{"paper":{"title":"On tau functions associated with linear systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Gordon Blower, Samantha L. Newsham","submitted_at":"2012-07-09T19:59:06Z","abstract_excerpt":"Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\\bf C}$ and state space $H$. The function $\\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\\Gamma_{\\phi_{(x)}}$ on $L^2((0, \\infty ); {\\bf C})$; if $\\Gamma_{\\phi_{(x)}}$ is trace class, then the Fredholm determinant $\\tau (x)=\\det (I+ \\Gamma_{\\phi_{(x)}})$ defines the tau function of $(-A,B,C)$. Such tau functions arise in Tracy and Widom's theory of matrix models, where they describe the fundamental probability distributions of random matrix theory. Dyson considered such tau functions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2143","kind":"arxiv","version":2},"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"}