Pub Date : 2023-12-22DOI: 10.1017/s0004972723001351
Nayandeep Deka Baruah
We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$ , $p(n)=0$ for $n<0$ ), then for all nonnegative integers m, $$ begin{align*}sum_{k=0}^infty p(24m+23-omega(-2k))+sum_{k=1}^infty p(24m+23-omega(2k))equiv 0~ (text{mod}~144),end{align*} $$ where $omega (k)=k(3k+1)/2$ .
{"title":"DIVISIBILITY OF SUMS OF PARTITION NUMBERS BY MULTIPLES OF 2 AND 3","authors":"Nayandeep Deka Baruah","doi":"10.1017/s0004972723001351","DOIUrl":"https://doi.org/10.1017/s0004972723001351","url":null,"abstract":"\u0000\t <jats:p>We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if <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=\"S0004972723001351_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$p(n)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> denotes the number of unrestricted partitions of a positive integer <jats:italic>n</jats:italic> (and <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=\"S0004972723001351_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$p(0)=1$\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=\"S0004972723001351_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$p(n)=0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> for <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=\"S0004972723001351_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$n<0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>), then for all nonnegative integers <jats:italic>m</jats:italic>, <jats:disp-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001351_eqnu1.png\" />\u0000\t\t<jats:tex-math>\u0000$$ begin{align*}sum_{k=0}^infty p(24m+23-omega(-2k))+sum_{k=1}^infty p(24m+23-omega(2k))equiv 0~ (text{mod}~144),end{align*} $$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:disp-formula></jats:p>\u0000\t <jats:p>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=\"S0004972723001351_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$omega (k)=k(3k+1)/2$\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":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138945112","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 : 2023-12-18DOI: 10.1017/s0004972723001156
C. C. CORRIGAN
We use the Weyl bound for Dirichlet L-functions to derive zero-density estimates for L-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when $sigma $ is sufficiently distant from the critical line.
我们利用狄利克特 L 函数的韦尔界，推导出与定阶狄利克特特征族相关的 L 函数的零密度估计。当 $sigma $ 离临界线足够远时，这些结果改进了作者之前给出的约束。
{"title":"A NOTE ON THE ZEROS OF L-FUNCTIONS ASSOCIATED TO FIXED-ORDER DIRICHLET CHARACTERS","authors":"C. C. CORRIGAN","doi":"10.1017/s0004972723001156","DOIUrl":"https://doi.org/10.1017/s0004972723001156","url":null,"abstract":"<p>We use the Weyl bound for Dirichlet <span>L</span>-functions to derive zero-density estimates for <span>L</span>-functions associated to families of fixed-order Dirichlet characters. The results improve on previous bounds given by the author when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216081215188-0622:S0004972723001156:S0004972723001156_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$sigma $</span></span></img></span></span> is sufficiently distant from the critical line.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138715123","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 : 2023-12-18DOI: 10.1017/s000497272300117x
BO YU, JIANKUI LI
Let H be a complex separable Hilbert space with $dim H geq 2$. Let $mathcal {N}$ be a nest on H such that $E_+ neq E$ for any $E neq H, E in mathcal {N}$. We prove that every 2-local isometry of $operatorname {Alg}mathcal {N}$ is a surjective linear isometry.
让 H 是一个复杂的可分离的希尔伯特空间，其中有 $dim H geq 2$。让 $mathcal {N}$ 是 H 上的一个巢，对于任意 $E neq H, E in mathcal {N}$ 来说，$E_+ neq E$ 是这样的。我们证明 $operatorname {Alg}mathcal {N}$ 的每一个 2 局部等势都是投射线性等势。
{"title":"2-LOCAL ISOMETRIES OF SOME NEST ALGEBRAS","authors":"BO YU, JIANKUI LI","doi":"10.1017/s000497272300117x","DOIUrl":"https://doi.org/10.1017/s000497272300117x","url":null,"abstract":"<p>Let <span>H</span> be a complex separable Hilbert space with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216090148228-0567:S000497272300117X:S000497272300117X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$dim H geq 2$</span></span></img></span></span>. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216090148228-0567:S000497272300117X:S000497272300117X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {N}$</span></span></img></span></span> be a nest on <span>H</span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216090148228-0567:S000497272300117X:S000497272300117X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$E_+ neq E$</span></span></img></span></span> for any <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216090148228-0567:S000497272300117X:S000497272300117X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$E neq H, E in mathcal {N}$</span></span></img></span></span>. We prove that every 2-local isometry of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231216090148228-0567:S000497272300117X:S000497272300117X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {Alg}mathcal {N}$</span></span></img></span></span> is a surjective linear isometry.</p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138714876","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 : 2023-12-12DOI: 10.1017/s0004972723001120
ZHE DONG, JINZE JIANG, YAFEI ZHAO
We introduce the notion of weakly local reflexivity in operator space theory and prove that any dual operator space is weakly locally reflexive.
我们在算子空间理论中引入了弱局部反折的概念，并证明任何对偶算子空间都是弱局部反折的。
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Pub Date : 2023-12-12DOI: 10.1017/s0004972723001144
Behrooz Niknami
{"title":"Stochastic Matching Models","authors":"Behrooz Niknami","doi":"10.1017/s0004972723001144","DOIUrl":"https://doi.org/10.1017/s0004972723001144","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139009806","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 : 2023-12-12DOI: 10.1017/s0004972723001132
A. Simonič
{"title":"NEW EFFECTIVE RESULTS IN THE THEORY OF THE RIEMANN ZETA-FUNCTION","authors":"A. Simonič","doi":"10.1017/s0004972723001132","DOIUrl":"https://doi.org/10.1017/s0004972723001132","url":null,"abstract":"","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138977190","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 : 2023-11-21DOI: 10.1017/s0004972723001119
SUMANDEEP KAUR, SURENDER KUMAR
Let $K={mathbb {Q}}(theta )$ be an algebraic number field with $theta $ satisfying a monic irreducible polynomial $f(x)$ of degree n over ${mathbb {Q}}.$ The polynomial $f(x)$ is said to be monogenic if ${1,theta ,ldots ,theta ^{n-1}}$ is an integral basis of K. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, Bull. Aust. Math. Soc.100 (2019), 239–244] conjectured that if $nge 3$ , $1le mle n-1$ , $gcd (n,mB)=1$ and A is a prime number, th
{"title":"ON A CONJECTURE OF LENNY JONES ABOUT CERTAIN MONOGENIC POLYNOMIALS","authors":"SUMANDEEP KAUR, SURENDER KUMAR","doi":"10.1017/s0004972723001119","DOIUrl":"https://doi.org/10.1017/s0004972723001119","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=\"S0004972723001119_inline1.png\" /> <jats:tex-math> $K={mathbb {Q}}(theta )$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be an algebraic number field with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline2.png\" /> <jats:tex-math> $theta $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> satisfying a monic irreducible polynomial <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline3.png\" /> <jats:tex-math> $f(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of degree <jats:italic>n</jats:italic> over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline4.png\" /> <jats:tex-math> ${mathbb {Q}}.$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> The polynomial <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline5.png\" /> <jats:tex-math> $f(x)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is said to be monogenic if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline6.png\" /> <jats:tex-math> ${1,theta ,ldots ,theta ^{n-1}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an integral basis of <jats:italic>K</jats:italic>. Deciding whether or not a monic irreducible polynomial is monogenic is an important problem in algebraic number theory. In an attempt to answer this problem for a certain family of polynomials, Jones [‘A brief note on some infinite families of monogenic polynomials’, <jats:italic>Bull. Aust. Math. Soc.</jats:italic>100 (2019), 239–244] conjectured that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline7.png\" /> <jats:tex-math> $nge 3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline8.png\" /> <jats:tex-math> $1le mle n-1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972723001119_inline9.png\" /> <jats:tex-math> $gcd (n,mB)=1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:italic>A</jats:italic> is a prime number, th","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495303","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 : 2023-11-15DOI: 10.1017/s0004972723001016
JIN-HUI FANG, CSABA SÁNDOR
Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/xrightarrow 1$ . For $A={a_1<a_2<cdots }$ , let $A(x)$ denote the counting function of A and let $a^*(x)$ denote the largest element in $Abigcap [1,x]$ . Following the work of Ruzsa [‘Exact additive complements’, Quart. J. Math.68 (2017) 227–235] and Chen and Fang [‘Additive complements with Narkiewicz’s condition’, Combinatorica39 (2019), 813–823], we prove that, for exact additive complements $A,B$ with ${a_{n+1}}/ {na_n}rightarrow infty $ ,
Pub Date : 2023-11-08DOI: 10.1017/s0004972722001538
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{"title":"AUTHOR INDEX FOR VOLUME 108","authors":"","doi":"10.1017/s0004972722001538","DOIUrl":"https://doi.org/10.1017/s0004972722001538","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343221","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 : 2023-11-08DOI: 10.1017/s0004972722001526
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
{"title":"BAZ volume 108 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0004972722001526","DOIUrl":"https://doi.org/10.1017/s0004972722001526","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135343235","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}
$sigma $ is sufficiently distant from the critical line.
$dim H geq 2$. Let 
$mathcal {N}$ be a nest on H such that 
$E_+ neq E$ for any 
$E neq H, E in mathcal {N}$. We prove that every 2-local isometry of 
$operatorname {Alg}mathcal {N}$ is a surjective linear isometry.
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