{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:4GYKTCS2Z5S5LFPDHIPAQHQK7V","short_pith_number":"pith:4GYKTCS2","canonical_record":{"source":{"id":"1310.6167","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T09:55:49Z","cross_cats_sorted":[],"title_canon_sha256":"add24f7a1c226f150dba723d8d12413cef6b3304521c6df4f16c5ab1e124c15f","abstract_canon_sha256":"5765fa40571f9f657c8b7db8f503b201eb587cf1298b5bd805bbd0a937b45dd1"},"schema_version":"1.0"},"canonical_sha256":"e1b0a98a5acf65d595e33a1e081e0afd43e9ae6e8a141b45431bd9c8040d5731","source":{"kind":"arxiv","id":"1310.6167","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6167","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6167v1","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6167","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"4GYKTCS2Z5S5","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4GYKTCS2Z5S5LFPD","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4GYKTCS2","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:4GYKTCS2Z5S5LFPDHIPAQHQK7V","target":"record","payload":{"canonical_record":{"source":{"id":"1310.6167","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T09:55:49Z","cross_cats_sorted":[],"title_canon_sha256":"add24f7a1c226f150dba723d8d12413cef6b3304521c6df4f16c5ab1e124c15f","abstract_canon_sha256":"5765fa40571f9f657c8b7db8f503b201eb587cf1298b5bd805bbd0a937b45dd1"},"schema_version":"1.0"},"canonical_sha256":"e1b0a98a5acf65d595e33a1e081e0afd43e9ae6e8a141b45431bd9c8040d5731","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:21.237659Z","signature_b64":"ptNyLs2uuLJNNnXSLfGYXkMlRDPL/6KG4gr3lpE2HrKbYCPh3Y/eFvS5KIEoafimouGaXLmXhr3yuV4veXs7Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1b0a98a5acf65d595e33a1e081e0afd43e9ae6e8a141b45431bd9c8040d5731","last_reissued_at":"2026-05-18T03:09:21.236785Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:21.236785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.6167","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":"2026-05-18T03:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ICvFvkBFZIHmReJVGBEk0aHAJVopvleRNNLpobBCB6pPlCbD14QajJhCZD6ToE7swSTK0wE+762qNKdfRJf7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T10:17:29.431987Z"},"content_sha256":"55df3bddf5806f74dabb352be45a384fdb3519daabc1bad0a64c226eaaaa58fc","schema_version":"1.0","event_id":"sha256:55df3bddf5806f74dabb352be45a384fdb3519daabc1bad0a64c226eaaaa58fc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:4GYKTCS2Z5S5LFPDHIPAQHQK7V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Monge-Amp\\`ere equation and its link to optimal transportation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Guido De Philippis","submitted_at":"2013-10-23T09:55:49Z","abstract_excerpt":"We survey the (old and new) regularity theory for the Monge-Amp\\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\\`ere type equations arising in that context."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6167","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":"2026-05-18T03:09:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uD71VundaYdQNNhdkmBKjvF3JyxnnivR7cXmMpFdvOZ4uS6C5Bew3us0lpCnQm5mnwzP7vrHIuo2+RY367/gBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T10:17:29.432524Z"},"content_sha256":"01ae5f80d5d6d8faff15838f212fcd81d8bf880340a8bbb190b924180dc9df93","schema_version":"1.0","event_id":"sha256:01ae5f80d5d6d8faff15838f212fcd81d8bf880340a8bbb190b924180dc9df93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/bundle.json","state_url":"https://pith.science/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/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":"2026-05-24T10:17:29Z","links":{"resolver":"https://pith.science/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V","bundle":"https://pith.science/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/bundle.json","state":"https://pith.science/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4GYKTCS2Z5S5LFPDHIPAQHQK7V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4GYKTCS2Z5S5LFPDHIPAQHQK7V","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":"5765fa40571f9f657c8b7db8f503b201eb587cf1298b5bd805bbd0a937b45dd1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T09:55:49Z","title_canon_sha256":"add24f7a1c226f150dba723d8d12413cef6b3304521c6df4f16c5ab1e124c15f"},"schema_version":"1.0","source":{"id":"1310.6167","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.6167","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"arxiv_version","alias_value":"1310.6167v1","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.6167","created_at":"2026-05-18T03:09:21Z"},{"alias_kind":"pith_short_12","alias_value":"4GYKTCS2Z5S5","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4GYKTCS2Z5S5LFPD","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4GYKTCS2","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:01ae5f80d5d6d8faff15838f212fcd81d8bf880340a8bbb190b924180dc9df93","target":"graph","created_at":"2026-05-18T03:09:21Z","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 survey the (old and new) regularity theory for the Monge-Amp\\`ere equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Amp\\`ere type equations arising in that context.","authors_text":"Alessio Figalli, Guido De Philippis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T09:55:49Z","title":"The Monge-Amp\\`ere equation and its link to optimal transportation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6167","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":"sha256:55df3bddf5806f74dabb352be45a384fdb3519daabc1bad0a64c226eaaaa58fc","target":"record","created_at":"2026-05-18T03:09:21Z","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":"5765fa40571f9f657c8b7db8f503b201eb587cf1298b5bd805bbd0a937b45dd1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-10-23T09:55:49Z","title_canon_sha256":"add24f7a1c226f150dba723d8d12413cef6b3304521c6df4f16c5ab1e124c15f"},"schema_version":"1.0","source":{"id":"1310.6167","kind":"arxiv","version":1}},"canonical_sha256":"e1b0a98a5acf65d595e33a1e081e0afd43e9ae6e8a141b45431bd9c8040d5731","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1b0a98a5acf65d595e33a1e081e0afd43e9ae6e8a141b45431bd9c8040d5731","first_computed_at":"2026-05-18T03:09:21.236785Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:21.236785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ptNyLs2uuLJNNnXSLfGYXkMlRDPL/6KG4gr3lpE2HrKbYCPh3Y/eFvS5KIEoafimouGaXLmXhr3yuV4veXs7Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:21.237659Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.6167","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:55df3bddf5806f74dabb352be45a384fdb3519daabc1bad0a64c226eaaaa58fc","sha256:01ae5f80d5d6d8faff15838f212fcd81d8bf880340a8bbb190b924180dc9df93"],"state_sha256":"c7974ec903915723bf9d587d51fb9aae15711404d1f5a2f98e772a8a6fc5b90c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OGoQNKDD7rr54udaRzS9RWfR0lIhuSmE2vIliojSErp51QfnLOF2uk4nlkVPwb0vcYD4XkzmIBlKTncFWDC8CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T10:17:29.435926Z","bundle_sha256":"92f559c2449679869e0c07155ab5dc517e0f94f3437555d8bbf7b88243ca310c"}}