{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:FW2SWANRHI6V7IPBL7MRGJM55T","short_pith_number":"pith:FW2SWANR","canonical_record":{"source":{"id":"math/0610927","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2006-10-30T13:00:55Z","cross_cats_sorted":[],"title_canon_sha256":"11e27e568a42130969aff107005188c52f1cc7b0b22be01f5e5d89589d4fe85b","abstract_canon_sha256":"59cae966615690f64c7ecff98f08738b81f1089409f54dfe751ada8787e2a765"},"schema_version":"1.0"},"canonical_sha256":"2db52b01b13a3d5fa1e15fd913259decf164c2de1779cc56c9d0ffa2020ee915","source":{"kind":"arxiv","id":"math/0610927","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610927","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610927v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610927","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"FW2SWANRHI6V","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"FW2SWANRHI6V7IPB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"FW2SWANR","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:FW2SWANRHI6V7IPBL7MRGJM55T","target":"record","payload":{"canonical_record":{"source":{"id":"math/0610927","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2006-10-30T13:00:55Z","cross_cats_sorted":[],"title_canon_sha256":"11e27e568a42130969aff107005188c52f1cc7b0b22be01f5e5d89589d4fe85b","abstract_canon_sha256":"59cae966615690f64c7ecff98f08738b81f1089409f54dfe751ada8787e2a765"},"schema_version":"1.0"},"canonical_sha256":"2db52b01b13a3d5fa1e15fd913259decf164c2de1779cc56c9d0ffa2020ee915","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:22.524111Z","signature_b64":"FGID9Eyu5RjJ4AZlcOGadlYSgPZmjO8JbelTv3O6BhIrgJt45tqYzrlbZWn56rFlvopehfDZSCrp21QAK+5uBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2db52b01b13a3d5fa1e15fd913259decf164c2de1779cc56c9d0ffa2020ee915","last_reissued_at":"2026-05-18T01:05:22.523418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:22.523418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0610927","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B2ZjbsbRF8+pTn6qD3jaHK+26VKAhUb0OLYjxP7VQam+HwKYmyLunZZqQuIXtL8J5I8QL99cUfLrKxIgw79TAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:16:17.765048Z"},"content_sha256":"c16220c2c1fa25954851764d1ef8e9bac1300b9ad8f735722962e45194674519","schema_version":"1.0","event_id":"sha256:c16220c2c1fa25954851764d1ef8e9bac1300b9ad8f735722962e45194674519"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:FW2SWANRHI6V7IPBL7MRGJM55T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Radon transform on real, complex and quaternionic Grassmannians","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Genkai Zhang","submitted_at":"2006-10-30T13:00:55Z","abstract_excerpt":"Let $G_{n,k}(\\bbK)$ be the\n Grassmannian manifold of $k$-dimensional $\\bbK$-subspaces in $\\bbK^n$ where $\\bbK=\\mathbb R, \\mathbb C, \\mathbb H$ is the field of real, complex or quaternionic numbers.\n For $1\\le k < k^\\prime \\le n-1$ we define the Radon transform $(\\mathcal R f)(\\eta)$, $\\eta \\in G_{n,k^\\prime}(\\bbK)$, for functions $f(\\xi)$ on $G_{n,k}(\\bbK)$ as an integration over all $\\xi \\subset \\eta$. When $k+k^\\prime \\le n$ we give an inversion formula in terms of the G\\aa{}rding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive $k\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610927","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":"2026-05-18T01:05:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CuAlXjlQYRtIdQ7kfb4sVHfIURpWPMdTh4L+/t8tcVVFVBNJzKSPzuW+KmgZJkYPefQPswM5PM56mUPAz7sfCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:16:17.765422Z"},"content_sha256":"a2cb5bb11e132e284ccac51accf1228b70a071896c0108b4d5427ae447d6ff8a","schema_version":"1.0","event_id":"sha256:a2cb5bb11e132e284ccac51accf1228b70a071896c0108b4d5427ae447d6ff8a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FW2SWANRHI6V7IPBL7MRGJM55T/bundle.json","state_url":"https://pith.science/pith/FW2SWANRHI6V7IPBL7MRGJM55T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FW2SWANRHI6V7IPBL7MRGJM55T/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":"2026-06-22T12:16:17Z","links":{"resolver":"https://pith.science/pith/FW2SWANRHI6V7IPBL7MRGJM55T","bundle":"https://pith.science/pith/FW2SWANRHI6V7IPBL7MRGJM55T/bundle.json","state":"https://pith.science/pith/FW2SWANRHI6V7IPBL7MRGJM55T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FW2SWANRHI6V7IPBL7MRGJM55T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:FW2SWANRHI6V7IPBL7MRGJM55T","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":"59cae966615690f64c7ecff98f08738b81f1089409f54dfe751ada8787e2a765","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2006-10-30T13:00:55Z","title_canon_sha256":"11e27e568a42130969aff107005188c52f1cc7b0b22be01f5e5d89589d4fe85b"},"schema_version":"1.0","source":{"id":"math/0610927","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610927","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610927v1","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610927","created_at":"2026-05-18T01:05:22Z"},{"alias_kind":"pith_short_12","alias_value":"FW2SWANRHI6V","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"FW2SWANRHI6V7IPB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"FW2SWANR","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:a2cb5bb11e132e284ccac51accf1228b70a071896c0108b4d5427ae447d6ff8a","target":"graph","created_at":"2026-05-18T01:05:22Z","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":"Let $G_{n,k}(\\bbK)$ be the\n Grassmannian manifold of $k$-dimensional $\\bbK$-subspaces in $\\bbK^n$ where $\\bbK=\\mathbb R, \\mathbb C, \\mathbb H$ is the field of real, complex or quaternionic numbers.\n For $1\\le k < k^\\prime \\le n-1$ we define the Radon transform $(\\mathcal R f)(\\eta)$, $\\eta \\in G_{n,k^\\prime}(\\bbK)$, for functions $f(\\xi)$ on $G_{n,k}(\\bbK)$ as an integration over all $\\xi \\subset \\eta$. When $k+k^\\prime \\le n$ we give an inversion formula in terms of the G\\aa{}rding-Gindikin fractional integration and the Cayley type differential operator on the symmetric cone of positive $k\\t","authors_text":"Genkai Zhang","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2006-10-30T13:00:55Z","title":"Radon transform on real, complex and quaternionic Grassmannians"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610927","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:c16220c2c1fa25954851764d1ef8e9bac1300b9ad8f735722962e45194674519","target":"record","created_at":"2026-05-18T01:05:22Z","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":"59cae966615690f64c7ecff98f08738b81f1089409f54dfe751ada8787e2a765","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"2006-10-30T13:00:55Z","title_canon_sha256":"11e27e568a42130969aff107005188c52f1cc7b0b22be01f5e5d89589d4fe85b"},"schema_version":"1.0","source":{"id":"math/0610927","kind":"arxiv","version":1}},"canonical_sha256":"2db52b01b13a3d5fa1e15fd913259decf164c2de1779cc56c9d0ffa2020ee915","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2db52b01b13a3d5fa1e15fd913259decf164c2de1779cc56c9d0ffa2020ee915","first_computed_at":"2026-05-18T01:05:22.523418Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:22.523418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FGID9Eyu5RjJ4AZlcOGadlYSgPZmjO8JbelTv3O6BhIrgJt45tqYzrlbZWn56rFlvopehfDZSCrp21QAK+5uBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:22.524111Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610927","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c16220c2c1fa25954851764d1ef8e9bac1300b9ad8f735722962e45194674519","sha256:a2cb5bb11e132e284ccac51accf1228b70a071896c0108b4d5427ae447d6ff8a"],"state_sha256":"a5f2f133bdf7aa1f52c3f7ca1a2091e311e7c6f42706696af9ee5128b4f8978b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7Rv3KFDApC/54Miqy2Ay7hnmEnXB2sSw6GzmlDMO6r5uaflwWy2/yBwXDUF1PNz/D5dEFSKcFAsUH0wEnlCLBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T12:16:17.767440Z","bundle_sha256":"66177bae45aa55f11685187f00d77c9e7e37a139d6c4b98e1fcf30a90dfe3963"}}