Bulletin of the Australian Mathematical Society最新文献

{"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>$min {q,beta }&gt;0$</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>$1&lt;gamma leq 2$</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>$beta gamma &lt;n$</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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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> $$ begin{align*}bigg(frac{det D_{(p-1)/2}}{p}bigg)= begin{cases} 1 &amp; mbox{if} pequiv1pmod4, (-1)^{(h(-p)+1)/2} &amp; mbox{if} pequiv 3pmod4 text{and} p&gt;3, end{cases}end{align*} $$ </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}