{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:43:22Z","timestamp":1776797002781,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,8,13]],"date-time":"2014-08-13T00:00:00Z","timestamp":1407888000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Positive cubature rules of degree\n \n \n \n 4<\/mml:mn>\n 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n 5<\/mml:mn>\n 5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on the\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional simplex are constructed for a range of dimensions\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and used to construct cubature rules of index\n \n \n \n 8<\/mml:mn>\n 8<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n or degree\n \n \n \n 9<\/mml:mn>\n 9<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on the unit sphere. The latter ones lead to explicit isometric embedding among the classical Banach spaces. Among other things, our results include several explicit representations of\n \n \n \n \n (<\/mml:mo>\n \n x<\/mml:mi>\n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:msubsup>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n x<\/mml:mi>\n d<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n \n )<\/mml:mo>\n t<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n (x_1^2+ \\cdots + x_d^2)^t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in terms of linear forms of degree\n \n \n \n \n 2<\/mml:mn>\n t<\/mml:mi>\n <\/mml:mrow>\n 2t<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with rational coefficients for\n \n \n \n \n t<\/mml:mi>\n =<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n t = 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n 5<\/mml:mn>\n 5<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2013-02762-7","type":"journal-article","created":{"date-parts":[[2013,8,13]],"date-time":"2013-08-13T12:54:37Z","timestamp":1376398477000},"page":"1251-1277","source":"Crossref","is-referenced-by-count":9,"title":["On positive cubature rules on the simplex and isometric embeddings"],"prefix":"10.1090","volume":"83","author":[{"given":"Masanori","family":"Sawa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,8,13]]},"reference":[{"key":"1","isbn-type":"print","first-page":"87","article-title":"Modular forms, lattices and spherical designs","author":"Bachoc, Christine","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/2940264023"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s10801-006-0042-3","article-title":"Orbits of the hyperoctahedral group as Euclidean designs","volume":"25","author":"Bajnok, B\u00e9la","year":"2007","journal-title":"J. Algebraic Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/0925-9899","issn-type":"print"},{"issue":"12","key":"3","doi-asserted-by":"publisher","first-page":"2990","DOI":"10.1016\/j.apnum.2009.07.005","article-title":"A fourth degree integration formula for the \ud835\udc5b-dimensional simplex","volume":"59","author":"Dan, Wei","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"4","unstructured":"L. E. Dickson, History of the Theory of Numbers, Vol. II. Carnegie Institution of Washington, 1923, pp. 717-725."},{"key":"5","series-title":"Encyclopedia of Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511565717","volume-title":"Orthogonal polynomials of several variables","volume":"81","author":"Dunkl, Charles F.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0521800439"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/BF01450405","article-title":"Beweis f\u00fcr die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl \ud835\udc5b^{\ud835\udc61\ud835\udc52\ud835\udc5f} Potenzen (Waringsches Problem)","volume":"67","author":"Hilbert, David","year":"1909","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"7","unstructured":"G. B. Khosrovshahi, R. Laue, \ud835\udc61-Designs with \ud835\udc61\u22653, in: C. J. Colbourn, J. H. Dinitz (eds.), Handbook of Combinatorial Designs (2nd ed.). CRC Press, Boca Raton, USA, 2007, pp. 79-101."},{"key":"8","isbn-type":"print","first-page":"203","article-title":"Cubature formulae, polytopes, and spherical designs","author":"Goethals, J.-M.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/0387905871"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1137\/0715019","article-title":"Invariant integration formulas for the \ud835\udc5b-simplex by combinatorial methods","volume":"15","author":"Grundmann, Axel","year":"1978","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1090\/S1061-0022-07-00946-6","article-title":"Construction of spherical cubature formulas using lattices","volume":"18","author":"de la Harpe, P.","year":"2006","journal-title":"Algebra i Analiz","ISSN":"https:\/\/id.crossref.org\/issn\/0234-0852","issn-type":"print"},{"key":"11","series-title":"Springer Series in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1478-6","volume-title":"Orthogonal arrays","author":"Hedayat, A. S.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0387987665"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"626","DOI":"10.1137\/S003614299935543X","article-title":"Invariant cubature formulae for spheres and balls by combinatorial methods","volume":"38","author":"Heo, Sangwoo","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/BF01263664","article-title":"Isometric embeddings between classical Banach spaces, cubature formulas, and spherical designs","volume":"47","author":"Lyubich, Yuri I.","year":"1993","journal-title":"Geom. Dedicata","ISSN":"https:\/\/id.crossref.org\/issn\/0046-5755","issn-type":"print"},{"issue":"463","key":"14","doi-asserted-by":"publisher","first-page":"viii+155","DOI":"10.1090\/memo\/0463","article-title":"Sums of even powers of real linear forms","volume":"96","author":"Reznick, Bruce","year":"1992","journal-title":"Mem. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-9266","issn-type":"print"},{"key":"15","unstructured":"G. N. Salihov, Cubature formulas for a hypersphere that are invariant with respect to the group of the regular 600-face, Soviet Math. Dokl., 16 (1975), 1046-1050."},{"key":"16","unstructured":"O. Shatalov, Isometric embeddings \ud835\udc59\u2082^{\ud835\udc5a}\u27f6\ud835\udc59_{\ud835\udc5d}\u207f and cubature formulas over classical fields, Doctor Thesis, Technion-Israel Institute of Technology, Haifa, Israel, 2001."},{"key":"17","first-page":"310","article-title":"Cubature formulas on the sphere which are invariant under transformations of finite rotation groups","volume":"146","author":"Sobolev, S. L.","year":"1962","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"18","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"Approximate calculation of multiple integrals","author":"Stroud, A. H.","year":"1971"},{"issue":"134","key":"19","doi-asserted-by":"publisher","first-page":"291","DOI":"10.2307\/2005970","article-title":"Some fourth degree integration formulas for simplexes","volume":"30","author":"Stroud, A. H.","year":"1976","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1137\/S0036142902407952","article-title":"Asymmetric cubature formulae with few points in high dimension for symmetric measures","volume":"42","author":"Victoir, Nicolas","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"169","DOI":"10.4310\/MAA.1998.v5.n2.a5","article-title":"Orthogonal polynomials and cubature formulae on spheres and on simplices","volume":"5","author":"Xu, Yuan","year":"1998","journal-title":"Methods Appl. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1073-2772","issn-type":"print"},{"issue":"1","key":"22","first-page":"89","article-title":"A set of symmetric quadrature rules on triangles and tetrahedra","volume":"27","author":"Zhang, Linbo","year":"2009","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02762-7\/S0025-5718-2013-02762-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02762-7\/S0025-5718-2013-02762-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:53:43Z","timestamp":1776794023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-287\/S0025-5718-2013-02762-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,8,13]]},"references-count":22,"journal-issue":{"issue":"287","published-print":{"date-parts":[[2014,5]]}},"alternative-id":["S0025-5718-2013-02762-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02762-7","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2013,8,13]]}}}