{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T13:51:57Z","timestamp":1761918717548},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,3,3]],"date-time":"2001-03-03T00:00:00Z","timestamp":983577600000},"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":"<p>We construct symmetric cubature formulae of degrees in the 13-39 range for the surface measure on the unit sphere. We exploit a recently published correspondence between cubature formulae on the sphere and on the triangle. Specifically, a fully symmetric cubature formula for the surface measure on the unit sphere corresponds to a symmetric cubature formula for the triangle with weight function <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis u 1 u 2 u 3 right-parenthesis Superscript negative 1 slash 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:msub>\n        <mml:mi>u<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>1<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:msub>\n        <mml:mi>u<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>2<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:msub>\n        <mml:mi>u<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mn>3<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:msup>\n        <mml:mo stretchy=\"false\">)<\/mml:mo>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mo>\/<\/mml:mo>\n          <\/mml:mrow>\n          <mml:mn>2<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msup>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">(u_{1}u_{2}u_{3})^{-1\/2}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u 1\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mn>1<\/mml:mn>\n      <\/mml:mrow>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_{1}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u 2\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mn>2<\/mml:mn>\n      <\/mml:mrow>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_{2}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, and <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"u 3\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>u<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mn>3<\/mml:mn>\n      <\/mml:mrow>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">u_{3}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> are homogeneous coordinates.<\/p>","DOI":"10.1090\/s0025-5718-00-01198-4","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T21:02:26Z","timestamp":1121115746000},"page":"269-279","source":"Crossref","is-referenced-by-count":54,"title":["Constructing fully symmetric cubature formulae for the sphere"],"prefix":"10.1090","volume":"70","author":[{"given":"Sangwoo","family":"Heo","sequence":"first","affiliation":[]},{"given":"Yuan","family":"Xu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,3,3]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0020-0190(86)90039-6","article-title":"A note on Presburger arithmetic with array segments, permutation and equality","volume":"22","author":"Rosier, Louis E.","year":"1986","journal-title":"Inform. 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Xu, Constructing cubature formulae for spheres and balls, J. Comp. Appl. Math. 12 (1999), 95\u2013119. .","DOI":"10.1016\/S0377-0427(99)00216-2"},{"key":"6","unstructured":"S. Heo and Y. Xu, Constructing cubature formulae for spheres and triangles, Technical Report, University of Oregon, 1998."},{"issue":"1-2","key":"7","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/0377-0427(87)90044-6","article-title":"Cubature formulas for the surface of the sphere","volume":"17","author":"Keast, Patrick","year":"1987","journal-title":"J. Comput. Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/0045-7825(86)90059-9","article-title":"Moderate-degree tetrahedral quadrature formulas","volume":"55","author":"Keast, Patrick","year":"1986","journal-title":"Comput. Methods Appl. Mech. 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