{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:00:20Z","timestamp":1753884020026,"version":"3.41.2"},"reference-count":14,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,5,21]],"date-time":"2012-05-21T00:00:00Z","timestamp":1337558400000},"content-version":"vor","delay-in-days":141,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Let <jats:italic>n<\/jats:italic> and <jats:italic>r<\/jats:italic> be two integers such that 0 &lt; <jats:italic>r<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic>; we denote by <jats:italic>\u03b3<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>)[<jats:italic>\u03b7<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>)] the minimum [maximum] number of the nonnegative partial sums of a sum , where <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> are <jats:italic>n<\/jats:italic> real numbers arbitrarily chosen in such a way that <jats:italic>r<\/jats:italic> of them are nonnegative and the remaining <jats:italic>n<\/jats:italic> \u2212 <jats:italic>r<\/jats:italic> are negative. We study the following two problems: (<jats:italic>P<\/jats:italic>1) <jats:italic>which are the values of<\/jats:italic> <jats:italic>\u03b3<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>) <jats:italic>and<\/jats:italic> <jats:italic>\u03b7<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>) <jats:italic>for each<\/jats:italic> <jats:italic>n<\/jats:italic> <jats:italic>and<\/jats:italic> <jats:italic>r<\/jats:italic>, 0 &lt; <jats:italic>r<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic><jats:italic>?<\/jats:italic> (<jats:italic>P<\/jats:italic>2) <jats:italic>if<\/jats:italic> <jats:italic>q<\/jats:italic> <jats:italic>is an integer such that<\/jats:italic> <jats:italic>\u03b3<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>) \u2264 <jats:italic>q<\/jats:italic> \u2264 <jats:italic>\u03b7<\/jats:italic>(<jats:italic>n<\/jats:italic>, <jats:italic>r<\/jats:italic>) <jats:italic>, can we find<\/jats:italic> <jats:italic>n<\/jats:italic> <jats:italic>real numbers<\/jats:italic> <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, <jats:italic>such that<\/jats:italic><jats:italic>r<\/jats:italic> <jats:italic>of them are nonnegative and the remaining<\/jats:italic> <jats:italic>n<\/jats:italic> \u2212 <jats:italic>r<\/jats:italic> <jats:italic>are negative with<\/jats:italic> , <jats:italic>such that the number of the nonnegative sums formed from these numbers is exactly<\/jats:italic> <jats:italic>q<\/jats:italic><jats:italic>?<\/jats:italic><\/jats:p>","DOI":"10.1155\/2012\/847958","type":"journal-article","created":{"date-parts":[[2012,5,21]],"date-time":"2012-05-21T21:06:23Z","timestamp":1337634383000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum"],"prefix":"10.1155","volume":"2012","author":[{"given":"G.","family":"Chiaselotti","sequence":"first","affiliation":[]},{"given":"G.","family":"Marino","sequence":"additional","affiliation":[]},{"given":"C.","family":"Nardi","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,5,21]]},"reference":[{"key":"e_1_2_5_1_2","first-page":"385","article-title":"On the number of nonnegative partial sums of a nonnegative sum","volume":"52","author":"Manickam N.","year":"1987","journal-title":"Colloquia Mathematica Societatis Janos Bolyai"},{"key":"e_1_2_5_2_2","unstructured":"BisiC.andChiaselottiG. A class of lattices and boolean functions related to a Manickam-Mikl\u00f6s-Singhi Conjecture. In press."},{"key":"e_1_2_5_3_2","unstructured":"BisiC.andChiaselottiG. Extension results for boolean maps and a class of systems of linear inequalities http:\/\/arxiv.org\/abs\/1012.5486."},{"key":"e_1_2_5_4_2","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.2000.0470"},{"key":"e_1_2_5_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00192-4"},{"key":"e_1_2_5_6_2","unstructured":"BhattacharyaA. Some problems in combinatorics Ph.D. thesis 2004 Indian Institute of Technology Bombay India."},{"key":"e_1_2_5_7_2","first-page":"61","article-title":"The first distribution invariant of the Johnson-scheme","volume":"11","author":"Bier T.","year":"1987","journal-title":"Southeast Asian Bulletin of Mathematics"},{"key":"e_1_2_5_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2007.03.002"},{"key":"e_1_2_5_9_2","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.2002.0587"},{"key":"e_1_2_5_10_2","first-page":"121","article-title":"Distribution invariants of association schemes","volume":"61","author":"Manickam N.","year":"1988","journal-title":"Congressus Numerantium"},{"key":"e_1_2_5_11_2","first-page":"465","article-title":"First distributed sets in the association scheme of bilinear forms","volume":"60","author":"Manickam N.","year":"1992","journal-title":"Colloquia Mathematica Societatis Janos Bolyai"},{"key":"e_1_2_5_12_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809088"},{"key":"e_1_2_5_13_2","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"e_1_2_5_14_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511805967"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/847958.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2012\/847958.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2012\/847958","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T06:52:45Z","timestamp":1718088765000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2012\/847958"}},"subtitle":[],"editor":[{"given":"Yonghong","family":"Yao","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1155\/2012\/847958"],"URL":"https:\/\/doi.org\/10.1155\/2012\/847958","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2012,1]]},"assertion":[{"value":"2012-02-06","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-03-02","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-05-21","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"847958"}}