{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:JOJGIKU2FGVANDHBQFMAHXQWR2","short_pith_number":"pith:JOJGIKU2","canonical_record":{"source":{"id":"1805.00602","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-02T02:16:03Z","cross_cats_sorted":[],"title_canon_sha256":"b4bf74b9242092abbf252a02e3326a571c1642e813b65da15347558ca00df609","abstract_canon_sha256":"648968e53679994829b5d69988de7d841fdd52dac9ac8e2504d3cd67c14c6df2"},"schema_version":"1.0"},"canonical_sha256":"4b92642a9a29aa068ce1815803de168e9e416b8a1b0e4a17c9898700c5446db3","source":{"kind":"arxiv","id":"1805.00602","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00602","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00602v1","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00602","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"JOJGIKU2FGVA","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"JOJGIKU2FGVANDHB","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"JOJGIKU2","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:JOJGIKU2FGVANDHBQFMAHXQWR2","target":"record","payload":{"canonical_record":{"source":{"id":"1805.00602","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-02T02:16:03Z","cross_cats_sorted":[],"title_canon_sha256":"b4bf74b9242092abbf252a02e3326a571c1642e813b65da15347558ca00df609","abstract_canon_sha256":"648968e53679994829b5d69988de7d841fdd52dac9ac8e2504d3cd67c14c6df2"},"schema_version":"1.0"},"canonical_sha256":"4b92642a9a29aa068ce1815803de168e9e416b8a1b0e4a17c9898700c5446db3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:58.376098Z","signature_b64":"3u/9ctvVzhv0rEt28283uFBORKOWTSBAB4S4pUf17E544zTeeqIhMIUaz4PBl7c+fRkGA//wGHU81orinBvXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b92642a9a29aa068ce1815803de168e9e416b8a1b0e4a17c9898700c5446db3","last_reissued_at":"2026-05-18T00:16:58.375385Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:58.375385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.00602","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":"2026-05-18T00:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCD6PnBLk0KILnblL28GTxEpnQ8KPgYi4AI0sOEjU2LIu1OTa6a2/jJ8KmzS0Rsw2FZxh5wvFc2YVZy4lVQ/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:08:32.217721Z"},"content_sha256":"8b21991cdf0ffddd62ff5d6d78c47aede9aa498740ea1454a8c61383e08b8aeb","schema_version":"1.0","event_id":"sha256:8b21991cdf0ffddd62ff5d6d78c47aede9aa498740ea1454a8c61383e08b8aeb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:JOJGIKU2FGVANDHBQFMAHXQWR2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The generalized numerical range of a set of matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chi-Kwong Li, Nung-sing Sze, Pan-Shun Lau, Yiu-Tung Poon","submitted_at":"2018-05-02T02:16:03Z","abstract_excerpt":"For a given set of $n\\times n$ matrices $\\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\\mathcal F$, denoted by $W_C({\\mathcal F})$. We obtain basic algebraic and topological properties of $W_C({\\mathcal F})$, and show that there are connections between the geometric properties of $W_C({\\mathcal F})$ and the algebraic properties of $C$ and the matrices in ${\\mathcal F}$. Furthermore, we consider the starshapedness and convexity of the set $W_C({\\mathcal F})$. In particular, we show that if ${\\mathcal F}$ is the convex hull of two matrices such that $W_C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00602","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":"2026-05-18T00:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C6JofD2+LzX0pYGx6Gcct32L2PFrQtRZN8w2RmEsxvvB//vFg9yue54qa6dkBtduMnhz29YRODmGqBhNQaRvDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T12:08:32.218074Z"},"content_sha256":"e488fdbadf12291eeffe5ebac05400950e0fb7cfb58926a01a1c30e691151ec1","schema_version":"1.0","event_id":"sha256:e488fdbadf12291eeffe5ebac05400950e0fb7cfb58926a01a1c30e691151ec1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/bundle.json","state_url":"https://pith.science/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/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":"2026-05-28T12:08:32Z","links":{"resolver":"https://pith.science/pith/JOJGIKU2FGVANDHBQFMAHXQWR2","bundle":"https://pith.science/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/bundle.json","state":"https://pith.science/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JOJGIKU2FGVANDHBQFMAHXQWR2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:JOJGIKU2FGVANDHBQFMAHXQWR2","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":"648968e53679994829b5d69988de7d841fdd52dac9ac8e2504d3cd67c14c6df2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-02T02:16:03Z","title_canon_sha256":"b4bf74b9242092abbf252a02e3326a571c1642e813b65da15347558ca00df609"},"schema_version":"1.0","source":{"id":"1805.00602","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00602","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00602v1","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00602","created_at":"2026-05-18T00:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"JOJGIKU2FGVA","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"JOJGIKU2FGVANDHB","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"JOJGIKU2","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:e488fdbadf12291eeffe5ebac05400950e0fb7cfb58926a01a1c30e691151ec1","target":"graph","created_at":"2026-05-18T00:16:58Z","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":"For a given set of $n\\times n$ matrices $\\mathcal F$, we study the union of the $C$-numerical ranges of the matrices in the set $\\mathcal F$, denoted by $W_C({\\mathcal F})$. We obtain basic algebraic and topological properties of $W_C({\\mathcal F})$, and show that there are connections between the geometric properties of $W_C({\\mathcal F})$ and the algebraic properties of $C$ and the matrices in ${\\mathcal F}$. Furthermore, we consider the starshapedness and convexity of the set $W_C({\\mathcal F})$. In particular, we show that if ${\\mathcal F}$ is the convex hull of two matrices such that $W_C","authors_text":"Chi-Kwong Li, Nung-sing Sze, Pan-Shun Lau, Yiu-Tung Poon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-02T02:16:03Z","title":"The generalized numerical range of a set of matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00602","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":"sha256:8b21991cdf0ffddd62ff5d6d78c47aede9aa498740ea1454a8c61383e08b8aeb","target":"record","created_at":"2026-05-18T00:16:58Z","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":"648968e53679994829b5d69988de7d841fdd52dac9ac8e2504d3cd67c14c6df2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-05-02T02:16:03Z","title_canon_sha256":"b4bf74b9242092abbf252a02e3326a571c1642e813b65da15347558ca00df609"},"schema_version":"1.0","source":{"id":"1805.00602","kind":"arxiv","version":1}},"canonical_sha256":"4b92642a9a29aa068ce1815803de168e9e416b8a1b0e4a17c9898700c5446db3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b92642a9a29aa068ce1815803de168e9e416b8a1b0e4a17c9898700c5446db3","first_computed_at":"2026-05-18T00:16:58.375385Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:58.375385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3u/9ctvVzhv0rEt28283uFBORKOWTSBAB4S4pUf17E544zTeeqIhMIUaz4PBl7c+fRkGA//wGHU81orinBvXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:58.376098Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.00602","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b21991cdf0ffddd62ff5d6d78c47aede9aa498740ea1454a8c61383e08b8aeb","sha256:e488fdbadf12291eeffe5ebac05400950e0fb7cfb58926a01a1c30e691151ec1"],"state_sha256":"3539e4774e6ab03b7a17ae9cd8f034a8e8bc36e1fad3303ac4890457c6ab44d6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mY+o0ESOk0HssgalRRHhqUlkb72tyONpT9GHMtjVI+85b09twnn4vNmB+kqz8G6yw31mKSHbNKtrA/RY+pqGCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T12:08:32.220060Z","bundle_sha256":"22f1e097c556bbb13a54bdd86e600bd5be54d2cc6557431d8b305addee46379d"}}