Pub Date : 2024-03-14DOI: 10.1017/s0004972724000170
ATHANASIOS BESLIKAS
We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols $varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.
{"title":"A CLASS OF SYMBOLS THAT INDUCE BOUNDED COMPOSITION OPERATORS FOR DIRICHLET-TYPE SPACES ON THE DISC","authors":"ATHANASIOS BESLIKAS","doi":"10.1017/s0004972724000170","DOIUrl":"https://doi.org/10.1017/s0004972724000170","url":null,"abstract":"<p>We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313131113149-0098:S0004972724000170:S0004972724000170_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$varphi $</span></span></img></span></span> that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"68 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-11DOI: 10.1017/s0004972724000091
DUC HIEP PHAM
We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math.40(2) (1977), 171–193].
我们讨论与代数数域上定义的椭圆曲线相关的 p-adic Weierstrass zeta 函数，以及它们在 p-adic 域中的值的线性关系。这些结果是 Wüstholz 在复数域给出的 p-adic 类似结果的扩展[见 A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3]，同时也将 Bertrand 的一个结果推广到更高维度['Sous-groupes à un paramètre p-adique de variétés de groupe', Invent.Math.40(2) (1977), 171-193].
{"title":"WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS","authors":"DUC HIEP PHAM","doi":"10.1017/s0004972724000091","DOIUrl":"https://doi.org/10.1017/s0004972724000091","url":null,"abstract":"<p>We discuss the <span>p</span>-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the <span>p</span>-adic domain. These results are extensions of the <span>p</span>-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, <span>Logarithmic Forms and Diophantine Geometry</span>, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre <span>p</span>-adique de variétés de groupe’, <span>Invent. Math.</span> <span>40</span>(2) (1977), 171–193].</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-11DOI: 10.1017/s0004972724000108
NIMISH KUMAR MAHAPATRA, PREM PRAKASH PANDEY
We construct a new family of quintic non-Pólya fields with large Pólya groups. We show that the Pólya number of such a field never exceeds five times the size of its Pólya group. Finally, we show that these non-Pólya fields are nonmonogenic of field index one.
{"title":"NON-PÓLYA FIELDS WITH LARGE PÓLYA GROUPS ARISING FROM LEHMER QUINTICS","authors":"NIMISH KUMAR MAHAPATRA, PREM PRAKASH PANDEY","doi":"10.1017/s0004972724000108","DOIUrl":"https://doi.org/10.1017/s0004972724000108","url":null,"abstract":"<p>We construct a new family of quintic non-Pólya fields with large Pólya groups. We show that the Pólya number of such a field never exceeds five times the size of its Pólya group. Finally, we show that these non-Pólya fields are nonmonogenic of field index one.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"126 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-08DOI: 10.1017/s0004972724000133
SAYAN GOSWAMI, WEN HUANG, XIAOSHENG WU
A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant k and a set D consisting of at most 720 computable Maillet numbers such that, for any integer n, $kn$ can be expressed as a product of a Kronecker number and a Maillet number in D. We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.
如果一个偶数正数可以写成两个素数之差，那么它就是麦列数；如果一个偶数正数可以用无限多种方法写成两个素数之差，那么它就是克罗内克数。一般认为，所有偶数都是克罗内克数。我们研究了克罗内克数的除法和乘法，并证明这些数相当丰富。我们证明了存在一个可计算常数 k 和一个由最多 720 个可计算的麦列特数组成的集合 D，对于任意整数 n，$kn$ 都可以表示为 D 中的一个克罗内克数和一个麦列特数的乘积。
{"title":"ON THE SET OF KRONECKER NUMBERS","authors":"SAYAN GOSWAMI, WEN HUANG, XIAOSHENG WU","doi":"10.1017/s0004972724000133","DOIUrl":"https://doi.org/10.1017/s0004972724000133","url":null,"abstract":"A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant <jats:italic>k</jats:italic> and a set <jats:italic>D</jats:italic> consisting of at most 720 computable Maillet numbers such that, for any integer <jats:italic>n</jats:italic>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000133_inline1.png\" /> <jats:tex-math> $kn$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be expressed as a product of a Kronecker number and a Maillet number in <jats:italic>D</jats:italic>. We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"276 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-08DOI: 10.1017/s0004972724000121
GUO-SHUAI MAO
We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, J. Number Theory13(11) (2011), 2219–2238]. Let p be an odd prime. Then $$ begin{align*} sum_{k=1}^{p-1}frac{binom{2k}k}{k2^k}equiv-frac12H_{{(p-1)}/2}+frac7{16}p^2B_{p-3}pmod{p^3}, end{align*} $$ where $H_n$ is the nth harmonic number and $B_n$ is the nth Bernoulli number. In addition, we evaluate $sum _{k=0}^{p-1}(ak+b)binom {2k}k/2^k$ modulo $p^3$ for any p-adic integers $a, b$ .
我们证明了 Z. -W.Sun ['On congruences related to central binomial coefficients', J. Number Theory13(11) (2011), 2219-2238].设 p 是奇素数。那么 $$ begin{align*}sum_{k=1}^{p-1}frac{binom{2k}k}{k2^k}equiv-frac12H_{{(p-1)}/2}+frac7{16}p^2B_{p-3}pmod{p^3}, end{align*}$$ 其中 $H_n$ 是第 n 次谐波数，$B_n$ 是第 n 次伯努利数。此外，对于任意 p-adic 整数 $a，b$，我们将对 $sum _{k=0}^{p-1}(ak+b)binom {2k}k/2^k$ modulo $p^3$ 进行求值。
{"title":"ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS","authors":"GUO-SHUAI MAO","doi":"10.1017/s0004972724000121","DOIUrl":"https://doi.org/10.1017/s0004972724000121","url":null,"abstract":"We prove the following conjecture of Z.-W. Sun [‘On congruences related to central binomial coefficients’, <jats:italic>J. Number Theory</jats:italic>13(11) (2011), 2219–2238]. Let <jats:italic>p</jats:italic> be an odd prime. Then <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000121_eqnu1.png\" /> <jats:tex-math> $$ begin{align*} sum_{k=1}^{p-1}frac{binom{2k}k}{k2^k}equiv-frac12H_{{(p-1)}/2}+frac7{16}p^2B_{p-3}pmod{p^3}, end{align*} $$ </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=\"S0004972724000121_inline1.png\" /> <jats:tex-math> $H_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:italic>n</jats:italic>th harmonic number and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000121_inline2.png\" /> <jats:tex-math> $B_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the <jats:italic>n</jats:italic>th Bernoulli number. In addition, we evaluate <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000121_inline3.png\" /> <jats:tex-math> $sum _{k=0}^{p-1}(ak+b)binom {2k}k/2^k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> modulo <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000121_inline4.png\" /> <jats:tex-math> $p^3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for any <jats:italic>p</jats:italic>-adic integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000121_inline5.png\" /> <jats:tex-math> $a, b$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-08DOI: 10.1017/s000497272400008x
SHUFEI WU, XIAOBEI XIONG
Let <jats:italic>G</jats:italic> be a graph with <jats:italic>m</jats:italic> edges, minimum degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline1.png" /> <jats:tex-math> $delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan <jats:italic>et al.</jats:italic> [‘Bisections of graphs without short cycles’, <jats:italic>Combinatorics, Probability and Computing</jats:italic>27(1) (2018), 44–59] showed that if (i) <jats:italic>G</jats:italic> is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline2.png" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-connected, or (ii) <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline3.png" /> <jats:tex-math> $delta ge 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, or (iii) <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline4.png" /> <jats:tex-math> $delta ge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and the girth of <jats:italic>G</jats:italic> is at least 5, then <jats:italic>G</jats:italic> admits a bisection such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline5.png" /> <jats:tex-math> $max {e(V_1),e(V_2)}le (1/4+o(1))m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline6.png" /> <jats:tex-math> $e(V_i)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the number of edges of <jats:italic>G</jats:italic> with both ends in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline7.png" /> <jats:tex-math> $V_i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline8.png" /> <jats:tex-math> $sge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an integer. In this note, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S000497272400008X_inline9.png" /> <jats:tex-math> $delta ge 2s-1$ </jats:tex-math> </jat
让 G 是一个有 m 条边、最小度为 $delta $ 且不包含长度为 4 的循环的图。['没有短周期的图的分叉'，Combinatorics, Probability and Computing27(1) (2018)，44-59]表明，如果（i）G 是 2 美元连接的，或者（ii）$delta ge 3$ 、或者(iii) $deltage 2$，并且 G 的周长至少为 5，那么 G 允许一个分段，使得 $max {e(V_1),e(V_2)}le (1/4+o(1))m$ ，其中 $e(V_i)$ 表示 G 中两端都在 $V_i$ 中的边的数量。让 $sge 2$ 为整数。在本注中，我们将证明如果 $delta ge 2s-1$并且 G 不包含 $K_{2,s}$ 作为子图，那么 G 允许有一个分段，使得 $max {e(V_1),e(V_2)}le (1/4+o(1))m$ 。
{"title":"A NOTE ON JUDICIOUS BISECTIONS OF GRAPHS","authors":"SHUFEI WU, XIAOBEI XIONG","doi":"10.1017/s000497272400008x","DOIUrl":"https://doi.org/10.1017/s000497272400008x","url":null,"abstract":"Let <jats:italic>G</jats:italic> be a graph with <jats:italic>m</jats:italic> edges, minimum degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline1.png\" /> <jats:tex-math> $delta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan <jats:italic>et al.</jats:italic> [‘Bisections of graphs without short cycles’, <jats:italic>Combinatorics, Probability and Computing</jats:italic>27(1) (2018), 44–59] showed that if (i) <jats:italic>G</jats:italic> is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline2.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-connected, or (ii) <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline3.png\" /> <jats:tex-math> $delta ge 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, or (iii) <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline4.png\" /> <jats:tex-math> $delta ge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and the girth of <jats:italic>G</jats:italic> is at least 5, then <jats:italic>G</jats:italic> admits a bisection such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline5.png\" /> <jats:tex-math> $max {e(V_1),e(V_2)}le (1/4+o(1))m$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline6.png\" /> <jats:tex-math> $e(V_i)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the number of edges of <jats:italic>G</jats:italic> with both ends in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline7.png\" /> <jats:tex-math> $V_i$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline8.png\" /> <jats:tex-math> $sge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an integer. In this note, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S000497272400008X_inline9.png\" /> <jats:tex-math> $delta ge 2s-1$ </jats:tex-math> </jat","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"15 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-06DOI: 10.1017/s0004972724000066
ELCHIN HASANALIZADE
<p>A generalisation of the well-known Pell sequence <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline2.png"><span data-mathjax-type="texmath"><span>${P_n}_{nge 0}$</span></span></img></span></span> given by <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline3.png"><span data-mathjax-type="texmath"><span>$P_0=0$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline4.png"><span data-mathjax-type="texmath"><span>$P_1=1$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline5.png"><span data-mathjax-type="texmath"><span>$P_{n+2}=2P_{n+1}+P_n$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline6.png"><span data-mathjax-type="texmath"><span>$nge 0$</span></span></img></span></span> is the <span>k</span>-generalised Pell sequence <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline7.png"><span data-mathjax-type="texmath"><span>${P^{(k)}_n}_{nge -(k-2)}$</span></span></img></span></span> whose first <span>k</span> terms are <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline8.png"><span data-mathjax-type="texmath"><span>$0,ldots ,0,1$</span></span></img></span></span> and each term afterwards is given by the linear recurrence <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline9.png"><span data-mathjax-type="texmath"><span>$P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+cdots +P^{(k)}_{n-k}$</span></span></img></span></span>. For the Pell sequence, the formula <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline10.png"><span data-mathjax-type="texmath"><span>$P^2_n+P^2_{n+1}=P_{2n+1}$</span></span></img></span></span> holds for all <span><span><img data-mime
{"title":"ON THE DIOPHANTINE EQUATION","authors":"ELCHIN HASANALIZADE","doi":"10.1017/s0004972724000066","DOIUrl":"https://doi.org/10.1017/s0004972724000066","url":null,"abstract":"<p>A generalisation of the well-known Pell sequence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${P_n}_{nge 0}$</span></span></img></span></span> given by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$P_0=0$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$P_1=1$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$P_{n+2}=2P_{n+1}+P_n$</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:20240305093818166-0200:S0004972724000066:S0004972724000066_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$nge 0$</span></span></img></span></span> is the <span>k</span>-generalised Pell sequence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline7.png\"><span data-mathjax-type=\"texmath\"><span>${P^{(k)}_n}_{nge -(k-2)}$</span></span></img></span></span> whose first <span>k</span> terms are <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$0,ldots ,0,1$</span></span></img></span></span> and each term afterwards is given by the linear recurrence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+cdots +P^{(k)}_{n-k}$</span></span></img></span></span>. For the Pell sequence, the formula <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240305093818166-0200:S0004972724000066:S0004972724000066_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$P^2_n+P^2_{n+1}=P_{2n+1}$</span></span></img></span></span> holds for all <span><span><img data-mime","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"138 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-05DOI: 10.1017/s0004972724000078
N. A. KOLEGOV
The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either $lceil log _2 nrceil $ or $lceil log _2 nrceil +1$, where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.
{"title":"IDEMPOTENT GENERATORS OF INCIDENCE ALGEBRAS","authors":"N. A. KOLEGOV","doi":"10.1017/s0004972724000078","DOIUrl":"https://doi.org/10.1017/s0004972724000078","url":null,"abstract":"<p>The minimum number of idempotent generators is calculated for an incidence algebra of a finite poset over a commutative ring. This quantity equals either <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$lceil log _2 nrceil $</span></span></img></span></span> or <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240304124645615-0035:S0004972724000078:S0004972724000078_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$lceil log _2 nrceil +1$</span></span></img></span></span>, where <span>n</span> is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"9 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140033940","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 : 2024-02-29DOI: 10.1017/s0004972724000030
R. J. HIGGS
Let $alpha $ be a complex-valued $2$ -cocycle of a finite group G with $alpha $ chosen so that the $alpha $ -characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $alpha $ -regular are characterised by the values that the irreducible $alpha $ -characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.
{"title":"PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS","authors":"R. J. HIGGS","doi":"10.1017/s0004972724000030","DOIUrl":"https://doi.org/10.1017/s0004972724000030","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline1.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a complex-valued <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline2.png\" /> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-cocycle of a finite group <jats:italic>G</jats:italic> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline3.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> chosen so that the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline4.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-characters of <jats:italic>G</jats:italic> are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of <jats:italic>G</jats:italic> that are also <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline5.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular are characterised by the values that the irreducible <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline6.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-characters of <jats:italic>G</jats:italic> take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline7.png\" /> <jats:tex-math> $G;$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> however, the initial choice of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000030_inline8.png\" /> <jats:tex-math> $alpha $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> from its cohomology class is not unique in general and it is shown the results can vary for a different choice.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"35 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006826","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 : 2024-02-29DOI: 10.1017/s0004972724000042
BENJAMIN STEINBERG
Carlsen [‘ $ast $ -isomorphism of Leavitt path algebras over $Bbb Z$ ’, Adv. Math.324 (2018), 326–335] showed that any $ast $ -homomorphism between Leavitt path algebras over $mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of $mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every $ast $ -homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.
{"title":"A NOTE ON PROJECTIONS IN ÉTALE GROUPOID ALGEBRAS AND DIAGONAL-PRESERVING HOMOMORPHISMS","authors":"BENJAMIN STEINBERG","doi":"10.1017/s0004972724000042","DOIUrl":"https://doi.org/10.1017/s0004972724000042","url":null,"abstract":"Carlsen [‘<jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline1.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-isomorphism of Leavitt path algebras over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline2.png\" /> <jats:tex-math> $Bbb Z$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>’, <jats:italic>Adv. Math.</jats:italic>324 (2018), 326–335] showed that any <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline3.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-homomorphism between Leavitt path algebras over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline4.png\" /> <jats:tex-math> $mathbb Z$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline5.png\" /> <jats:tex-math> $mathbb C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000042_inline6.png\" /> <jats:tex-math> $ast $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006698","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}
$varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.
$lceil log _2 nrceil $ or 
$lceil log _2 nrceil +1$, where n is the cardinality of the poset. The two cases are separated in terms of the embedding of the Hasse diagram of the poset into the complement of the hypercube graph.
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