{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,3]],"date-time":"2024-04-03T19:48:04Z","timestamp":1712173684238},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M equals 2 Superscript w Baseline comma\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>M<\/mml:mi>\n      <mml:mo>=<\/mml:mo>\n      <mml:msup>\n        <mml:mn>2<\/mml:mn>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>w<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msup>\n      <mml:mo>,<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">M = 2^{w},<\/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=\"upper G Subscript upper M Baseline equals StartSet 1 comma 3 comma period period period comma upper M minus 1 EndSet period\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>G<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>M<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo>,<\/mml:mo>\n      <mml:mn>3<\/mml:mn>\n      <mml:mo>,<\/mml:mo>\n      <mml:mo>.<\/mml:mo>\n      <mml:mo>.<\/mml:mo>\n      <mml:mo>.<\/mml:mo>\n      <mml:mo>,<\/mml:mo>\n      <mml:mi>M<\/mml:mi>\n      <mml:mo>\u2212<!-- \u2212 --><\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n      <mml:mo>.<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">G_{M} = \\{1,3,...,M-1 \\}.<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> A sequence <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace y Subscript n Baseline right-brace comma y Subscript n Baseline element-of upper G Subscript upper M Baseline comma\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:msub>\n        <mml:mi>y<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n      <mml:mo>,<\/mml:mo>\n      <mml:msub>\n        <mml:mi>y<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>\u2208<!-- \u2208 --><\/mml:mo>\n      <mml:msub>\n        <mml:mi>G<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>M<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>,<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\{y_{n} \\}, y_{n} \\in G_{M},<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is obtained by the formula <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y Subscript n plus 1 Baseline identical-to a y overbar Subscript n Baseline plus b plus c y Subscript n Baseline normal m normal o normal d upper M period\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msub>\n        <mml:mi>y<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n          <mml:mo>+<\/mml:mo>\n          <mml:mn>1<\/mml:mn>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>\u2261<!-- \u2261 --><\/mml:mo>\n      <mml:mi>a<\/mml:mi>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:msub>\n          <mml:mover>\n            <mml:mi>y<\/mml:mi>\n            <mml:mo accent=\"false\">\u00af<!-- \u00af --><\/mml:mo>\n          <\/mml:mover>\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mi>n<\/mml:mi>\n          <\/mml:mrow>\n        <\/mml:msub>\n      <\/mml:mrow>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>b<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>c<\/mml:mi>\n      <mml:msub>\n        <mml:mi>y<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mspace width=\"thickmathspace\" \/>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"normal\">m<\/mml:mi>\n        <mml:mi mathvariant=\"normal\">o<\/mml:mi>\n        <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mspace width=\"thickmathspace\" \/>\n      <mml:mi>M<\/mml:mi>\n      <mml:mo>.<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">y_{n+1} \\equiv a{\\overline {y}_{n}} + b + cy_{n} \\; \\mathrm {mod} \\; M.<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> The sequence <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-brace x Subscript n Baseline right-brace comma x Subscript n Baseline equals y Subscript n Baseline slash upper M comma\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n      <mml:msub>\n        <mml:mi>x<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n      <mml:mo>,<\/mml:mo>\n      <mml:msub>\n        <mml:mi>x<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mo>=<\/mml:mo>\n      <mml:msub>\n        <mml:mi>y<\/mml:mi>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi>n<\/mml:mi>\n        <\/mml:mrow>\n      <\/mml:msub>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mi>M<\/mml:mi>\n      <mml:mo>,<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\{x_{n} \\}, x_{n}=y_{n}\/M,<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a sequence of pseudorandom numbers of the maximal period length <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M slash 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>M<\/mml:mi>\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:annotation encoding=\"application\/x-tex\">M\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> if and only if <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a plus c identical-to 1\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>a<\/mml:mi>\n      <mml:mo>+<\/mml:mo>\n      <mml:mi>c<\/mml:mi>\n      <mml:mo>\u2261<!-- \u2261 --><\/mml:mo>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">a+c \\equiv 1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> (mod 4), <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b identical-to 2\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>b<\/mml:mi>\n      <mml:mo>\u2261<!-- \u2261 --><\/mml:mo>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">b \\equiv 2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> (mod 4). In this note, the uniformity is investigated by the 2-dimensional serial test for the sequence. We follow closely the method of papers by Eichenauer-Herrmann and Niederreiter.<\/p>","DOI":"10.1090\/s0025-5718-96-00712-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:44Z","timestamp":1027721684000},"page":"761-769","source":"Crossref","is-referenced-by-count":3,"title":["The serial test for a nonlinear pseudorandom number generator"],"prefix":"10.1090","volume":"65","author":[{"given":"Takashi","family":"Kato","sequence":"first","affiliation":[]},{"given":"Li-Ming","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Niro","family":"Yanagihara","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"issue":"193","key":"1","doi-asserted-by":"publisher","first-page":"297","DOI":"10.2307\/2008543","article-title":"Inversive congruential pseudorandom numbers avoid the planes","volume":"56","author":"Eichenauer-Herrmann, J\u00fcrgen","year":"1991","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"201","key":"2","doi-asserted-by":"publisher","first-page":"375","DOI":"10.2307\/2153174","article-title":"Statistical independence of a new class of inversive congruential pseudorandom numbers","volume":"60","author":"Eichenauer-Herrmann, J\u00fcrgen","year":"1993","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"207","key":"3","doi-asserted-by":"publisher","first-page":"293","DOI":"10.2307\/2153575","article-title":"On generalized inversive congruential pseudorandom numbers","volume":"63","author":"Eichenauer-Herrmann, J\u00fcrgen","year":"1994","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/0377-0427(90)90338-Z","article-title":"On the lattice structure of a nonlinear generator with modulus 2^{\ud835\udefc}","volume":"31","author":"Eichenauer-Herrmann, J.","year":"1990","journal-title":"J. Comput. Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"184","key":"5","doi-asserted-by":"publisher","first-page":"757","DOI":"10.2307\/2008776","article-title":"A nonlinear congruential pseudorandom number generator with power of two modulus","volume":"51","author":"Eichenauer, J\u00fcrgen","year":"1988","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"198","key":"6","doi-asserted-by":"publisher","first-page":"775","DOI":"10.2307\/2153216","article-title":"Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus","volume":"58","author":"Eichenauer-Herrmann, J\u00fcrgen","year":"1992","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"T. Kato, L.-M. Wu, and N. Yanagihara, On a nonlinear congruential pseudorandom number generator, Math. Comp. 65 (1996) (to appear).","DOI":"10.1090\/S0025-5718-96-00694-1"},{"key":"8","series-title":"Addison-Wesley Series in Computer Science and Information Processing","isbn-type":"print","volume-title":"The art of computer programming. Vol. 2","author":"Knuth, Donald E.","year":"1981","ISBN":"http:\/\/id.crossref.org\/isbn\/0201038226","edition":"2"},{"issue":"185","key":"9","doi-asserted-by":"publisher","first-page":"135","DOI":"10.2307\/2008658","article-title":"The serial test for congruential pseudorandom numbers generated by inversions","volume":"52","author":"Niederreiter, Harald","year":"1989","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-4","key":"10","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/BF02204856","article-title":"Recent trends in random number and random vector generation","volume":"31","author":"Niederreiter, Harald","year":"1991","journal-title":"Ann. Oper. Res.","ISSN":"http:\/\/id.crossref.org\/issn\/0254-5330","issn-type":"print"},{"key":"11","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random number generation and quasi-Monte Carlo methods","volume":"63","author":"Niederreiter, Harald","year":"1992","ISBN":"http:\/\/id.crossref.org\/isbn\/0898712955"},{"key":"12","doi-asserted-by":"crossref","unstructured":"H. Sali\u00e9, \u00dcber die Kloostermanschen Summen S(u,v;q), Math. Z. 34 (1932), 91\u2013109.","DOI":"10.1007\/BF01180579"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00712-0\/S0025-5718-96-00712-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00712-0\/S0025-5718-96-00712-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T23:01:20Z","timestamp":1635894080000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-214\/S0025-5718-96-00712-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":12,"journal-issue":{"issue":"214","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["S0025-5718-96-00712-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00712-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996]]}}}