Bulletin of the Australian Mathematical Society最新文献_第5页

BAZ volume 109 issue 2 Cover and Back matter BAZ 第 109 卷第 2 期封面和封底

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-11 DOI: 10.1017/s0004972723001211

引用次数: 0

BAZ volume 109 issue 2 Cover and Front matter BAZ 第 109 卷第 2 期封面和封底

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-11 DOI: 10.1017/s000497272300120x

引用次数: 0

WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS 韦尔斯特拉斯 ZETA 函数和 p-ADIC 线性关系

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-11 DOI: 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].

引用次数: 0

NON-PÓLYA FIELDS WITH LARGE PÓLYA GROUPS ARISING FROM LEHMER QUINTICS 由雷汞五元组产生的具有大波利亚群的非波利亚场

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-11 DOI: 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.

我们构建了一个具有大波利亚群的五元非波利亚场新家族。我们证明，这种场的波利亚数永远不会超过其波利亚群大小的五倍。最后，我们证明了这些非波利亚场是场指数为 1 的非单源场。

引用次数: 0

ON THE SET OF KRONECKER NUMBERS 上的克朗克尔数集

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-08 DOI: 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。

引用次数: 0

ON SOME CONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS 关于涉及中心二项式系数的一些同余式

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-08 DOI: 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。

引用次数: 0

A NOTE ON JUDICIOUS BISECTIONS OF GRAPHS 关于图形的明智平分的说明

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-08 DOI: 10.1017/s000497272400008x

SHUFEI WU, XIAOBEI XIONG

Let G be a graph with m edges, minimum degree

$delta $

and containing no cycle of length 4. Answering a question of Bollobás and Scott, Fan et al. [‘Bisections of graphs without short cycles’, Combinatorics, Probability and Computing27(1) (2018), 44–59] showed that if (i) G is

$2$

-connected, or (ii)

$delta ge 3$

, or (iii)

$delta ge 2$

and the girth of G is at least 5, then G admits a bisection such that

$max {e(V_1),e(V_2)}le (1/4+o(1))m$

, where

$e(V_i)$

denotes the number of edges of G with both ends in

$V_i$

. Let

$sge 2$

be an integer. In this note, we prove that if

$delta ge 2s-1$

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 $max {e(V_1),e(V_2)}le (1/4+o(1))m$ 。

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引用次数: 0

EXTRACTION OF DENSITY-LAYERED FLUID FROM A POROUS MEDIUM 从多孔介质中提取密度层流体

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-07 DOI: 10.1017/s000497272300000x

Jyothi Jose

引用次数: 0

ON THE DIOPHANTINE EQUATION 关于二次方程

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-06 DOI: 10.1017/s0004972724000066

ELCHIN HASANALIZADE

A generalisation of the well-known Pell sequence ${P_n}_{nge 0}$ given by $P_0=0$, $P_1=1$ and $P_{n+2}=2P_{n+1}+P_n$ for all $nge 0$ is the k-generalised Pell sequence ${P^{(k)}_n}_{nge -(k-2)}$ whose first k terms are $0,ldots ,0,1$ and each term afterwards is given by the linear recurrence $P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+cdots +P^{(k)}_{n-k}$. For the Pell sequence, the formula $P^2_n+P^2_{n+1}=P_{2n+1}$ holds for all

对于所有 $nge 0$，由 $P_0=0$，$P_1=1$ 和 $P_{n+2}=2P_{n+1}+P_n$ 给出的众所周知的 Pell 序列 ${P_n}_{nge 0}$的广义化是 k 个广义 Pell 序列 ${P^{(k)}_n}_{nge -(k-2)}$ ，其前 k 项为 $0、ldots ,0,1$，之后的每项由线性递推公式$P^{(k)}_n=2P^{(k)}_{n-1}+P^{(k)}_{n-2}+cdots +P^{(k)}_{n-k}$给出。对于佩尔序列，公式 $P^2_n+P^2_{n+1}=P_{2n+1}$ 对于所有 $nge 0$ 都成立。本文将证明 Diophantine 方程 $$ begin{align*} (P^{(k)}_n)^2+(P^{(k)}_{n+1})^2=P^{(k)}_m end{align*}在正整数 $k、m$ 和 n 中，$n>1$ 和 $kge 3$ 无解。

引用次数: 0

IDEMPOTENT GENERATORS OF INCIDENCE ALGEBRAS 幂等生成数

IF 0.7 4区数学 Q3 Mathematics

Bulletin of the Australian Mathematical Society

Pub Date : 2024-03-05 DOI: 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.

计算了交换环上有限正集的入射代数的最小幂生子数。这个数量要么等于 $lceil log _2 nrceil $，要么等于 $lceil log _2 nrceil +1 $，其中 n 是正集的万有引力。这两种情况可以通过将正集的哈塞图嵌入超立方图的补集来区分。

引用次数: 0

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