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For $n \\geq 3$, the set of refined inertias $\\mathbb{H}_n=\\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\\}$ is important for the onset of Hopf bifurcation in dynamical systems. We say that an $n\\times n$ sign pattern ${\\cal A}$ requires $\\mathbb{H}_n$ if $\\mathbb{H}_n=\\{\\text{"},"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":"1710.08955","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-24T19:12:49Z","cross_cats_sorted":[],"title_canon_sha256":"219ec8d5a6acbd2740ddc368eb6f0d84b6e9120962ee46a2b71a60b3a24b2e6d","abstract_canon_sha256":"5e4ebcb4aaf51b8b0085c4132599a88797cf561148547f4287af403f173b5477"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:00.689677Z","signature_b64":"sIHJBlgMt+llGDpHxEOezkYH3aaw07Mk4beSPsVjMaV7rZJg4wh80h05/kKwHCx00iSKGgrHZ31JsWLqmN7CDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a93fa47efd7a0865fe16cd4f5d81237c5bf960c3f558c1a66fe069bc0679ba62","last_reissued_at":"2026-05-18T00:32:00.689198Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:00.689198Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sign patterns that require $\\mathbb{H}_n$ exist for each $n\\geq 4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lihua Zhang, Wei Gao, Zhongshan Li","submitted_at":"2017-10-24T19:12:49Z","abstract_excerpt":"The refined inertia of a square real matrix $A$ is the ordered $4$-tuple $(n_+, n_-, n_z, 2n_p)$, where $n_+$ (resp., $n_-$) is the number of eigenvalues of $A$ with positive (resp., negative) real part, $n_z$ is the number of zero eigenvalues of $A$, and $2n_p$ is the number of nonzero pure imaginary eigenvalues of $A$. 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