{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:48:21Z","timestamp":1776721701568,"version":"3.51.2"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper we develop a theory of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t\">\n                        <mml:semantics>\n                          <mml:mi>t<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -cycle\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D minus upper Z\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>D<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>Z<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">D-Z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    representations for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper D\">\n                        <mml:semantics>\n                          <mml:mi>D<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">D<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -matrix consisting of the nontrivial invariants. Among these is a family of\n                    <italic>triangular<\/italic>\n                    forms, which, besides being canonical, have the defining property that their\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Z\">\n                        <mml:semantics>\n                          <mml:mi>Z<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">Z<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. Our main result is to define a unique canonical form for prime-power rules. This\n                    <italic>ultratriangular<\/italic>\n                    form is a triangular form, is easy to recognize, and may be derived in a straightforward manner.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00691-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"165-178","source":"Crossref","is-referenced-by-count":4,"title":["Triangular canonical forms for lattice rules of prime-power order"],"prefix":"10.1090","volume":"65","author":[{"given":"J.","family":"Lyness","sequence":"first","affiliation":[]},{"given":"S.","family":"Joe","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1007\/BF01387711","article-title":"Zur angen\u00e4herten Berechnung mehrfacher Integrale","volume":"66","author":"Hlawka, Edmund","year":"1962","journal-title":"Monatsh. 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