Bulletin of the Australian Mathematical Society最新文献

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GENERATING FUNCTIONS FOR THE QUOTIENTS OF NUMERICAL SEMIGROUPS 数值半群商数的生成函数

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-02-29 DOI: 10.1017/s0004972724000054

FEIHU LIU

We propose generating functions,

$textrm {RGF}_p(x)$

, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain

$textrm {RGF}_p(x)$

by extracting the constant term of a rational function. We use

$textrm {RGF}_p(x)$

to give a system of generators for the quotient of the numerical semigroup

$langle a_1,a_2,a_3rangle $

by p for a small positive integer p, and we characterise the generators of

${langle Arangle }/{p}$

for a general numerical semigroup A and any positive integer p.

我们提出了与西尔维斯特数数相关的数值半群商数的生成函数 $textrm {RGF}_p(x)$ 。利用麦克马洪的分割分析，我们可以通过提取有理函数的常数项得到 $textrm {RGF}_p(x)$ 。我们利用 $textrm {RGF}_p(x)$ 给出了一个小正整数 p 的数值半群 $langle a_1,a_2,a_3rangle $ 的商的生成器系统，并描述了一般数值半群 A 和任意正整数 p 的 ${langle Arangle }/{p}$ 的生成器的特征。

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引用次数: 0

DIOPHANTINE TRANSFERENCE PRINCIPLE OVER FUNCTION FIELDS 函数域上的二刁移原则

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-02-28 DOI: 10.1017/s0004972724000029

SOURAV DAS, ARIJIT GANGULY

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3)101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

我们研究的是函数域上的 Diophantine 转移原理。通过改编 Beresnevich 和 Velani 的方法['不均匀转移原理和 Diophantine 近似'，Proc.Lond.Math.(3)101 (2010)，821-851] 到函数域，我们将许多结果从同质扩展到非同质 Diophantine 近似。这也产生了函数场的非均质贝克-斯普林茹克猜想和一般非极端情形的上界。

引用次数: 0

GRAPH CHARACTERISATION OF THE ANNIHILATOR IDEALS OF LEAVITT PATH ALGEBRAS Leavitt 路径代数的湮没子表征图

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-02-15 DOI: 10.1017/s0004972723001466

LIA VAŠ

If E is a graph and K is a field, we consider an ideal I of the Leavitt path algebra

$L_K(E)$

of E over K. We describe the admissible pair corresponding to the smallest graded ideal which contains I where the grading in question is the natural grading of

$L_K(E)$

${mathbb {Z}}$

. Using this description, we show that the right and the left annihilators of I are equal (which may be somewhat surprising given that I may not be self-adjoint). In particular, we establish that both annihilators correspond to the same admissible pair and its description produces the characterisation from the title. Then, we turn to the property that the right (equivalently left) annihilator of any ideal is a direct summand and recall that a unital ring with this property is said to be quasi-Baer. We exhibit a condition on E which is equivalent to unital

$L_K(E)$

having this property.

如果 E 是一个图，K 是一个域，我们将考虑 E 在 K 上的莱维特路径代数 $L_K(E)$ 的理想 I。我们将描述与包含 I 的最小分级理想相对应的可容许对，其中的分级是 $L_K(E)$ 的自然分级 ${mathbb {Z}}$ 。利用这一描述，我们可以证明 I 的右湮和左湮是相等的（鉴于 I 可能不是自结的）。特别是，我们确定这两个湮没器对应于同一可容许对，并且其描述产生了标题中的特征。然后，我们将讨论任意理想的右湮没器（等同于左湮没器）是直接和这一性质，并回顾具有这一性质的单素环被称为准巴环。我们将展示 E 的一个条件，它等价于具有这一性质的单素 $L_K(E)$。

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引用次数: 0

EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS 无爪图半总支配集上的边加权函数

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-02-12 DOI: 10.1017/s0004972724000017

JIE CHEN, HONGZHANG CHEN, SHOU-JUN XU

In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by

$gamma _{t2}(G)$

, is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if

$Gneq N_2$

is a connected claw-free graph of order

$ngeq 6$

with minimum degree

$delta (G)geq 3$

, then

$gamma _{t2}(G)leq frac{4}{11}n$

and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin.33(5) (2017), 1119–1130].

在无孤立图 G 中，如果顶点子集 S 是 G 的支配集，且 S 中的每个顶点与 S 中另一个顶点的距离都在 2 以内，则该顶点子集 S 是 G 的半总支配集。G 的半总支配数用 $gamma _{t2}(G)$ 表示，是 G 中半总支配集的最小心数。利用半总支配集上的边加权函数，我们证明了如果 $Gneq N_2$ 是一个阶数为 $n/geq 6$ 且最小度数为 $delta (G)geq 3$ 的无连接爪图，那么 $gamma _{t2}(G)leq frac{4}{11}n$ 并且这个约束是尖锐的，推翻了 Zhu 等人提出的猜想。['无爪立方图中的半总支配'，Graphs Combin.33(5) (2017), 1119-1130].

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引用次数: 0

ON THE LIMIT SET OF A COMPLEX HYPERBOLIC TRIANGLE GROUP 上的复双曲三角形群的极限集

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-02-02 DOI: 10.1017/s0004972723001478

MENGQI SHI, JIEYAN WANG

Let

$Gamma =langle I_{1}, I_{2}, I_{3}rangle $

be the complex hyperbolic

$(4,4,infty )$

triangle group with

$I_1I_3I_2I_3$

being unipotent. We show that the limit set of

$Gamma $

is connected and the closure of a countable union of

$mathbb {R}$

-circles.

让 $Gamma =langle I_{1}, I_{2}, I_{3}rangle $ 是复双曲 $(4,4,infty )$ 三角形群，其中 $I_1I_3I_2I_3$ 是单能的。我们证明了 $Gamma $ 的极限集是连通的，并且是 $mathbb {R}$ - 圆的可数联盟的闭合。

引用次数: 0

GENERALISED MUTUALLY PERMUTABLE PRODUCTS AND SATURATED FORMATIONS, II 广义互变积和饱和形成，ii

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-30 DOI: 10.1017/s0004972723001430

ADOLFO BALLESTER-BOLINCHES, SESUAI Y. MADANHA, TENDAI M. MUDZIIRI SHUMBA, MARÍA C. PEDRAZA-AGUILERA

A group <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline1.png">$G=AB$</img> is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline2.png">$A cap B$</img> and B permutes with every subgroup of A containing <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline3.png">$A cap B$</img>. Weakly mutually permutable products were introduced by the first, second and fourth authors [‘Generalised mutually permutable products and saturated formations’, J. Algebra 595 (2022), 434–443] who showed that if <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline4.png">$G'$</img> is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, then <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline5.png">$G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}} $</img>, where <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline6.png">$ mathfrak {F} $</img> is a saturated formation containing <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240129062406367-0762:S0004972723001430:S0004972723001430_inline7.png">$ mathfrak {U} $</img>, the class of supersoluble groups. In this article we prove results on weakly mutually permutable products concerning <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:

如果 A 与 B 的每一个包含 $A cap B$ 的子群发生互变，而 B 与 A 的每一个包含 $A cap B$ 的子群发生互变，那么一个群 $G=AB$ 是子群 A 和 B 的弱互变积。第一、第二和第四作者提出了弱互变积['广义互变积与饱和形式'，《代数学报》595 (2022)，434-443]，他们证明了如果 $G'$ 是零幂的，A 与 B 的每个 Sylow 子群互变，B 与 A 的每个 Sylow 子群互变，那么 $G^{mathfrak {F}}=A^{mathfrak {F}}B^{mathfrak {F}}.$, 其中 $ mathfrak {F} $ 是包含 $ mathfrak {U} $ 的饱和形成，即超可溶群类。在这篇文章中，我们证明了关于$ mathfrak {F} $残差、$ mathfrak {F} $投影和$ mathfrak {F}$ 归一的弱互变积的结果。作为我们一些论证的应用，我们统一了关于弱互斥 $sn$ 积的一些结果。

引用次数: 0

ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION 关于方差-伽马分布的累积分布函数

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-29 DOI: 10.1017/s0004972723001387

ROBERT E. GAUNT

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

我们得到了方差-伽马分布的累积分布函数的精确公式，即涉及修正的第二类贝塞尔函数和修正的第一类洛梅尔函数的无穷级数。根据这些公式，我们推导出两个相关零均值正态分布随机变量乘积的累积分布函数的精确公式。

引用次数: 0

MULTIPLE SOLUTIONS FOR -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO 非线性为零的-拉普拉斯方程的多重解

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-29 DOI: 10.1017/s0004972723001405

SHIBO LIU

We consider the Dirichlet problem for

$p(x)$

-Laplacian equations of the form

$$ begin{align*} -Delta_{p(x)}u+b(x)vert uvert ^{p(x)-2}u=f(x,u),quad uin W_{0}^{1,p(x)}(Omega). end{align*} $$

The odd nonlinearity

$f(x,u)$

$p(x)$

-sublinear at

$u=0$

but the related limit need not be uniform for

$xin Omega $

. Except being subcritical, no additional assumption is imposed on

$f(x,u)$

for

$|u|$

large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function

$u=0$

我们考虑形式为 $$ 的 $p(x)$ 拉普拉斯方程的 Dirichlet 问题。-Delta_{p(x)}u+b(x)vert uvert ^{p(x)-2}u=f(x,u),quad uin W_{0}^{1,p(x)}(Omega).end{align*}$$ 奇数非线性 $f(x,u)$ 在 $u=0$ 时是 $p(x)$ - 次线性的，但对于 $xin Omega $ 而言，相关极限不一定是均匀的。除了是次临界外，在 $|u|$ 较大时，对 $f(x,u)$没有额外的假设。通过应用克拉克定理和截断方法，我们得到了一系列具有负能量并接近零函数 $u=0$ 的解。

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引用次数: 0

CONGRUENCES FOR RANKS OF PARTITIONS 分区等级的同余

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-29 DOI: 10.1017/s0004972723001454

RENRONG MAO

Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of

$5$

for ranks of partitions.

分区秩在分区理论中发挥着重要作用。它们为拉马努扬著名的分区函数全等提供了组合解释。我们为分区的秩建立了一个调制 5$ 的幂的全等族。

引用次数: 0

HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS 图的同源线性商和边沿理想

IF 0.7 4区数学 Q3 MATHEMATICS

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-29 DOI: 10.1017/s0004972723001363

NADIA TAGHIPOUR, SHAMILA BAYATI, FARHAD RAHMATI

It is well known that the edge ideal $I(G)$ of a simple graph G has linear quotients if and only if $G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph $G^c$ when $I(G)$ has homological linear quotients results in a graph with the same property. In particular, $I(G)$ has homological linear quotients when $G^c$ is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, $I(G)$ has homological linear quotients for every graph G such that $G^c$ is a $lambda $ -min

众所周知，简单图 G 的边理想 $I(G)$ 具有线性商，当且仅当 $G^c$ 是弦性的。我们将研究边理想的同调移动理想何时继承了线性商的性质。我们将看到，当 $I(G)$ 具有同调线性商时，给图 $G^c$ 添加一个簇会得到具有相同性质的图。特别是，当 $G^c$ 是块图时，$I(G)$ 具有同调线性商。我们还证明，将尖顶添加到树中会保留其补集的边理想具有同调线性商的特性。此外，$I(G)$ 对每个图 G 都有同调线性商，这样 $G^c$ 就是一个 $lambda $ 最小弦图。

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