Pub Date : 2024-05-14DOI: 10.1017/s0004972724000364
CHUNHONG LI, YUTIAN LEI
We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation $$ begin{align*} w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n. end{align*} $$
Here, $n geq 1$, $min {q,beta }>0$, $1<gamma leq 2$ and $beta gamma <n$. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of $w(i)$ when $|i| to infty $.
我们使用势分析来研究离散的沃尔夫型方程 $$ 的正解的性质： w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n.end{align*}$$Here, $n geq 1$, $min {q,beta }>0$, $1<gamma leq 2$ and $beta gamma <n$.这样的方程可以用来研究晶格、神经网络和其他离散模型研究中出现的图形上的非线性问题。我们利用正则性提升的方法获得了方程正解的最优求和性。根据这一结果，我们得到了当 $|i| to infty $ 时 $w(i)$ 的衰减率。
{"title":"SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE","authors":"CHUNHONG LI, YUTIAN LEI","doi":"10.1017/s0004972724000364","DOIUrl":"https://doi.org/10.1017/s0004972724000364","url":null,"abstract":"<p>We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514060414573-0998:S0004972724000364:S0004972724000364_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n. end{align*} $$</span></span></img></span></p><p>Here, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514060414573-0998:S0004972724000364:S0004972724000364_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$n geq 1$</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:20240514060414573-0998:S0004972724000364:S0004972724000364_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$min {q,beta }>0$</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:20240514060414573-0998:S0004972724000364:S0004972724000364_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$1<gamma leq 2$</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:20240514060414573-0998:S0004972724000364:S0004972724000364_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$beta gamma <n$</span></span></img></span></span>. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514060414573-0998:S0004972724000364:S0004972724000364_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$w(i)$</span></span></img></span></span> when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514060414573-0998:S0004972724000364:S0004972724000364_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$|i| to infty $</span></span></img></span></span>.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942073","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 : 2024-05-13DOI: 10.1017/s0004972724000376
COLIN PETITJEAN
We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if $(x_n) subset X$ and $(y_n) subset Y$ are two weakly null sequences such that $(x_n otimes y_n)$ converges weakly in $X widehat {otimes }_pi Y$ , then $(x_n otimes y_n)$ is also weakly null.
{"title":"THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS","authors":"COLIN PETITJEAN","doi":"10.1017/s0004972724000376","DOIUrl":"https://doi.org/10.1017/s0004972724000376","url":null,"abstract":"We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, <jats:italic>Indag. Math. (N.S.)</jats:italic>35(1) (2024), 60–75], proving that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000376_inline1.png\"/> <jats:tex-math> $(x_n) subset X$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000376_inline2.png\"/> <jats:tex-math> $(y_n) subset Y$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are two weakly null sequences such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000376_inline3.png\"/> <jats:tex-math> $(x_n otimes y_n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> converges weakly in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000376_inline4.png\"/> <jats:tex-math> $X widehat {otimes }_pi Y$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000376_inline5.png\"/> <jats:tex-math> $(x_n otimes y_n)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is also weakly null.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931572","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 : 2024-05-09DOI: 10.1017/s0004972724000248
XUJIAN HUANG, JIABIN LIU, SHUMING WANG
We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping $f: Xrightarrow Y$ satisfying $$ begin{align*} {|f(x)+alpha f(y)|: alphain mathbb{T}}={|x+alpha y|: alphain mathbb{T}}, quad x,yin X, end{align*} $$ where $mathbb {T}$ is the unit circle of the complex plane, there exists a function $sigma : Xrightarrow mathbb {T}$ such that $sigma cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.
{"title":"THE WIGNER PROPERTY OF SMOOTH NORMED SPACES","authors":"XUJIAN HUANG, JIABIN LIU, SHUMING WANG","doi":"10.1017/s0004972724000248","DOIUrl":"https://doi.org/10.1017/s0004972724000248","url":null,"abstract":"We prove that every smooth complex normed space <jats:italic>X</jats:italic> has the Wigner property. That is, for any complex normed space <jats:italic>Y</jats:italic> and every surjective mapping <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000248_inline1.png\"/> <jats:tex-math> $f: Xrightarrow Y$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000248_eqnu1.png\"/> <jats:tex-math> $$ begin{align*} {|f(x)+alpha f(y)|: alphain mathbb{T}}={|x+alpha y|: alphain mathbb{T}}, quad x,yin X, end{align*} $$ </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=\"S0004972724000248_inline2.png\"/> <jats:tex-math> $mathbb {T}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the unit circle of the complex plane, there exists a function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000248_inline3.png\"/> <jats:tex-math> $sigma : Xrightarrow mathbb {T}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000248_inline4.png\"/> <jats:tex-math> $sigma cdot f$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941701","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 : 2024-05-06DOI: 10.1017/s0004972724000339
YOSHINORI HAMAHATA
In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.
{"title":"NORMAL BASES FOR FUNCTION FIELDS","authors":"YOSHINORI HAMAHATA","doi":"10.1017/s0004972724000339","DOIUrl":"https://doi.org/10.1017/s0004972724000339","url":null,"abstract":"<p>In function fields in positive characteristic, we provide a concrete example of completely normal elements for a finite Galois extension. More precisely, for a nonabelian extension, we construct completely normal elements for Drinfeld modular function fields using Siegel functions in function fields. For an abelian extension, we construct completely normal elements for cyclotomic function fields.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885642","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 : 2024-05-06DOI: 10.1017/s0004972724000327
PENG GAO, LIANGYI ZHAO
We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, Tr. Mat. Inst. Steklova132 (1973), 247–250].
我们建立了广义黎曼假设下平滑二次狄利克特特征和的矩上限，证实了 M. 尤蒂拉的猜想['论实数特征和'，Tr. Mat. Inst. Steklova 132 (1973), 247-250]。
{"title":"BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS","authors":"PENG GAO, LIANGYI ZHAO","doi":"10.1017/s0004972724000327","DOIUrl":"https://doi.org/10.1017/s0004972724000327","url":null,"abstract":"<p>We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila [‘On sums of real characters’, <span>Tr. Mat. Inst. Steklova</span> <span>132</span> (1973), 247–250].</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885300","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 : 2024-04-25DOI: 10.1017/s0004972724000285
D. A. Wolfram
We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.
{"title":"INVERSE CONNECTION FORMULAE FOR GENERALISED BESSEL POLYNOMIALS","authors":"D. A. Wolfram","doi":"10.1017/s0004972724000285","DOIUrl":"https://doi.org/10.1017/s0004972724000285","url":null,"abstract":"\u0000 We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140655585","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 : 2024-04-25DOI: 10.1017/s0004972724000273
PEIYI WU
We give a sharp estimate for the first eigenvalue of the Schrödinger operator $L:=-Delta -sigma $ which is defined on the closed minimal submanifold $M^{n}$ in the unit sphere $mathbb {S}^{n+m}$ , where $sigma $ is the square norm of the second fundamental form.
{"title":"FIRST EIGENVALUE CHARACTERISATION OF CLIFFORD HYPERSURFACES AND VERONESE SURFACES","authors":"PEIYI WU","doi":"10.1017/s0004972724000273","DOIUrl":"https://doi.org/10.1017/s0004972724000273","url":null,"abstract":"\u0000\t <jats:p>We give a sharp estimate for the first eigenvalue of the Schrödinger operator <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000273_inline1.png\"/>\u0000\t\t<jats:tex-math>\u0000$L:=-Delta -sigma $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> which is defined on the closed minimal submanifold <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000273_inline2.png\"/>\u0000\t\t<jats:tex-math>\u0000$M^{n}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> in the unit sphere <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000273_inline3.png\"/>\u0000\t\t<jats:tex-math>\u0000$mathbb {S}^{n+m}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>, where <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000273_inline4.png\"/>\u0000\t\t<jats:tex-math>\u0000$sigma $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> is the square norm of the second fundamental form.</jats:p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140653557","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 : 2024-04-22DOI: 10.1017/s0004972724000297
MARCIO COLOMBO FENILLE
Given maps $f_1,ldots ,f_n:Xto Y$ between (finite and connected) graphs, with $ngeq 3$ (the case $n=2$ is well known), we say that they are loose if they can be deformed by homotopy to coincidence free maps, and totally loose if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if Y is not homeomorphic to the circle, then any maps are totally loose; (ii) otherwise, any maps are loose and they are totally loose if and only if they are homotopic.
{"title":"MULTIPLE LOOSE MAPS BETWEEN GRAPHS","authors":"MARCIO COLOMBO FENILLE","doi":"10.1017/s0004972724000297","DOIUrl":"https://doi.org/10.1017/s0004972724000297","url":null,"abstract":"Given maps <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000297_inline1.png\" /> <jats:tex-math> $f_1,ldots ,f_n:Xto Y$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> between (finite and connected) graphs, with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000297_inline2.png\" /> <jats:tex-math> $ngeq 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> (the case <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000297_inline3.png\" /> <jats:tex-math> $n=2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is well known), we say that they are <jats:italic>loose</jats:italic> if they can be deformed by homotopy to coincidence free maps, and <jats:italic>totally loose</jats:italic> if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if <jats:italic>Y</jats:italic> is not homeomorphic to the circle, then any maps are totally loose; (ii) otherwise, any maps are loose and they are totally loose if and only if they are homotopic.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636457","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 : 2024-04-22DOI: 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":null,"pages":null},"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}
Pub Date : 2024-04-22DOI: 10.1017/s0004972724000303
NING-LIU WEI, YU-BO LI, HAI-LIANG WU
Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith.115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting $a_1,ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field $mathbb {F}_q$ containing q elements with $2nmid q$ , we give the explicit value of the determinant $D_{(q-1)/2}=det [(a_i+a_j)^{(q-3)/2}]_{1le i,jle (q-1)/2}$ . In particular, if $q=p$ is a prime greater than $3$ , then $$ begin{align*}bigg(frac{det D_{(p-1)/2}}{p}bigg)= begin{cases} 1 & mbox{if} pequiv1pmod4, (-1)^{(h(-p)+1)/2} & mbox{if} pequiv 3pmod4 text{and} p>3, end{cases}end{align*} $$ where $(frac {cdot }{p})$ is the Legendre symbol and $h(-p)$
{"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":null,"pages":null},"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}
$$ begin{align*} w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n. end{align*} $$
$n geq 1$, 
$min {q,beta }>0$, 
$1<gamma leq 2$ and 
$beta gamma <n$. Such an equation can be used to study nonlinear problems on graphs appearing in the study of crystal lattices, neural networks and other discrete models. We use the method of regularity lifting to obtain an optimal summability of positive solutions of the equation. From this result, we obtain the decay rate of 
$w(i)$ when 
$|i| to infty $.
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