{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7LPM5ICCA5VZUY4SGN2NWIGES6","short_pith_number":"pith:7LPM5ICC","canonical_record":{"source":{"id":"1701.04779","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-17T17:34:53Z","cross_cats_sorted":[],"title_canon_sha256":"be747833cd0113517aaa5094d089ac96416b0e9596d3ee4cd82d956028ab878d","abstract_canon_sha256":"c444e88ea73423e02959515e5eb7660c7b937d590fe788b5e1794e4fce7bcaf6"},"schema_version":"1.0"},"canonical_sha256":"fadecea042076b9a63923374db20c4978f9ae472d76aea42cfdebba14a26c01e","source":{"kind":"arxiv","id":"1701.04779","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.04779","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1701.04779v3","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04779","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"7LPM5ICCA5VZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7LPM5ICCA5VZUY4S","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7LPM5ICC","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7LPM5ICCA5VZUY4SGN2NWIGES6","target":"record","payload":{"canonical_record":{"source":{"id":"1701.04779","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-17T17:34:53Z","cross_cats_sorted":[],"title_canon_sha256":"be747833cd0113517aaa5094d089ac96416b0e9596d3ee4cd82d956028ab878d","abstract_canon_sha256":"c444e88ea73423e02959515e5eb7660c7b937d590fe788b5e1794e4fce7bcaf6"},"schema_version":"1.0"},"canonical_sha256":"fadecea042076b9a63923374db20c4978f9ae472d76aea42cfdebba14a26c01e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:23.925306Z","signature_b64":"SbYqEqh03LJLCMBk95zb5nOLiEuCEXp1q/mR9c5LxIAvacfOEugywpputJKHNQOkCiOh2Y234bb/sj5g/5y2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fadecea042076b9a63923374db20c4978f9ae472d76aea42cfdebba14a26c01e","last_reissued_at":"2026-05-18T00:11:23.924826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:23.924826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.04779","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GpQQIVIQZ6JeEQB7t+c7mL1J1O6pdWDoEP7pClFVvU1ZZ7v2nsx8+VpcUe9dzz+6KUPVlznY+ZhBiF2D5SOuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:04:41.839984Z"},"content_sha256":"1ae1212025ae93549274fe5cf5035d6e450679866f4aef5c31e8f9a38456c5fa","schema_version":"1.0","event_id":"sha256:1ae1212025ae93549274fe5cf5035d6e450679866f4aef5c31e8f9a38456c5fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7LPM5ICCA5VZUY4SGN2NWIGES6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convexity theorems for the gradient map on probability measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alberto Raffero, Leonardo Biliotti","submitted_at":"2017-01-17T17:34:53Z","abstract_excerpt":"Given a K\\\"ahler manifold $(Z,J,\\omega)$ and a compact real submanifold $M\\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\\rm G}$ on the space of probability measures on $M.$ In particular, we prove convexity results for such map when ${\\rm G}$ is Abelian and we investigate how to extend them to the non-Abelian case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04779","kind":"arxiv","version":3},"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":"2026-05-18T00:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OzAgry2HB3is0bxlAbS5cruJJkmvkjJAmaiYjxzKok8GHAu+1fQ1juNAWfI/CYPwTwjNBHS/kha5jB/1zEwMAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T17:04:41.840338Z"},"content_sha256":"4df28cbfbc4835d4e00ff279016b168bad2dbcf517df115ff2ca64c6f276c03b","schema_version":"1.0","event_id":"sha256:4df28cbfbc4835d4e00ff279016b168bad2dbcf517df115ff2ca64c6f276c03b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/bundle.json","state_url":"https://pith.science/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/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":"2026-06-03T17:04:41Z","links":{"resolver":"https://pith.science/pith/7LPM5ICCA5VZUY4SGN2NWIGES6","bundle":"https://pith.science/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/bundle.json","state":"https://pith.science/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7LPM5ICCA5VZUY4SGN2NWIGES6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7LPM5ICCA5VZUY4SGN2NWIGES6","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":"c444e88ea73423e02959515e5eb7660c7b937d590fe788b5e1794e4fce7bcaf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-17T17:34:53Z","title_canon_sha256":"be747833cd0113517aaa5094d089ac96416b0e9596d3ee4cd82d956028ab878d"},"schema_version":"1.0","source":{"id":"1701.04779","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.04779","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"1701.04779v3","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04779","created_at":"2026-05-18T00:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"7LPM5ICCA5VZ","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7LPM5ICCA5VZUY4S","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7LPM5ICC","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:4df28cbfbc4835d4e00ff279016b168bad2dbcf517df115ff2ca64c6f276c03b","target":"graph","created_at":"2026-05-18T00:11:23Z","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 a K\\\"ahler manifold $(Z,J,\\omega)$ and a compact real submanifold $M\\subset Z$, we study the properties of the gradient map associated with the action of a noncompact real reductive Lie group ${\\rm G}$ on the space of probability measures on $M.$ In particular, we prove convexity results for such map when ${\\rm G}$ is Abelian and we investigate how to extend them to the non-Abelian case.","authors_text":"Alberto Raffero, Leonardo Biliotti","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-17T17:34:53Z","title":"Convexity theorems for the gradient map on probability measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04779","kind":"arxiv","version":3},"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":"sha256:1ae1212025ae93549274fe5cf5035d6e450679866f4aef5c31e8f9a38456c5fa","target":"record","created_at":"2026-05-18T00:11:23Z","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":"c444e88ea73423e02959515e5eb7660c7b937d590fe788b5e1794e4fce7bcaf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-01-17T17:34:53Z","title_canon_sha256":"be747833cd0113517aaa5094d089ac96416b0e9596d3ee4cd82d956028ab878d"},"schema_version":"1.0","source":{"id":"1701.04779","kind":"arxiv","version":3}},"canonical_sha256":"fadecea042076b9a63923374db20c4978f9ae472d76aea42cfdebba14a26c01e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fadecea042076b9a63923374db20c4978f9ae472d76aea42cfdebba14a26c01e","first_computed_at":"2026-05-18T00:11:23.924826Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:23.924826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SbYqEqh03LJLCMBk95zb5nOLiEuCEXp1q/mR9c5LxIAvacfOEugywpputJKHNQOkCiOh2Y234bb/sj5g/5y2Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:23.925306Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.04779","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1ae1212025ae93549274fe5cf5035d6e450679866f4aef5c31e8f9a38456c5fa","sha256:4df28cbfbc4835d4e00ff279016b168bad2dbcf517df115ff2ca64c6f276c03b"],"state_sha256":"4384ff8385950b455e169745f9875228c9e0031eaedebb72338105353339062e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o+qwRpcl7CVaQcE+Bw0V40dA53uInVFKYkr5X7s2WQbg9oBwiVlIVnHxg2nG4yg8iWphIJbmoo0H4ocGmw+bBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T17:04:41.842232Z","bundle_sha256":"80f272fa00a150c1f8b42053ad1b8a4d5e6fa8909d95a94c75bb3ada0e02df33"}}