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Among other things, we prove the following properties of its numerical range: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\\neq 0$, (2) its boundary $\\partial W(J_n(a))$ contains a line segment if and only if $n\\ge 3$ and $|a|=1$, and (3) the intersection of the boundaries $\\partial W(J_n(a))$ and $\\partial W(J_n(a)[j])$ is either the singleton $\\{\\min\\sigma(\\re J_n(a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.0295","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-04-01T04:30:04Z","cross_cats_sorted":[],"title_canon_sha256":"cda7afdcaa9a4b73ba8ec3221d76c9e9953cfaa97e83e683a1aa9d21ca7bcaab","abstract_canon_sha256":"dcf725fa494e02c3188a26bf89327230c78f3fe25cf221637a58a29a8cdf4eff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:17.364797Z","signature_b64":"mROWL5y+XrbWr/x88wNiKYjDjzdi0HAV3ULoum0aDDwss7EnPH+sTN2IW/RLKlUy4YKNHvznb8WlNH0mppFtAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86128e8cb0fed34c190992880c8a7f3693b91785895c4580d8fce7a8c587ac5f","last_reissued_at":"2026-05-18T03:29:17.364246Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:17.364246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Numerical Ranges of KMS Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hwa-Long Gau, Pei Yuan Wu","submitted_at":"2013-04-01T04:30:04Z","abstract_excerpt":"A KMS matrix is one of the form $$J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \\ddots & \\vdots & & \\ddots & \\ddots & a^2 & & & \\ddots & a 0 & & & & 0{array}]$$ for $n\\ge 1$ and $a$ in $\\mathbb{C}$. Among other things, we prove the following properties of its numerical range: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\\neq 0$, (2) its boundary $\\partial W(J_n(a))$ contains a line segment if and only if $n\\ge 3$ and $|a|=1$, and (3) the intersection of the boundaries $\\partial W(J_n(a))$ and $\\partial W(J_n(a)[j])$ is either the singleton $\\{\\min\\sigma(\\re J_n(a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.0295","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1304.0295","created_at":"2026-05-18T03:29:17.364328+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.0295v1","created_at":"2026-05-18T03:29:17.364328+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.0295","created_at":"2026-05-18T03:29:17.364328+00:00"},{"alias_kind":"pith_short_12","alias_value":"QYJI5DFQ73JU","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"QYJI5DFQ73JUYGIJ","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"QYJI5DFQ","created_at":"2026-05-18T12:27:57.521954+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2","json":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2.json","graph_json":"https://pith.science/api/pith-number/QYJI5DFQ73JUYGIJSKEAZCT7G2/graph.json","events_json":"https://pith.science/api/pith-number/QYJI5DFQ73JUYGIJSKEAZCT7G2/events.json","paper":"https://pith.science/paper/QYJI5DFQ"},"agent_actions":{"view_html":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2","download_json":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2.json","view_paper":"https://pith.science/paper/QYJI5DFQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.0295&json=true","fetch_graph":"https://pith.science/api/pith-number/QYJI5DFQ73JUYGIJSKEAZCT7G2/graph.json","fetch_events":"https://pith.science/api/pith-number/QYJI5DFQ73JUYGIJSKEAZCT7G2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2/action/storage_attestation","attest_author":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2/action/author_attestation","sign_citation":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2/action/citation_signature","submit_replication":"https://pith.science/pith/QYJI5DFQ73JUYGIJSKEAZCT7G2/action/replication_record"}},"created_at":"2026-05-18T03:29:17.364328+00:00","updated_at":"2026-05-18T03:29:17.364328+00:00"}