Bulletin of the Australian Mathematical Society最新文献

{"title":"COMPUTING CENTRALISERS IN [FINITELY GENERATED FREE]-BY-CYCLIC GROUPS","authors":"ANDRÉ CARVALHO","doi":"10.1017/s0004972724000443","DOIUrl":"https://doi.org/10.1017/s0004972724000443","url":null,"abstract":"<p>We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"55 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255302","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":"CHARACTERISTIC POLYNOMIALS OF THE MATRICES WITH","authors":"HAN WANG, ZHI-WEI SUN","doi":"10.1017/s000497272400039x","DOIUrl":"https://doi.org/10.1017/s000497272400039x","url":null,"abstract":"&lt;p&gt;We determine the characteristic polynomials of the matrices &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$[q^{,j-k}+t]_{1le ,j,kle n}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$[q^{,j+k}+t]_{1le ,j,kle n}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; for any complex number &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$qnot =0,1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. As an application, for complex numbers &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$a,b,c$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; with &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$bnot =0$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline8.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$a^2not =4b$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, and the sequence &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline9.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(w_m)_{min mathbb Z}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; with &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline10.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$w_{m+1}=aw_m-bw_{m-1}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; for all &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111027021-0956:S000497272400039X:S000497272400039X_inline11.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$min mathbb Z$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, we determine the exact value of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255240","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":"ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS","authors":"PRANJAL TALUKDAR","doi":"10.1017/s000497272400042x","DOIUrl":"https://doi.org/10.1017/s000497272400042x","url":null,"abstract":"&lt;p&gt;An integer partition of a positive integer &lt;span&gt;n&lt;/span&gt; is called &lt;span&gt;t&lt;/span&gt;-core if none of its hook lengths is divisible by &lt;span&gt;t&lt;/span&gt;. Gireesh &lt;span&gt;et al.&lt;/span&gt; [‘A new analogue of &lt;span&gt;t&lt;/span&gt;-core partitions’, &lt;span&gt;Acta Arith.&lt;/span&gt; &lt;span&gt;199&lt;/span&gt; (2021), 33–53] introduced an analogue &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$overline {a}_t(n)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of the &lt;span&gt;t&lt;/span&gt;-core partition function. They obtained multiplicative formulae and arithmetic identities for &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$overline {a}_t(n)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$t in {3,4,5,8}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and studied the arithmetic density of &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$overline {a}_t(n)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; modulo &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$p_i^{,j}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$t=p_1^{a_1}cdots p_m^{a_m}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$p_igeq 5$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, &lt;span&gt;J. Integer Seq.&lt;/span&gt; &lt;span&gt;27&lt;/span&gt; (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline8","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255701","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":"NOTES ON FERMAT-TYPE DIFFERENCE EQUATIONS","authors":"ILPO LAINE, ZINELAABIDINE LATREUCH","doi":"10.1017/s0004972724000406","DOIUrl":"https://doi.org/10.1017/s0004972724000406","url":null,"abstract":"<p>We consider the existence problem of meromorphic solutions of the Fermat-type difference equation <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111227055-0589:S0004972724000406:S0004972724000406_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$ begin{align*} f(z)^p+f(z+c)^q=h(z), end{align*} $$</span></span></img></span></p><p>where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111227055-0589:S0004972724000406:S0004972724000406_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$p,q$</span></span></img></span></span> are positive integers, and <span>h</span> has few zeros and poles in the sense that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111227055-0589:S0004972724000406:S0004972724000406_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$N(r,h) + N(r,1/h) = S(r,h)$</span></span></img></span></span>. As a particular case, we consider <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111227055-0589:S0004972724000406:S0004972724000406_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$h=e^g$</span></span></img></span></span>, where <span>g</span> is an entire function. Additionally, we briefly discuss the case where <span>h</span> is small with respect to <span>f</span> in the standard sense <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111227055-0589:S0004972724000406:S0004972724000406_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$T(r,h)=S(r,f)$</span></span></img></span></span>.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"69 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255312","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":"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":"1 1","pages":""},"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":"1153 1","pages":""},"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":"2 1","pages":""},"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":"64 1","pages":""},"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":"161 1","pages":""},"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":"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":"211 1","pages":""},"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}