{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:E6YH4YDLY46UU7JMCOLDGDTFV7","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":"818efd2dd34b5b4a96e0eb3e9f011aa631008eb7772dc13f08c14d5df9db239c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-04-12T11:39:57Z","title_canon_sha256":"405b0165211f8776ea1e3f76e83d4dc66d923598a69cf97ca77897edaf2b5dda"},"schema_version":"1.0","source":{"id":"0904.1852","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0904.1852","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"arxiv_version","alias_value":"0904.1852v4","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0904.1852","created_at":"2026-05-18T03:19:57Z"},{"alias_kind":"pith_short_12","alias_value":"E6YH4YDLY46U","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"E6YH4YDLY46UU7JM","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"E6YH4YDL","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:5b2fb9cd05da6c2ae6381019f2697e12374cda51b24980765e1d24e093522ea2","target":"graph","created_at":"2026-05-18T03:19:57Z","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":"Given two probability measures $\\mu$ and $\\nu$ we consider a mass transportation mapping $T$ satisfying 1) $T$ sends $\\mu$ to $\\nu$, 2) $T$ has the form $T = \\phi \\frac{\\nabla \\phi}{|\\nabla \\phi|}$, where $\\phi$ is a function with convex sublevel sets.\n  We prove a change of variables formula for $T$. We also establish Sobolev estimates for $\\phi$, and a new form of the parabolic maximum principle.\n  In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.","authors_text":"Alexander V. Kolesnikov","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-04-12T11:39:57Z","title":"Weak regularity of Gauss mass transport"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.1852","kind":"arxiv","version":4},"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":"sha256:2e019b47868bd1ce8cffe59a2feccf4f3d22d39b2d5a6dfcd4d267941f052ce1","target":"record","created_at":"2026-05-18T03:19:57Z","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":"818efd2dd34b5b4a96e0eb3e9f011aa631008eb7772dc13f08c14d5df9db239c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-04-12T11:39:57Z","title_canon_sha256":"405b0165211f8776ea1e3f76e83d4dc66d923598a69cf97ca77897edaf2b5dda"},"schema_version":"1.0","source":{"id":"0904.1852","kind":"arxiv","version":4}},"canonical_sha256":"27b07e606bc73d4a7d2c1396330e65afe89fdbecd9e14f4a4e554b6a962dcfdf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27b07e606bc73d4a7d2c1396330e65afe89fdbecd9e14f4a4e554b6a962dcfdf","first_computed_at":"2026-05-18T03:19:57.190475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:57.190475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sbj8JtkC8U1XAcD6i9u1jI8u802jLt/Oy2iFjRz6+n0qFsH85rYhTV9iAtubNrGbYk+HftO90dcrq0a/No2GDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:57.191194Z","signed_message":"canonical_sha256_bytes"},"source_id":"0904.1852","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e019b47868bd1ce8cffe59a2feccf4f3d22d39b2d5a6dfcd4d267941f052ce1","sha256:5b2fb9cd05da6c2ae6381019f2697e12374cda51b24980765e1d24e093522ea2"],"state_sha256":"888f413788a7f3aa753d1f32642b3df5ad152bf0eaabf223b2f4058a92e7b347"}