{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GPWDZQFZCWETARXFCR4LHASXBQ","short_pith_number":"pith:GPWDZQFZ","canonical_record":{"source":{"id":"1207.5927","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-25T08:58:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d6cfee7b912b9eb50dbca946323d3200b6e61b627bb4d13ff537293f69c3b73d","abstract_canon_sha256":"1e5be40476b266ed8705b6696be2c5d17d413b9524b256a3ac1e321a2af0c58b"},"schema_version":"1.0"},"canonical_sha256":"33ec3cc0b915893046e51478b382570c2f8332ae191c160a2c8b8deeafb4063c","source":{"kind":"arxiv","id":"1207.5927","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5927","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5927v2","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5927","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"GPWDZQFZCWET","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GPWDZQFZCWETARXF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GPWDZQFZ","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GPWDZQFZCWETARXFCR4LHASXBQ","target":"record","payload":{"canonical_record":{"source":{"id":"1207.5927","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-25T08:58:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d6cfee7b912b9eb50dbca946323d3200b6e61b627bb4d13ff537293f69c3b73d","abstract_canon_sha256":"1e5be40476b266ed8705b6696be2c5d17d413b9524b256a3ac1e321a2af0c58b"},"schema_version":"1.0"},"canonical_sha256":"33ec3cc0b915893046e51478b382570c2f8332ae191c160a2c8b8deeafb4063c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:52.595803Z","signature_b64":"65VNdYhW3Y6n8xlBvCbjH3iWDUowsG4MHMXkvsqNa7CblFnrtBSKe6k/JPUsw7YVeI82Ic+KXYDINAjRsR1vDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33ec3cc0b915893046e51478b382570c2f8332ae191c160a2c8b8deeafb4063c","last_reissued_at":"2026-05-18T02:17:52.595095Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:52.595095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.5927","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xuzIYu50xp54PlxzPskdVEh6PL+arK4E4QjckEVdpVykMKsAWQXHJbrRGdL1DwL86jP2Np/3uzqpa0M54NFMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:40:46.995053Z"},"content_sha256":"16fbc6255e307733724d6a8d647140055cae674e269c80dca4f846e94dd38f02","schema_version":"1.0","event_id":"sha256:16fbc6255e307733724d6a8d647140055cae674e269c80dca4f846e94dd38f02"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GPWDZQFZCWETARXFCR4LHASXBQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Claude Bardos, Fran\\c{c}ois Golse, Peter Markowich, Thierry Paul","submitted_at":"2012-07-25T08:58:36Z","abstract_excerpt":"Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5927","kind":"arxiv","version":2},"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":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pvHYWRUUpCN6iAcpFWodaGTgWcnm9z0qfzqCZdwt+tu39eU2oaMXWZtokc9iV0VNU+oRrw1cPVJ7bz4Een9gAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:40:46.995697Z"},"content_sha256":"bca00401b50bbb28c85e85dc5329b124633e5ebac6f27b9c63e5e3dea3c4d71b","schema_version":"1.0","event_id":"sha256:bca00401b50bbb28c85e85dc5329b124633e5ebac6f27b9c63e5e3dea3c4d71b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GPWDZQFZCWETARXFCR4LHASXBQ/bundle.json","state_url":"https://pith.science/pith/GPWDZQFZCWETARXFCR4LHASXBQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GPWDZQFZCWETARXFCR4LHASXBQ/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":"2026-05-25T11:40:46Z","links":{"resolver":"https://pith.science/pith/GPWDZQFZCWETARXFCR4LHASXBQ","bundle":"https://pith.science/pith/GPWDZQFZCWETARXFCR4LHASXBQ/bundle.json","state":"https://pith.science/pith/GPWDZQFZCWETARXFCR4LHASXBQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GPWDZQFZCWETARXFCR4LHASXBQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GPWDZQFZCWETARXFCR4LHASXBQ","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":"1e5be40476b266ed8705b6696be2c5d17d413b9524b256a3ac1e321a2af0c58b","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-25T08:58:36Z","title_canon_sha256":"d6cfee7b912b9eb50dbca946323d3200b6e61b627bb4d13ff537293f69c3b73d"},"schema_version":"1.0","source":{"id":"1207.5927","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.5927","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1207.5927v2","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5927","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"GPWDZQFZCWET","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GPWDZQFZCWETARXF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GPWDZQFZ","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:bca00401b50bbb28c85e85dc5329b124633e5ebac6f27b9c63e5e3dea3c4d71b","target":"graph","created_at":"2026-05-18T02:17:52Z","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":"Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions","authors_text":"Claude Bardos, Fran\\c{c}ois Golse, Peter Markowich, Thierry Paul","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-25T08:58:36Z","title":"Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5927","kind":"arxiv","version":2},"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":"sha256:16fbc6255e307733724d6a8d647140055cae674e269c80dca4f846e94dd38f02","target":"record","created_at":"2026-05-18T02:17:52Z","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":"1e5be40476b266ed8705b6696be2c5d17d413b9524b256a3ac1e321a2af0c58b","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-25T08:58:36Z","title_canon_sha256":"d6cfee7b912b9eb50dbca946323d3200b6e61b627bb4d13ff537293f69c3b73d"},"schema_version":"1.0","source":{"id":"1207.5927","kind":"arxiv","version":2}},"canonical_sha256":"33ec3cc0b915893046e51478b382570c2f8332ae191c160a2c8b8deeafb4063c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33ec3cc0b915893046e51478b382570c2f8332ae191c160a2c8b8deeafb4063c","first_computed_at":"2026-05-18T02:17:52.595095Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:52.595095Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"65VNdYhW3Y6n8xlBvCbjH3iWDUowsG4MHMXkvsqNa7CblFnrtBSKe6k/JPUsw7YVeI82Ic+KXYDINAjRsR1vDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:52.595803Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.5927","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16fbc6255e307733724d6a8d647140055cae674e269c80dca4f846e94dd38f02","sha256:bca00401b50bbb28c85e85dc5329b124633e5ebac6f27b9c63e5e3dea3c4d71b"],"state_sha256":"cd237c1299bd03a0c1e4515baee9b16212fbabf13d4367f5f0c33b08a6142519"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2btcq/+dADwZqEEH7UJD9VjCwCDuZShUwOzbjRhenOb7rfID7LCSx4RKjgbarWSmTy7ehsaJCtpFTCgB1S3/BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T11:40:46.999445Z","bundle_sha256":"0c2a0580e359a1a98f1f6c36c534489a75424146ec4d18fb438f6377cc8ec5cb"}}