Bulletin of the Australian Mathematical Society最新文献_第7页

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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引用次数: 0

MAXIMAL SUBSEMIGROUPS OF INFINITE SYMMETRIC GROUPS 无限对称群的最大子群

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-29 DOI: 10.1017/s0004972723001375

SUZANA MENDES-GONÇALVES, R. P. SULLIVAN

Brazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3)68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group

$G(X)$

defined on an infinite set X. It is easy to see that, in this case,

$G(X)$

contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of

$G(X)$

. We provide infinitely many examples of such semigroups.

Brazil et al. ['Maximal subgroups of infinite symmetric groups', Proc.Lond.Math.(3)68(1) (1994), 77-111] 提供了定义在无限集 X 上的对称群 $G(X)$ 的最大子群的新族。我们提供了无限多此类半群的例子。

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引用次数: 0

ON THE CHARACTERISATION OF ALTERNATING GROUPS BY CODEGREES 关于交替群的编码度特征

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-26 DOI: 10.1017/s0004972723001429

MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, YONG YANG

Let G be a finite group and

$mathrm {Irr}(G)$

the set of all irreducible complex characters of G. Define the codegree of

$chi in mathrm {Irr}(G)$

as

$mathrm {cod}(chi ):={|G:mathrm {ker}(chi ) |}/{chi (1)}$

and let

$mathrm {cod}(G):={mathrm {cod}(chi ) mid chi in mathrm {Irr}(G)}$

be the codegree set of G. Let

$mathrm {A}_n$

be an alternating group of degree

$n ge 5$

. We show that

$mathrm {A}_n$

is determined up to isomorphism by

$operatorname {cod}(mathrm {A}_n)$

.

让 G 是一个有限群，$mathrm {Irrr}(G)$ 是 G 所有不可还原复字符的集合。定义 $chi 在 mathrm {Irr}(G)$ 中的codegree为 $mathrm {cod}(chi ):={|G：|}/{chi (1)}$ 并且让 $mathrm {cod}(G):={mathrm {cod}(chi ) mid chi in mathrm {Irrr}(G)}$ 是 G 的codegree集合。让 $mathrm {A}_n$ 是一个度数为 $nge 5$ 的交替群。我们证明 $mathrm {A}_n$ 是由 $operatorname {cod}(mathrm {A}_n)$ 同构决定的。

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引用次数: 0

COMPLETE EMBEDDINGS OF GROUPS 群的完全嵌入

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-26 DOI: 10.1017/s0004972723001442

MARTIN R. BRIDSON, HAMISH SHORT

Every countable group G can be embedded in a finitely generated group

$G^*$

that is hopfian and complete, that is,

$G^*$

has trivial centre and every epimorphism

$G^*to G^*$

is an inner automorphism. Every finite subgroup of

$G^*$

is conjugate to a finite subgroup of G. If G has a finite presentation (respectively, a finite classifying space), then so does

$G^*$

. Our construction of

$G^*$

relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.

每个可数群 G 都可以嵌入一个有限生成的群 $G^*$，而这个群是跳化的和完全的，也就是说，$G^*$ 有微不足道的中心，并且每个外变 $G^*to G^*$ 都是一个内自变。$G^*$ 的每个有限子群都与 G 的一个有限子群共轭。如果 G 有一个有限的呈现（分别是有限的分类空间），那么 $G^*$ 也是如此。我们对 $G^*$ 的构造依赖于非对称和非哈肯的封闭双曲 3-manifolds。

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引用次数: 0

NEAR OPTIMAL THRESHOLDS FOR EXISTENCE OF DILATED CONFIGURATIONS IN 中存在扩张构型的近似最佳阈值

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-26 DOI: 10.1017/s0004972723001399

PABLO BHOWMIK, FIRDAVS RAKHMONOV

Let

$mathbb {F}_q^d$

denote the d-dimensional vector space over the finite field

$mathbb {F}_q$

with q elements. Define for

$alpha = (alpha _1, dots , alpha _d) in mathbb {F}_q^d$

. Let

$kin mathbb {N}$

, A be a nonempty subset of

${(i, j): 1 leq i < j leq k + 1}$

and

$rin (mathbb {F}_q)^2setminus {0}$

, where

$(mathbb {F}_q)^2={a^2:ain mathbb {F}_q}$

. If

$Esubset mathbb {F}_q^d$

, our main result demonstrates that when the size of the set E satisfies

$|E| geq C_k q^{d/2}$

, where

让 $mathbb {F}_q^d$ 表示有限域 $mathbb {F}_q$ 上有 q 个元素的 d 维向量空间。在 mathbb {F}_q^d$ 中定义 $alpha = (alpha _1, dots , alpha _d) 。让 $kin mathbb {N}$ , A 是 ${(i, j) 的一个非空子集：1 leq i < j leq k + 1}$ 和 $rin (mathbb {F}_q)^2setminus {0}$ , 其中 $(mathbb {F}_q)^2={a^2:a in mathbb {F}_q}$ .如果 $Esubset mathbb {F}_q^d$ ，我们的主要结果表明，当集合 E 的大小满足 $|E| geq C_k q^{d/2}$ 时，其中 $C_k$ 是一个完全取决于 k 的常数，那么就有可能在 E 中找到两个 $(k+1)$ 图元，使得其中一个图元相对于另一个图元扩张了 r，但只沿着 $|A|$ 边。更准确地说，我们确定在 E^{k+1}$ 中存在 $(x_1, dots , x_{k+1}) ，在 E^{k+1}$ 中存在 $(y_1, dots , y_{k+1}) ，这样，对于 $(i, j) in A$ 、我们有 $lVert y_i - y_j rVert = r lVert x_i - x_j rVert $ ，条件是 $x_i neq x_j$ 和 $y_i neq y_j$ for $1 leq i <；j leq k + 1$ 条件是 $|E| geq C_k q^{d/2}$ 和 $rin (mathbb {F}_q)^2setminus {0}$ 。我们为这一结果提供了两个不同的证明。第一个证明使用了群作用技术，这是解决此类问题的一种强有力的方法；第二个证明则基于基本的组合推理。此外，我们还证明了在维度 2 中，当 $q equiv 3 pmod 4$ 时，阈值 $d/2$ 是尖锐的。作为主要结果的推论，通过改变底层集合 A，我们确定了扩张 k 循环、k 路径和 k 星（其中 $k geq 3$ ）存在的阈值，其扩张比为 $rin (mathbb {F}_q)^2setminus {0}$ 。这些结果改进并概括了 Xie 和 Ge 的发现['Some results on similar configurations in subsets of $mathbb {F}_q^d$ ', Finite Fields Appl.91 (2023), Article no.102252, 20 pages].

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引用次数: 0

BOUNDING ZETA ON THE 1-LINE UNDER THE PARTIAL RIEMANN HYPOTHESIS 部分里曼假设下的单线上的Zeta界限

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-01-10 DOI: 10.1017/s0004972723001338

ANDRÉS CHIRRE

We provide explicit bounds for the Riemann zeta-function on the line $mathrm {Re},{s}=1$, assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.

我们为直线 $mathrm {Re},{s}=1$ 上的黎曼zeta函数提供了明确的边界，假定黎曼假设在高度 T 上成立。特别是，我们改进了黎曼zeta函数的对数导数和倒数在有限区域内的一些边界。

引用次数: 0

COUNTING UNIONS OF SCHREIER SETS 施赖尔集合的计数联合

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2023-12-27 DOI: 10.1017/s0004972723001326

KEVIN BEANLAND, DMITRIY GOROVOY, JĘDRZEJ HODOR, DANIIL HOMZA

A subset of positive integers F is a Schreier set if it is nonempty and

$|F|leqslant min F$

(here

$|F|$

is the cardinality of F). For each positive integer k, we define

$kmathcal {S}$

as the collection of all the unions of at most k Schreier sets. Also, for each positive integer n, let

$(kmathcal {S})^n$

be the collection of all sets in

$kmathcal {S}$

with maximum element equal to n. It is well known that the sequence

$(|(1mathcal {S})^n|)_{n=1}^infty $

is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence

$(|(kmathcal {S})^n|)_{n=1}^infty $

satisfies a linear recurrence for every positive k.

如果一个正整数子集 F 是非空的，并且 $|F|leqslant min F$（这里 $|F|$ 是 F 的万有引力），那么它就是施赖耶集。对于每个正整数 k，我们定义 $kmathcal {S}$ 为最多 k 个施赖尔集合的所有联合的集合。另外，对于每个正整数 n，让 $(kmathcal {S})^n$ 成为 $kmathcal {S}$ 中最大元素等于 n 的所有集合的集合。众所周知，序列 $(|(1mathcal {S})^n|)_{n=1}^infty $ 就是斐波那契序列。特别是，该序列满足线性递推。我们证明了序列 $(|(kmathcal {S})^n|)_{n=1}^infty $ 满足每一个正 k 的线性递归。

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引用次数: 0

SOLVABLE GROUPS WHOSE NONNORMAL SUBGROUPS HAVE FEW ORDERS 非正常子群阶数少的可解群

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2023-12-27 DOI: 10.1017/s0004972723001168

LIJUAN HE, HENG LV, GUIYUN CHEN

Suppose that G is a finite solvable group. Let

$t=n_c(G)$

denote the number of orders of nonnormal subgroups of G. We bound the derived length

$dl(G)$

in terms of

$n_c(G)$

. If G is a finite p-group, we show that

$|G'|leq p^{2t+1}$

and

$dl(G)leq lceil log _2(2t+3)rceil $

. If G is a finite solvable nonnilpotent group, we prove that the sum of the powers of the prime divisors of

$|G'|$

is less than t and that

$dl(G)leq lfloor 2(t+1)/3rfloor +1$

.

假设 G 是一个有限可解群。让 $t=n_c(G)$ 表示 G 的非正则子群的阶数。我们用 $n_c(G)$ 约束派生长度 $dl(G)$ 。如果 G 是有限 p 群，我们证明 $|G'|leq p^{2t+1}$ 和 $dl(G)leq lceil log _2(2t+3)rceil $ 。如果 G 是一个有限可解的非幂群，我们证明 $|G'|$ 的素除数的幂和小于 t 并且 $dl(G)leq lfloor 2(t+1)/3rfloor +1$ .

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引用次数: 0

INTERSECTING THE TORSION OF ELLIPTIC CURVES 与椭圆曲线的扭转相交

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2023-12-27 DOI: 10.1017/s000497272300134x

NATALIA GARCIA-FRITZ, HECTOR PASTEN

Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves

$E_1$

and

$E_2$

along with even degree-

$2$

maps

$pi _jcolon E_jto mathbb {P}^1$

having different branch loci, the intersection of the image of the torsion points of

$E_1$

and

$E_2$

under their respective

$pi _j$

is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-

$2$

maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna th

), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73-91]证明，给定两条复椭圆曲线 $E_1$ 和 $E_2$ 以及具有不同分支位置的偶数阶-2$ 映射 $pi _jcolon E_jto mathbb {P}^1$, $E_1$ 和 $E_2$ 在各自的 $pi _j$ 下的扭转点的像的交集是有限的。他们猜想（也是在与傅雷的合作中），这个交点的心率是均匀有界的，与椭圆曲线无关。最近对均匀马宁-芒福德猜想的证明意味着博戈莫洛夫-傅-茨钦克尔猜想的全解。在本文中，我们证明了博戈莫洛夫-傅-茨钦克尔猜想的广义化，即我们可以使用椭圆曲线上任何有界度的有理函数，只要它们有不同的分支位置，而不是偶数度-2$映射。我们的方法结合了内万林纳理论和统一马宁-芒福德猜想。通过类似的技术，我们还证明了一个关于数域上椭圆曲线秩下界的结果。

{"title":"INTERSECTING THE TORSION OF ELLIPTIC CURVES","authors":"NATALIA GARCIA-FRITZ, HECTOR PASTEN","doi":"10.1017/s000497272300134x","DOIUrl":"https://doi.org/10.1017/s000497272300134x","url":null,"abstract":"Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $pi _jcolon E_jto mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree- $2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna th","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139053000","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}

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