Pub Date : 2024-04-22DOI: 10.1017/s0004972724000261
SATOSHI KUMABE, HASAN SAAD
Lehmer [‘On certain character matrices’, Pacific J. Math.6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math.10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.
雷默['论某些特征矩阵'，《太平洋数学杂志》，6 (1956)，491-499，以及'幂特征矩阵'，《太平洋数学杂志》，10 (1960)，895-907]定义了四类由统一根构造的矩阵，它们的特征多项式和第 k 次幂都可以明确确定。我们研究了一类在有限域超几何函数的变换公式中自然出现的矩阵，它们的条目是合一根和零。我们确定了这些矩阵的特征多项式、特征值、特征向量和 k 次方。特征值是雅可比和积的自然族。
{"title":"ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS","authors":"SATOSHI KUMABE, HASAN SAAD","doi":"10.1017/s0004972724000261","DOIUrl":"https://doi.org/10.1017/s0004972724000261","url":null,"abstract":"Lehmer [‘On certain character matrices’, <jats:italic>Pacific J. Math.</jats:italic>6 (1956), 491–499, and ‘Power character matrices’, <jats:italic>Pacific J. Math.</jats:italic>10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the <jats:italic>k</jats:italic>th powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and <jats:italic>k</jats:italic>th powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"48 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ON GENERALISED LEGENDRE MATRICES INVOLVING ROOTS OF UNITY OVER FINITE FIELDS","authors":"NING-LIU WEI, YU-BO LI, HAI-LIANG WU","doi":"10.1017/s0004972724000303","DOIUrl":"https://doi.org/10.1017/s0004972724000303","url":null,"abstract":"Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, <jats:italic>Acta Arith.</jats:italic>115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline1.png\" /> <jats:tex-math> $a_1,ldots ,a_{(q-1)/2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be all the nonzero squares in the finite field <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline2.png\" /> <jats:tex-math> $mathbb {F}_q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> containing <jats:italic>q</jats:italic> elements with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline3.png\" /> <jats:tex-math> $2nmid q$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, we give the explicit value of the determinant <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline4.png\" /> <jats:tex-math> $D_{(q-1)/2}=det [(a_i+a_j)^{(q-3)/2}]_{1le i,jle (q-1)/2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline5.png\" /> <jats:tex-math> $q=p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a prime greater than <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline6.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_eqnu1.png\" /> <jats:tex-math> $$ begin{align*}bigg(frac{det D_{(p-1)/2}}{p}bigg)= begin{cases} 1 & mbox{if} pequiv1pmod4, (-1)^{(h(-p)+1)/2} & mbox{if} pequiv 3pmod4 text{and} p>3, end{cases}end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline7.png\" /> <jats:tex-math> $(frac {cdot }{p})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the Legendre symbol and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000303_inline8.png\" /> <jats:tex-math> $h(-p)$ </jats:tex-","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"39 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-18DOI: 10.1017/s0004972724000200
VASUDEVARAO ALLU, AMAL SHAJI
We obtain sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for starlike and convex functions.
我们得到了星形函数和凸函数对数反系数的第二汉克尔行列式的尖锐边界。
{"title":"SECOND HANKEL DETERMINANT FOR LOGARITHMIC INVERSE COEFFICIENTS OF CONVEX AND STARLIKE FUNCTIONS","authors":"VASUDEVARAO ALLU, AMAL SHAJI","doi":"10.1017/s0004972724000200","DOIUrl":"https://doi.org/10.1017/s0004972724000200","url":null,"abstract":"<p>We obtain sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for starlike and convex functions.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140617598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-18DOI: 10.1017/s000497272400025x
XIAO CHEN, LULU FANG, JUNJIE LI, LEI SHANG, XIN ZENG
Let $[a_1(x),a_2(x),a_3(x),ldots ]$ be the continued fraction expansion of an irrational number $xin [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all $xin [0,1)$, $$ begin{align*} liminf_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n} = 0quad text{and}quad limsup_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n}=1. end{align*} $$
We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.
让 $[a_1(x),a_2(x),a_3(x),ldots ]$ 是无理数 $xin [0,1)$ 的连续分数展开。我们关注的是 x 的连续部分商乘积的渐近行为。我们证明，对于 Lebesgue 几乎所有的 $xin [0,1)$, $$ （begin{align*}）。liminf_{ntoinfty}{log (a_n(x)a_{n+1}(x))}{log n} = 0（四边形）{text{and}（四边形） limsup_{n toinfty}frac{log (a_n(x)a_{n+1}(x))}{log n}=1.end{align*}$$We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.
{"title":"ASYMPTOTIC BEHAVIOUR FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS","authors":"XIAO CHEN, LULU FANG, JUNJIE LI, LEI SHANG, XIN ZENG","doi":"10.1017/s000497272400025x","DOIUrl":"https://doi.org/10.1017/s000497272400025x","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417094313995-0120:S000497272400025X:S000497272400025X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$[a_1(x),a_2(x),a_3(x),ldots ]$</span></span></img></span></span> be the continued fraction expansion of an irrational number <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417094313995-0120:S000497272400025X:S000497272400025X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$xin [0,1)$</span></span></img></span></span>. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of <span>x</span>. We prove that, for Lebesgue almost all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417094313995-0120:S000497272400025X:S000497272400025X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$xin [0,1)$</span></span></img></span></span>, <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417094313995-0120:S000497272400025X:S000497272400025X_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} liminf_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n} = 0quad text{and}quad limsup_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n}=1. end{align*} $$</span></span></img></span></p><p>We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-18DOI: 10.1017/s0004972724000236
XIAOXIA WANG, WENJIE YU
Liu [‘Supercongruences for truncated Appell series’, Colloq. Math.158(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$’, Integral Transforms Spec. Funct.31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series $F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.
Liu ['Supercongruences for truncated Appell series', Colloq.Math.158(2) (2019), 255-263] and Lin and Liu ['Congruences for the truncated Appell series $F_3$ and $F_4$', Integral Transforms Spec.Funct.31(1) (2020), 10-17] 确认了截断阿贝尔数列的四个超级共轭。受他们工作的启发，我们给出了截断阿贝尔数列 $F_{1}$ 的新超共假，并通过建立其 q-analogues ，给出了该超共假的两个广义。
{"title":"NEW CONGRUENCES FOR THE TRUNCATED APPELL SERIES","authors":"XIAOXIA WANG, WENJIE YU","doi":"10.1017/s0004972724000236","DOIUrl":"https://doi.org/10.1017/s0004972724000236","url":null,"abstract":"<p>Liu [‘Supercongruences for truncated Appell series’, <span>Colloq. Math.</span> <span>158</span>(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417092342642-0507:S0004972724000236:S0004972724000236_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$F_3$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417092342642-0507:S0004972724000236:S0004972724000236_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$F_4$</span></span></img></span></span>’, <span>Integral Transforms Spec. Funct.</span> <span>31</span>(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240417092342642-0507:S0004972724000236:S0004972724000236_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$F_{1}$</span></span></img></span></span>, together with two generalisations of this supercongruence, by establishing its <span>q</span>-analogues.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"440 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"APPROXIMATION OF IRRATIONAL NUMBERS BY PAIRS OF INTEGERS FROM A LARGE SET","authors":"ARTŪRAS DUBICKAS","doi":"10.1017/s0004972724000194","DOIUrl":"https://doi.org/10.1017/s0004972724000194","url":null,"abstract":"We show that there is a set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline1.png\" /> <jats:tex-math> $S subseteq {mathbb N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with lower density arbitrarily close to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline2.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that, for each sufficiently large real number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline3.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the inequality <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline4.png\" /> <jats:tex-math> $|malpha -n| geq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> holds for every pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline5.png\" /> <jats:tex-math> $(m,n) in S^2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. On the other hand, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline6.png\" /> <jats:tex-math> $S subseteq {mathbb N}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> has density <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline7.png\" /> <jats:tex-math> $1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then, for each irrational <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline8.png\" /> <jats:tex-math> $alpha>0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and any positive <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline9.png\" /> <jats:tex-math> $varepsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there exist <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000194_inline10.png\" /> <jats:tex-math> $m,n in S$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for which <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140594963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-01DOI: 10.1017/s0004972724000182
ANUJ JAKHAR
<p>Let <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline1.png"><span data-mathjax-type="texmath"><span>${mathbb {Z}}_{K}$</span></span></img></span></span> denote the ring of algebraic integers of an algebraic number field <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline2.png"><span data-mathjax-type="texmath"><span>$K = {mathbb Q}(theta )$</span></span></img></span></span>, where <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline3.png"><span data-mathjax-type="texmath"><span>$theta $</span></span></img></span></span> is a root of a monic irreducible polynomial <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline4.png"><span data-mathjax-type="texmath"><span>$f(x) = x^n + a(bx+c)^m in {mathbb {Z}}[x]$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline5.png"><span data-mathjax-type="texmath"><span>$1leq m<n$</span></span></img></span></span>. We say <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline6.png"><span data-mathjax-type="texmath"><span>$f(x)$</span></span></img></span></span> is monogenic if <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline7.png"><span data-mathjax-type="texmath"><span>${1, theta , ldots , theta ^{n-1}}$</span></span></img></span></span> is a basis for <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline8.png"><span data-mathjax-type="texmath"><span>${mathbb {Z}}_K$</span></span></img></span></span>. We give necessary and sufficient conditions involving only <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline9.png"><span data-mathjax-type="texmath"><span>$a, b, c, m, n$</span></span></img></span></span> for <span><span><img data-mimesubtype="png" data-type="" src="https://static.
{"title":"CHARACTERISATION OF PRIMES DIVIDING THE INDEX OF A CLASS OF POLYNOMIALS AND ITS APPLICATIONS","authors":"ANUJ JAKHAR","doi":"10.1017/s0004972724000182","DOIUrl":"https://doi.org/10.1017/s0004972724000182","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb {Z}}_{K}$</span></span></img></span></span> denote the ring of algebraic integers of an algebraic number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$K = {mathbb Q}(theta )$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$theta $</span></span></img></span></span> is a root of a monic irreducible polynomial <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$f(x) = x^n + a(bx+c)^m in {mathbb {Z}}[x]$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$1leq m<n$</span></span></img></span></span>. We say <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$f(x)$</span></span></img></span></span> is monogenic if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${1, theta , ldots , theta ^{n-1}}$</span></span></img></span></span> is a basis for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb {Z}}_K$</span></span></img></span></span>. We give necessary and sufficient conditions involving only <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328105942060-0650:S0004972724000182:S0004972724000182_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$a, b, c, m, n$</span></span></img></span></span> for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"301 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-25DOI: 10.1017/s0004972724000169
MARTIN BUNDER, BRUCE BATES, STEPHEN ARNOLD
The sequence $a( 1) ,a( 2) ,a( 3) ,ldots, $ labelled A088431 in the Online Encyclopedia of Integer Sequences, is defined by: $a( n) $ is half of the $( n+1) $ th component, that is, the $( n+2) $ th term, of the continued fraction expansion of $$ begin{align*} sum_{k=0}^{infty }frac{1}{2^{2^{k}}}. end{align*} $$ Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.
{"title":"THE SUMMED PAPERFOLDING SEQUENCE","authors":"MARTIN BUNDER, BRUCE BATES, STEPHEN ARNOLD","doi":"10.1017/s0004972724000169","DOIUrl":"https://doi.org/10.1017/s0004972724000169","url":null,"abstract":"The sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000169_inline1.png\" /> <jats:tex-math> $a( 1) ,a( 2) ,a( 3) ,ldots, $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> labelled A088431 in the <jats:italic>Online Encyclopedia of Integer Sequences</jats:italic>, is defined by: <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000169_inline2.png\" /> <jats:tex-math> $a( n) $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is half of the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000169_inline3.png\" /> <jats:tex-math> $( n+1) $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>th component, that is, the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000169_inline4.png\" /> <jats:tex-math> $( n+2) $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>th term, of the continued fraction expansion of <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000169_eqnu1.png\" /> <jats:tex-math> $$ begin{align*} sum_{k=0}^{infty }frac{1}{2^{2^{k}}}. end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula> Dimitri Hendriks has suggested that it is the sequence of run lengths of the paperfolding sequence, A014577. This paper proves several results for this summed paperfolding sequence and confirms Hendriks’s conjecture.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the fractional integral operator $I_{alpha }$ , $0<alpha <n$ , and the fractional maximal operator $M_{alpha }$ , $0le alpha <n$ , are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_alpha $ are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, Georgian Math. J.18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{alpha }$ are essentially new.
{"title":"CHOQUET INTEGRALS, HAUSDORFF CONTENT AND FRACTIONAL OPERATORS","authors":"NAOYA HATANO, RYOTA KAWASUMI, HIROKI SAITO, HITOSHI TANAKA","doi":"10.1017/s000497272400011x","DOIUrl":"https://doi.org/10.1017/s000497272400011x","url":null,"abstract":"We show that the fractional integral operator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline1.png\" /> <jats:tex-math> $I_{alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline2.png\" /> <jats:tex-math> $0<alpha <n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and the fractional maximal operator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline3.png\" /> <jats:tex-math> $M_{alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline4.png\" /> <jats:tex-math> $0le alpha <n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline5.png\" /> <jats:tex-math> $M_alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, <jats:italic>Georgian Math. J.</jats:italic>18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400011X_inline6.png\" /> <jats:tex-math> $I_{alpha }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are essentially new.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"101 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140170704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-15DOI: 10.1017/s0004972724000157
ZHIGANG WANG, A-MING LIU, VASILY G. SAFONOV, ALEXANDER N. SKIBA
Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$ . Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$ , that is, $G^{mathfrak {N}}cap A^{sG}= G^{mathfrak {N}}cap A_{sG}$ for every subnormal subgroup A of G.
设 G 是一个有限群。如果 G 的子群 A 与 G 的每个 Sylow 子群 P 都发生包络，即 $AP=PA$ ，则称 G 的子群 A 在 G 中是 S 可包络的。设 $A_{sG}$ 是由包含在 A 中的所有 G 的 S-permutable 子群生成的 A 子群，而 $A^{sG}$ 是包含 A 的所有 G 的 S-permutable 子群的交集。我们证明，如果 G 是可解群，那么当且仅当 G 的零能残差 $G^{mathfrak {N}}$ 避免了一对 $(A^{s G}、A_{sG})$ ，也就是说，对于 G 的每个子正常子群 A，$G^{mathfrak {N}cap A^{sG}= G^{mathfrak {N}cap A_{sG}$ 。
{"title":"A CHARACTERISATION OF SOLUBLE -GROUPS","authors":"ZHIGANG WANG, A-MING LIU, VASILY G. SAFONOV, ALEXANDER N. SKIBA","doi":"10.1017/s0004972724000157","DOIUrl":"https://doi.org/10.1017/s0004972724000157","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a finite group. A subgroup <jats:italic>A</jats:italic> of <jats:italic>G</jats:italic> is said to be <jats:italic>S-permutable</jats:italic> in <jats:italic>G</jats:italic> if <jats:italic>A</jats:italic> permutes with every Sylow subgroup <jats:italic>P</jats:italic> of <jats:italic>G</jats:italic>, that is, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline2.png\" /> <jats:tex-math> $AP=PA$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline3.png\" /> <jats:tex-math> $A_{sG}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the subgroup of <jats:italic>A</jats:italic> generated by all <jats:italic>S</jats:italic>-permutable subgroups of <jats:italic>G</jats:italic> contained in <jats:italic>A</jats:italic> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline4.png\" /> <jats:tex-math> $A^{sG}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be the intersection of all <jats:italic>S</jats:italic>-permutable subgroups of <jats:italic>G</jats:italic> containing <jats:italic>A</jats:italic>. We prove that if <jats:italic>G</jats:italic> is a soluble group, then <jats:italic>S</jats:italic>-permutability is a transitive relation in <jats:italic>G</jats:italic> if and only if the nilpotent residual <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline5.png\" /> <jats:tex-math> $G^{mathfrak {N}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of <jats:italic>G</jats:italic> avoids the pair <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline6.png\" /> <jats:tex-math> $(A^{s G}, A_{sG})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, that is, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000157_inline7.png\" /> <jats:tex-math> $G^{mathfrak {N}}cap A^{sG}= G^{mathfrak {N}}cap A_{sG}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for every subnormal subgroup <jats:italic>A</jats:italic> of <jats:italic>G</jats:italic>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"28 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$[a_1(x),a_2(x),a_3(x),ldots ]$ be the continued fraction expansion of an irrational number 
$xin [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all 
$xin [0,1)$, 
$$ begin{align*} liminf_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n} = 0quad text{and}quad limsup_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n}=1. end{align*} $$
$F_3$ and 
$F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series 
$F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.
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