Bulletin of the Australian Mathematical Society最新文献

英文中文

COMPUTING CENTRALISERS IN [FINITELY GENERATED FREE]-BY-CYCLIC GROUPS 计算[有限生成的自由]旁循环群的中心点

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-06-04 DOI: 10.1017/s0004972724000443

ANDRÉ CARVALHO

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.

我们证明了[有限生成的自由]旁循环群中元素的中心子是可计算的。作为推论，给定[有限生成的自由]-旁循环群中的两个共轭元素，可以计算共轭物的集合，并且无上下文约束的共轭问题是可解的。我们提出了几个由这项工作自然产生的问题。

引用次数: 0

CHARACTERISTIC POLYNOMIALS OF THE MATRICES WITH 矩阵的特征多项式与

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-06-03 DOI: 10.1017/s000497272400039x

HAN WANG, ZHI-WEI SUN

We determine the characteristic polynomials of the matrices <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">$[q^{,j-k}+t]_{1le ,j,kle n}$</img> and <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">$[q^{,j+k}+t]_{1le ,j,kle n}$</img> for any complex number <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">$qnot =0,1$</img>. As an application, for complex numbers <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">$a,b,c$</img> with <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">$bnot =0$</img> and <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">$a^2not =4b$</img>, and the sequence <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">$(w_m)_{min mathbb Z}$</img> with <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">$w_{m+1}=aw_m-bw_{m-1}$</img> for all <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">$min mathbb Z$</img>, we determine the exact value of <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.

我们确定了任意复数 $qnot =0,1$ 的矩阵 $[q^{,j-k}+t]_{1le,j,kle n}$ 和 $[q^{,j+k}+t]_{1le,j,kle n}$ 的特征多项式。作为应用，对于复数 $a,b,c$，其中 $bnot =0$ 和 $a^2not=4b$，以及序列 $(w_m)_{min mathbb Z}$，其中对于所有 $min mathbb Z$，$w_{m+1}=aw_m-bw_{m-1}$、我们确定 $det [w_{,j-k}+cdelta _{jk}]_{1le ,j,kle n}$ 的精确值。

引用次数: 0

ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS t-CORE 分段模拟的算术特性

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-06-03 DOI: 10.1017/s000497272400042x

PRANJAL TALUKDAR

An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith. 199 (2021), 33–53] introduced an analogue <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">$overline {a}_t(n)$</img> of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for <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">$overline {a}_t(n)$</img> where <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">$t in {3,4,5,8}$</img> and studied the arithmetic density of <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">$overline {a}_t(n)$</img> modulo <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">$p_i^{,j}$</img> where <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">$t=p_1^{a_1}cdots p_m^{a_m}$</img> and <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">$p_igeq 5$</img> are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq. 27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240531111127147-0430:S000497272400042X:S000497272400042X_inline8

Gireesh 等人['A new analogue of t-core partitions', Acta Arith.他们得到了 $overline {a}_t(n)$ 的乘法公式和算术等式，其中 $t in {3,4,5,8}$ 并研究了 $overline {a}_t(n)$ modulo $p_i^{,j}$ 的算术密度，其中 $t=p_1^{a_1}cdots p_m^{a_m}$ 和 $p_igeq 5$ 都是素数。Bandyopadhyay 和 Baruah [' Arithmetic identities for some analogs of the 5-core partition function', J. Integer Seq.27 (2024), 文章编号 24.4.5]证明了 $overline {a}_5(n)$ 所满足的新算术等式。我们研究了 $/overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^alpha m$ 的算术密度，其中 $gcd (m,6)$=1.另外，利用小野和田口的一个结果['某些模块形式的 2-adic 属性及其在算术函数中的应用'，Int.J. Number Theory 1 (2005), 75-101]关于赫克算子零点性的结果，我们证明了 $overline {a}_3(n)$ modulo arbitrary powers of 2 的无穷同余族。

引用次数: 0

NOTES ON FERMAT-TYPE DIFFERENCE EQUATIONS 费马型差分方程注释

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-06-03 DOI: 10.1017/s0004972724000406

ILPO LAINE, ZINELAABIDINE LATREUCH

We consider the existence problem of meromorphic solutions of the Fermat-type difference equation $$ begin{align*} f(z)^p+f(z+c)^q=h(z), end{align*} $$

where $p,q$ are positive integers, and h has few zeros and poles in the sense that $N(r,h) + N(r,1/h) = S(r,h)$. As a particular case, we consider $h=e^g$, where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense $T(r,h)=S(r,f)$.

我们考虑费马型差分方程 $$ begin{align*}f(z)^p+f(z+c)^q=h(z), end{align*}的同态解的存在性问题。其中，$p,q$ 为正整数，而 h 的零点和极点很少，即 $N(r,h) + N(r,1/h) = S(r,h)$。作为一种特殊情况，我们考虑 $h=e^g$，其中 g 是一次函数。此外，我们还简要讨论了 h 相对于 f 较小的情况，即标准意义上的 $T(r,h)=S(r,f)$。

引用次数: 0

SUMMABILITY AND ASYMPTOTICS OF POSITIVE SOLUTIONS OF AN EQUATION OF WOLFF TYPE 沃尔夫方程正解的求和性和渐近性

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-05-14 DOI: 10.1017/s0004972724000364

CHUNHONG LI, YUTIAN LEI

We use potential analysis to study the properties of positive solutions of a discrete Wolff-type equation $$ begin{align*} w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n. end{align*} $$

Here, $n geq 1$, $min {q,beta }>0$, $1<gamma leq 2$ and $beta gamma <n$. 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 $w(i)$ when $|i| to infty $.

我们使用势分析来研究离散的沃尔夫型方程 $$ 的正解的性质： w(i)=W_{beta,gamma}(w^q)(i), quad i in mathbb{Z}^n.end{align*}$$Here, $n geq 1$, $min {q,beta }>0$, $1<gamma leq 2$ and $beta gamma <n$.这样的方程可以用来研究晶格、神经网络和其他离散模型研究中出现的图形上的非线性问题。我们利用正则性提升的方法获得了方程正解的最优求和性。根据这一结果，我们得到了当 $|i| to infty $ 时 $w(i)$ 的衰减率。

引用次数: 0

THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS 基本张量集合在射影张量积中是弱闭合的

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-05-13 DOI: 10.1017/s0004972724000376

COLIN PETITJEAN

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’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if

$(x_n) subset X$

and

$(y_n) subset Y$

are two weakly null sequences such that

$(x_n otimes y_n)$

converges weakly in

$X widehat {otimes }_pi Y$

, then

$(x_n otimes y_n)$

is also weakly null.

我们证明了基本张量集合在两个巴拿赫空间的射影张量积中是弱封闭的。因此，我们回答了罗德里格斯（Rodríguez）和鲁埃达-佐卡（Rueda Zoca）的一个问题['射影张量积中的弱前封闭性'，Indag.(N.S.)35(1) (2024), 60-75)]，证明如果 $(x_n) subset X$ 和 $(y_n) subset Y$ 是两个弱空序列，使得 $(x_n otimes y_n)$ 在 $X widehat {otimes }_pi Y$ 中弱收敛，那么 $(x_n otimes y_n)$ 也是弱空的。

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引用次数: 0

THE WIGNER PROPERTY OF SMOOTH NORMED SPACES 光滑规范空间的维格纳特性

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-05-09 DOI: 10.1017/s0004972724000248

XUJIAN HUANG, JIABIN LIU, SHUMING WANG

We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping

$f: Xrightarrow Y$

satisfying

$$ begin{align*} {|f(x)+alpha f(y)|: alphain mathbb{T}}={|x+alpha y|: alphain mathbb{T}}, quad x,yin X, end{align*} $$

where

$mathbb {T}$

is the unit circle of the complex plane, there exists a function

$sigma : Xrightarrow mathbb {T}$

such that

$sigma cdot f$

is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.

我们证明每个光滑复规范空间 X 都具有维格纳特性。也就是说，对于任何复规范空间 Y 和每一个投射映射 $f：Xrightarrow Y$ 满足 $$ （开始{align*}）f(x)+α f(y)||：=（x+y）：alphain mathbb{T}}, quad x,yin X, end{align*}$$ 其中 $mathbb {T}$ 是复平面的单位圆，存在一个函数 $sigma : Xrightarrow mathbb {T}$ 使得 $sigma cdot f$ 是线性或反线性等距。这是复数规范空间的维格纳定理的变种。

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引用次数: 0

NORMAL BASES FOR FUNCTION FIELDS 功能域的正态基

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-05-06 DOI: 10.1017/s0004972724000339

YOSHINORI HAMAHATA

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.

在正特征的函数场中，我们为有限伽罗瓦扩展提供了一个完全正元的具体例子。更确切地说，对于非阿贝尔扩展，我们利用函数场中的西格尔函数，为 Drinfeld 模块化函数场构造了完全正元。对于非阿贝尔扩展，我们为回旋函数场构造了完全正元。

引用次数: 0

BOUNDS FOR MOMENTS OF QUADRATIC DIRICHLET CHARACTER SUMS 二次迪里夏特特征和的时刻界限

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-05-06 DOI: 10.1017/s0004972724000327

PENG GAO, LIANGYI ZHAO

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’, Tr. Mat. Inst. Steklova 132 (1973), 247–250].

我们建立了广义黎曼假设下平滑二次狄利克特特征和的矩上限，证实了 M. 尤蒂拉的猜想['论实数特征和'，Tr. Mat. Inst. Steklova 132 (1973), 247-250]。

引用次数: 0

MULTIPLE LOOSE MAPS BETWEEN GRAPHS 图形之间的多重松散映射

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-04-22 DOI: 10.1017/s0004972724000297

MARCIO COLOMBO FENILLE

Given maps

$f_1,ldots ,f_n:Xto Y$

between (finite and connected) graphs, with

$ngeq 3$

(the case

$n=2$

is well known), we say that they are loose if they can be deformed by homotopy to coincidence free maps, and totally loose if they can be deformed by homotopy to maps which are two by two coincidence free. We prove that: (i) if Y 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.

给定（有限连通）图之间的映射 $f_1,ldots ,f_n:Xto Y$，且 $ngeq 3$（众所周知的情况是 $n=2$），如果它们可以通过同调变形为无重合映射，我们就说它们是松散的；如果它们可以通过同调变形为二乘二无重合映射，我们就说它们是完全松散的。我们证明(i) 如果 Y 与圆不同构，那么任何映射都是完全松散的；(ii) 否则，任何映射都是松散的，而且只有当它们是同构的时候，它们才是完全松散的。

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引用次数: 0

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