{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T15:52:36Z","timestamp":1746805956413,"version":"3.40.5"},"reference-count":0,"publisher":"Informa UK Limited","issue":"2","license":[{"start":{"date-parts":[[1998,1,1]],"date-time":"1998-01-01T00:00:00Z","timestamp":883612800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics and Decision Sciences"],"published-print":{"date-parts":[[1998,1,1]]},"abstract":"<jats:p>The equation <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$x^a  + x^b  + x = x^c  + x^d  + 1$\"><mml:msup><mml:mi>x<\/mml:mi><mml:mi>a<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>b<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>=<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>c<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math> considered in this paper is a particular\nequiponderate equation. The number and location of the roots (w.r.t. x = 1) of this equation\nare determined in case <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mrow><mml:mo>(<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>b<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>c<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>d<\/mml:mi><\/mml:mrow><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mo>\u2208<\/mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>]<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mo>[<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mn>4<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math>. Based on these results, it is shown that any weight\nquadruplet, a basic tool in fuzzy preference modelling, admits an interesting expression in terms\nof Frank t-norms with reciprocal parameters.<\/jats:p>","DOI":"10.1155\/s117391269800008x","type":"journal-article","created":{"date-parts":[[2007,3,8]],"date-time":"2007-03-08T07:44:28Z","timestamp":1173339868000},"page":"147-158","source":"Crossref","is-referenced-by-count":1,"title":["On the equiponderate equation <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"$x^a  + x^b  + x = x^c  + x^d  + 1$\"><mml:msup><mml:mi>x<\/mml:mi><mml:mi>a<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>b<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi>x<\/mml:mi><mml:mo>=<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>c<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msup><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math> and a representation of weight quadruplets"],"prefix":"10.1080","volume":"2","author":[{"given":"Bernard","family":"de Baets","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, University of Gent, Krijgslaan 281 (S9), Gent B-9000, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hans","family":"de Meyer","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Computer Science, University of Gent, Krijgslaan 281 (S9), Gent B-9000, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"301","container-title":["Journal of Applied Mathematics and Decision Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/archive\/1998\/697927.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/archive\/1998\/697927.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T14:22:55Z","timestamp":1723126975000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ads\/1998\/697927\/abs\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1,1]]},"references-count":0,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1998,1,1]]}},"alternative-id":["697927"],"URL":"https:\/\/doi.org\/10.1155\/s117391269800008x","relation":{},"ISSN":["1173-9126"],"issn-type":[{"type":"print","value":"1173-9126"}],"subject":[],"published":{"date-parts":[[1998,1,1]]}}}