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":null,"pages":null},"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":"<p>We determine the characteristic polynomials of the matrices <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$[q^{,j-k}+t]_{1le ,j,kle n}$</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:20240531111027021-0956:S000497272400039X:S000497272400039X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$[q^{,j+k}+t]_{1le ,j,kle n}$</span></span></img></span></span> for any complex number <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$qnot =0,1$</span></span></img></span></span>. As an application, for complex numbers <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$a,b,c$</span></span></img></span></span> with <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$bnot =0$</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:20240531111027021-0956:S000497272400039X:S000497272400039X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$a^2not =4b$</span></span></img></span></span>, and the sequence <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$(w_m)_{min mathbb Z}$</span></span></img></span></span> with <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$w_{m+1}=aw_m-bw_{m-1}$</span></span></img></span></span> for all <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$min mathbb Z$</span></span></img></span></span>, we determine the exact value of <span><span><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":null,"pages":null},"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":"<p>An integer partition of a positive integer <span>n</span> is called <span>t</span>-core if none of its hook lengths is divisible by <span>t</span>. Gireesh <span>et al.</span> [‘A new analogue of <span>t</span>-core partitions’, <span>Acta Arith.</span> <span>199</span> (2021), 33–53] introduced an analogue <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$overline {a}_t(n)$</span></span></img></span></span> of the <span>t</span>-core partition function. They obtained multiplicative formulae and arithmetic identities for <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$overline {a}_t(n)$</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:20240531111127147-0430:S000497272400042X:S000497272400042X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$t in {3,4,5,8}$</span></span></img></span></span> and studied the arithmetic density of <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$overline {a}_t(n)$</span></span></img></span></span> modulo <span><span><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\"><span data-mathjax-type=\"texmath\"><span>$p_i^{,j}$</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:20240531111127147-0430:S000497272400042X:S000497272400042X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$t=p_1^{a_1}cdots p_m^{a_m}$</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:20240531111127147-0430:S000497272400042X:S000497272400042X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p_igeq 5$</span></span></img></span></span> are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, <span>J. Integer Seq.</span> <span>27</span> (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by <span><span><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":null,"pages":null},"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":null,"pages":null},"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":"A TROPICAL ANALOGUE OF THE LEMMA ON THE LOGARITHMIC DERIVATIVE","authors":"Juho Halonen","doi":"10.1017/s0004972724000388","DOIUrl":"https://doi.org/10.1017/s0004972724000388","url":null,"abstract":"\u0000 The tropical analogue of the lemma on the logarithmic derivative is generalised for noncontinuous tropical meromorphic functions, that is, piecewise linear functions that may have discontinuities. In addition, two Borel type results are generalised for piecewise continuous functions. With the generalisation of the tropical analogue of the lemma on the logarithmic derivative, several tropical analogues of Clunie and Mohon’ko type results are also automatically generalised for noncontinuous tropical meromorphic functions.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121241","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 THE SET OF BETTI ELEMENTS OF A PUISEUX MONOID","authors":"Scott T. Chapman, Joshua Jang, JASON MAO, Skyler Mao","doi":"10.1017/s0004972724000352","DOIUrl":"https://doi.org/10.1017/s0004972724000352","url":null,"abstract":"\u0000 Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of \u0000 \u0000 \u0000 \u0000$b in M$\u0000\u0000 \u0000 is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121709","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":"SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS","authors":"FLAVIEN MABILAT","doi":"10.1017/s0004972724000340","DOIUrl":"https://doi.org/10.1017/s0004972724000340","url":null,"abstract":"\u0000\t <jats:p>The <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=\"S0004972724000340_inline3.png\"/>\u0000\t\t<jats:tex-math>\u0000$lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-quiddities of size <jats:italic>n</jats:italic> are <jats:italic>n</jats:italic>-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline4.png\"/>\u0000\t\t<jats:tex-math>\u0000$lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-quiddities of odd size, and a lower and upper bound for the number of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline5.png\"/>\u0000\t\t<jats:tex-math>\u0000$lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-quiddities of even size, over the rings <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline6.png\"/>\u0000\t\t<jats:tex-math>\u0000${mathbb {Z}}/2^{m}{mathbb {Z}}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> (<jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline7.png\"/>\u0000\t\t<jats:tex-math>\u0000$m geq 2$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>). We also give explicit formulae for the number of <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline8.png\"/>\u0000\t\t<jats:tex-math>\u0000$lambda $\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>-quiddities of size <jats:italic>n</jats:italic> over <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline9.png\"/>\u0000\t\t<jats:tex-math>\u0000${mathbb {Z}}/8{mathbb {Z}}$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>.</jats:p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140966944","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":"BAZ volume 109 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0004972723001223","DOIUrl":"https://doi.org/10.1017/s0004972723001223","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968979","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":"AUTHOR INDEX FOR VOLUME 109","authors":"","doi":"10.1017/s0004972723001247","DOIUrl":"https://doi.org/10.1017/s0004972723001247","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140971304","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":"BAZ volume 109 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0004972723001235","DOIUrl":"https://doi.org/10.1017/s0004972723001235","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968005","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}