{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7O6JUOJCX2TZDOXRLTKE7UUFBW","short_pith_number":"pith:7O6JUOJC","canonical_record":{"source":{"id":"1712.07528","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-20T15:36:17Z","cross_cats_sorted":[],"title_canon_sha256":"635faf46bc81e0e2dfe62c4903a238b15b9896fced5630aed4eafc98e9d9b208","abstract_canon_sha256":"fbefd7c84f579a872603c032145f5a0fde6edc42ecf13a78fe4c7534c39bd22a"},"schema_version":"1.0"},"canonical_sha256":"fbbc9a3922bea791baf15cd44fd2850d91ca05818ad0de905eaff42cfee510fc","source":{"kind":"arxiv","id":"1712.07528","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.07528","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1712.07528v2","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.07528","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"7O6JUOJCX2TZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7O6JUOJCX2TZDOXR","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7O6JUOJC","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7O6JUOJCX2TZDOXRLTKE7UUFBW","target":"record","payload":{"canonical_record":{"source":{"id":"1712.07528","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-20T15:36:17Z","cross_cats_sorted":[],"title_canon_sha256":"635faf46bc81e0e2dfe62c4903a238b15b9896fced5630aed4eafc98e9d9b208","abstract_canon_sha256":"fbefd7c84f579a872603c032145f5a0fde6edc42ecf13a78fe4c7534c39bd22a"},"schema_version":"1.0"},"canonical_sha256":"fbbc9a3922bea791baf15cd44fd2850d91ca05818ad0de905eaff42cfee510fc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:59.913030Z","signature_b64":"k1zpujzqDTS4zHPZm1/klLFpP1OTk2W9dgIVUKEfERTPnF6aW6akreEpJon4+HreBwuJRMnH4F7GtOSe1mBvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbbc9a3922bea791baf15cd44fd2850d91ca05818ad0de905eaff42cfee510fc","last_reissued_at":"2026-05-17T23:44:59.912312Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:59.912312Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.07528","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":"2026-05-17T23:44:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oCdjPJcnkfCXTeOmjA0wK91J/gmAjpbKakptwsuhhSPzjHloaLjeX8vRuliKFWOILEJqzGZ32n6GDGFfr07nBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T05:09:21.434132Z"},"content_sha256":"29b09ef16a4d44f56ce93e4ec2ccc9ef23e67dbb55507062ddcd6e3f1d6b99c3","schema_version":"1.0","event_id":"sha256:29b09ef16a4d44f56ce93e4ec2ccc9ef23e67dbb55507062ddcd6e3f1d6b99c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7O6JUOJCX2TZDOXRLTKE7UUFBW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic mappings valued in the Wasserstein space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hugo Lavenant (LM-Orsay)","submitted_at":"2017-12-20T15:36:17Z","abstract_excerpt":"We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W\\_2) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07528","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":"2026-05-17T23:44:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JrluKTFFXu1zXDVAVk0667mB9NpleGtv7wsX5jULkq3EZveyykD51gy+6Ng8GNs1ah/YDSZNnfGImnn6fiVIBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T05:09:21.434514Z"},"content_sha256":"9d22ff6d6b8975e454ad46543c15313f89abc0c2e4b8c1a6f10a200fc30ca3fa","schema_version":"1.0","event_id":"sha256:9d22ff6d6b8975e454ad46543c15313f89abc0c2e4b8c1a6f10a200fc30ca3fa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/bundle.json","state_url":"https://pith.science/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/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":"2026-06-04T05:09:21Z","links":{"resolver":"https://pith.science/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW","bundle":"https://pith.science/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/bundle.json","state":"https://pith.science/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7O6JUOJCX2TZDOXRLTKE7UUFBW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7O6JUOJCX2TZDOXRLTKE7UUFBW","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":"fbefd7c84f579a872603c032145f5a0fde6edc42ecf13a78fe4c7534c39bd22a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-20T15:36:17Z","title_canon_sha256":"635faf46bc81e0e2dfe62c4903a238b15b9896fced5630aed4eafc98e9d9b208"},"schema_version":"1.0","source":{"id":"1712.07528","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.07528","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1712.07528v2","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.07528","created_at":"2026-05-17T23:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"7O6JUOJCX2TZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7O6JUOJCX2TZDOXR","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7O6JUOJC","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:9d22ff6d6b8975e454ad46543c15313f89abc0c2e4b8c1a6f10a200fc30ca3fa","target":"graph","created_at":"2026-05-17T23:44:59Z","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 definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W\\_2) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetn","authors_text":"Hugo Lavenant (LM-Orsay)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-20T15:36:17Z","title":"Harmonic mappings valued in the Wasserstein space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07528","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":"sha256:29b09ef16a4d44f56ce93e4ec2ccc9ef23e67dbb55507062ddcd6e3f1d6b99c3","target":"record","created_at":"2026-05-17T23:44:59Z","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":"fbefd7c84f579a872603c032145f5a0fde6edc42ecf13a78fe4c7534c39bd22a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-12-20T15:36:17Z","title_canon_sha256":"635faf46bc81e0e2dfe62c4903a238b15b9896fced5630aed4eafc98e9d9b208"},"schema_version":"1.0","source":{"id":"1712.07528","kind":"arxiv","version":2}},"canonical_sha256":"fbbc9a3922bea791baf15cd44fd2850d91ca05818ad0de905eaff42cfee510fc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbbc9a3922bea791baf15cd44fd2850d91ca05818ad0de905eaff42cfee510fc","first_computed_at":"2026-05-17T23:44:59.912312Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:59.912312Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k1zpujzqDTS4zHPZm1/klLFpP1OTk2W9dgIVUKEfERTPnF6aW6akreEpJon4+HreBwuJRMnH4F7GtOSe1mBvAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:59.913030Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.07528","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29b09ef16a4d44f56ce93e4ec2ccc9ef23e67dbb55507062ddcd6e3f1d6b99c3","sha256:9d22ff6d6b8975e454ad46543c15313f89abc0c2e4b8c1a6f10a200fc30ca3fa"],"state_sha256":"b1a3bbc3dfa2d2f68dbb329dfc6a9c63f557b6559094560203770797bee23677"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jEYbhcG+ze9+n3CFnfLDov59/JIujssLDvIxmIoYBXFVe0pA0mxvusWMPFu5YjLnPJw5StVjfLtxUOFgqnL4DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T05:09:21.436482Z","bundle_sha256":"f866a4800164fc2fa88a5a2ec96a044af598d2e330ac66502021ebf7a245ddd8"}}