{"paper":{"title":"On Bi-free Multiplicative Convolution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Mingchu Gao","submitted_at":"2017-10-13T22:03:25Z","abstract_excerpt":"In this paper, we study the partial bi-free $S$-transform of a pair $(a,b)$ of random variables, and the $S$-transform of the $2\\times 2$ matrix-valued random variable $\\left(\\begin{matrix}a&0\\\\0&b\\end{matrix}\\right)$ associated with $(a,b)$ when restricted to upper triangular $2\\times 2$ matrices. We first derive an explicit expression of bi-free multiplicative convolution (of probability measures on the bi-unit-sphere $\\mathbb{T}^2$ of $\\mathbb{C}^2$, or on $\\mathbb{R}^2_+$ in $\\mathbb{C}^2$) from a subordination equation for bi-free multiplicative convolution. We then show that, when $(a_1,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.05087","kind":"arxiv","version":3},"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"}