{"paper":{"title":"A matricial view of the Karpelevi\\v{c} Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Charles R. Johnson, Pietro Paparella","submitted_at":"2016-11-21T19:44:54Z","abstract_excerpt":"The question of the exact region in the complex plane of the possible single eigenvalues of all $n$-by-$n$ stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevi\\v{c} in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevi\\v{c} result is unwieldy, but a simplification was given by {\\DJ}okovi\\'c in 1990 and Ito in 1997. The Karpelevi\\v{c} region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than $n$. It is shown here that each of these arcs is realized by a single, somewhat simple, parametrized stocha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06970","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"}