84ad937631c2827194513ea5d683c","target":"record","created_at":"2026-07-05T01:47:58Z","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":"9fb83e2918f9dd82de7128e314de10071ff8bceccafd87cdcb15cdb949bd54d9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2020-10-31T03:50:32Z","title_canon_sha256":"176df4278d494de203a4fcff5abfa6be567f944faf285082867e5e05fdd2bd4a"},"schema_version":"1.0","source":{"id":"2011.00180","kind":"arxiv","version":1}},"canonical_sha256":"7ee019cbcfaf7c52d73c118d1adc525b9a88d384645a776d0422838d805d4854","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ee019cbcfaf7c52d73c118d1adc525b9a88d384645a776d0422838d805d4854","first_computed_at":"2026-07-05T01:47:58.143105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:47:58.143105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FvYstYpxpo/+5un5L26MqB6LxbT+37Gkk19WLw2pDuWe3gb3M5sOfUhnWUkt3Hjlyak2SEKSHoYxl1iLIWDTBg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:47:58.143593Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.00180","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1f3a477d8241c8b2d048692d55458f4add84ad937631c2827194513ea5d683c","sha256:96c0bb15f1bda03ccc14ffd4c886e2d1302f625ceb8db2644bbf06f0d5dfb725"],"state_sha256":"154039aa2102047ac8fa0c3d759ec23194129ad7edce8c05e7ab754b43ff0cea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xRJc/26XhmGNYcR9u40kayjxpDOlTsH2nZv9rHc7aZMtbv4MjPrHSc0QsnkuOIYz2dG+BF+fshBdX5Hz7NheBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T03:09:28.810088Z","bundle_sha256":"68c9fba4e16ff75fc7bb55a6408937072af25a6db4e4032598fb2caedbd18115"}}