{"paper":{"title":"A Minimum problem for finite sets of real numbers with non-negative sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Caterina Nardi, Giampiero Chiaselotti, Giuseppe Marino","submitted_at":"2011-02-23T15:17:30Z","abstract_excerpt":"Let $n$ and $r$ be two integers such that $0 < r \\le n$; we denote by $\\gamma(n,r)$ [$\\eta(n,r)$] the minimum [maximum] number of the non-negative partial sums of a sum $\\sum_{1=1}^n a_i \\ge 0$, where $a_1, \\cdots, a_n$ are $n$ real numbers arbitrarily chosen in such a way that $r$ of them are non-negative and the remaining $n-r$ are negative. Inspired by some interesting extremal combinatorial sum problems raised by Manickam, Mikl\\\"os and Singhi in 1987 \\cite{ManMik87} and 1988 \\cite{ManSin88} we study the following two problems:\n  \\noindent$(P1)$ {\\it which are the values of $\\gamma(n,r)$ an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4761","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"}