{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZUZ2NLUCZKY6K7ET4N5IU3G4NJ","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":"5599d1ce6e763849ccf081f55d8b0acef8e1ba0afc87dcd0f2c7f548d3142e85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-26T17:21:20Z","title_canon_sha256":"b31581d9cc576c9c28913f552833d120808d7bdb66ec57c80ca854564325c414"},"schema_version":"1.0","source":{"id":"1811.10518","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.10518","created_at":"2026-05-17T23:59:54Z"},{"alias_kind":"arxiv_version","alias_value":"1811.10518v1","created_at":"2026-05-17T23:59:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.10518","created_at":"2026-05-17T23:59:54Z"},{"alias_kind":"pith_short_12","alias_value":"ZUZ2NLUCZKY6","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZUZ2NLUCZKY6K7ET","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZUZ2NLUC","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:2e6278eb4c8abe32fb953ec3a415582318970d8b797eaf36ea37c684bbbc38af","target":"graph","created_at":"2026-05-17T23:59:54Z","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 use principal angles between two subspaces to define Jordan planes. Jordan planes provide an optimal way to decompose $\\mathbb{C}^n$ in relation to given two subspaces. We apply Jordan planes to show that two pairs of of subspaces $(M,N)$ and $(M^{\\perp},N^{\\perp})$ are unitarily equivalent if $M$ and $N$ are subspaces of $\\mathbb{C}^n$ in generic position. We compute numerical ranges of sum and product of two orthogonal projections by using Jordan planes.","authors_text":"Jaedeok Kim, Youngmi Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-26T17:21:20Z","title":"Jordan Plane and Numerical Range of Operators Involving Two Projections"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10518","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:bdb52561f3ce84809f679cf85316b99bd7e79315f345c0527e3a2b2e16b9e0f3","target":"record","created_at":"2026-05-17T23:59:54Z","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":"5599d1ce6e763849ccf081f55d8b0acef8e1ba0afc87dcd0f2c7f548d3142e85","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-26T17:21:20Z","title_canon_sha256":"b31581d9cc576c9c28913f552833d120808d7bdb66ec57c80ca854564325c414"},"schema_version":"1.0","source":{"id":"1811.10518","kind":"arxiv","version":1}},"canonical_sha256":"cd33a6ae82cab1e57c93e37a8a6cdc6a4402dbc81c0f7ae9cc74c2d2ff557b3e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd33a6ae82cab1e57c93e37a8a6cdc6a4402dbc81c0f7ae9cc74c2d2ff557b3e","first_computed_at":"2026-05-17T23:59:54.463222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:54.463222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fUIRfSR2UX6KZwvAdB1l4wwnDr6mm/oJsf4IAt1tGxTw+Mjaag+vzJHGuwA44DFuR1w25B4ZNojD341sKQ7IDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:54.463774Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.10518","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bdb52561f3ce84809f679cf85316b99bd7e79315f345c0527e3a2b2e16b9e0f3","sha256:2e6278eb4c8abe32fb953ec3a415582318970d8b797eaf36ea37c684bbbc38af"],"state_sha256":"8d8b51346cb7ea34f43cc5c1412930e9c244115d01159c9d7b25c9e07a8e159d"}