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It is shown that the numerical radii $w(\\cdot)$ of $A$, $A'$ and $A''$ are related"},"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":"1410.0339","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-01T19:22:00Z","cross_cats_sorted":[],"title_canon_sha256":"ebb245eb2430fea2b1c00bd9bd6fc4aae49e7334a619a6cdb48a350125e5da69","abstract_canon_sha256":"41054e4a7087050f2e295bad78e26ddd7ad434dfa90cc81c047d9ed6f875a0c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:17.508375Z","signature_b64":"ce8dmBmMq7am6pP59YYWzk3BVMt1LEUPAOEf6emxs5iErCIeXk9Zwd1FJRJQUQBerULJ/sS1ZeAZhKqNkS99Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2a5dd4d9d13b43375a83adf12c6912dcfe025fed8416009fe23501a2fc24434","last_reissued_at":"2026-05-18T02:41:17.507826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:17.507826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper and Lower Bounds for Numerical Radii of Block Shifts","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":"2014-10-01T19:22:00Z","abstract_excerpt":"For any $n$-by-$n$ matrix $A$ of the form \\[[\\begin{array}{cccc} 0 & A_1 & & \\\\ & 0 & \\ddots & \\\\ & & \\ddots & A_{k-1} \\\\ & & & 0\\end{array}],\\] we consider two $k$-by-$k$ matrices \\[A'=[\\begin{array}{cccc} 0 & \\|A_1\\| & & \\\\ & 0 & \\ddots & \\\\ & & \\ddots & \\|A_{k-1}\\| \\\\ & & & 0\\end{array}] \\ {and} \\ A''=[\\begin{array}{cccc} 0 & m(A_1) & & \\\\ & 0 & \\ddots & \\\\ & & \\ddots & m(A_{k-1}) \\\\ & & & 0\\end{array}],\\] where $\\|\\cdot\\|$ and $m(\\cdot)$ denote the operator norm and minimum modulus of a matrix, respectively. 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