{"paper":{"title":"Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Edward B. Saff, Johann S. Brauchart, Peter D. Dragnev, Robert S. Womersley","submitted_at":"2017-06-28T16:18:10Z","abstract_excerpt":"We consider the minimal discrete and continuous energy problems on the unit sphere $\\mathbb{S}^d$ in the Euclidean space $\\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on $\\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$ is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d \\geq 3$ and the logarithmic potential $\\log(1/r)$ if $d = 2$. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the conti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09346","kind":"arxiv","version":1},"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"}