Mathematical Proceedings of the Cambridge Philosophical Society最新文献

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Generalised knotoids 普通节肢动物

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-09-19 DOI: 10.1017/s0305004124000148

COLIN ADAMS, ZACHARY ROMRELL, ALEXANDRA BONAT, MAYA CHANDE, JOYE CHEN, MAXWELL JIANG, DANIEL SANTIAGO, BENJAMIN SHAPIRO, DORA WOODRUFF

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

2010 年，图拉耶夫提出了 "结"（knotoids）作为 "结 "的一种变体，它将圆的嵌入替换为具有两个端点的封闭区间的嵌入，在这里我们称之为 "极点"。我们定义的广义旋结允许任意多的极点、区间和圆，每个极点对应任意数量的区间端点，包括零。这一理论包含了其他各种相关的拓扑对象，并引入了一些特别有趣的新情况。我们探索了各种类似的knotoid不变式，包括高度、指数多项式、括号多项式和双曲性。我们进一步将其推广到knotoid图，这是空间图的自然扩展，同时允许极点和顶点。

引用次数: 0

Domination inequalities and dominating graphs 支配不等式和支配图

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-09-19 DOI: 10.1017/s0305004124000185

DAVID CONLON, JOONKYUNG LEE

We say that a graph H dominates another graph H′ if the number of homomorphisms from H′ to any graph G is dominated, in an appropriate sense, by the number of homomorphisms from H to G. We study the family of dominating graphs, those graphs with the property that they dominate all of their subgraphs. It has long been known that even-length paths are dominating in this sense and a result of Hatami implies that all weakly norming graphs are dominating. In a previous paper, we showed that every finite reflection group gives rise to a family of weakly norming, and hence dominating, graphs. Here we revisit this connection to show that there is a much broader class of dominating graphs.

如果从 H′ 到任何图 G 的同构数在适当意义上被从 H 到 G 的同构数所支配，我们就说一个图 H 支配另一个图 H′。众所周知，偶数长度的路径在这个意义上具有支配性，而 Hatami 的一个结果意味着所有弱规范图都具有支配性。在之前的一篇论文中，我们证明了每个有限反射群都会产生一个弱规范图族，因此也是支配图族。在此，我们将重新探讨这一联系，以证明存在一类更广泛的支配图。

引用次数: 0

Multiplicative dependence of rational values modulo approximate finitely generated groups 有理值模近似有限生成群的乘法依赖性

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-09-19 DOI: 10.1017/s0305004124000173

ATTILA BÉRCZES, YANN BUGEAUD, KÁLMÁN GYŐRY, JORGE MELLO, ALINA OSTAFE, MIN SHA

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, ldots, f_nin K(X)$, there are only finitely many elements $alpha in K$ such that $f_1(alpha),ldots,f_n(alpha)$ are multiplicatively dependent modulo such sets.

在本文中，我们建立了一些关于有理值模数集乘法依赖性的有限性结果，这些有理值模数集 "接近"（关于魏尔高度）数域 K 的有限生成乘法群的除法群。例如，我们证明了在有理函数 $f_1,ldots,f_n/in K(X)$ 的某些条件下，K$ 中只有有限多个元素 $alpha inK$使得 $f_1(alpha),ldots,f_n(alpha)$ 与这些集合具有乘法依赖性。

引用次数: 0

Tropical curves in abelian surfaces I: enumeration of curves passing through points 无性曲面中的热带曲线 I：通过点的曲线枚举

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-09-19 DOI: 10.1017/s0305004124000161

THOMAS BLOMME

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the tropical multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of the Block–Göttsche refined multiplicity to get tropical refined invariants.

本文是通过热带方法研究无性曲面枚举不变式的系列论文的第一部分。在本文中，我们考虑枚举通过 g 个点的固定阶数的 g 属曲线。我们计算了由 T. Nishinou 提出的对应定理所提供的热带多重性，并证明可以按照 Block-Göttsche 精炼多重性的方式对这一多重性进行精炼，从而得到热带精炼不变式。

引用次数: 0

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves 椭圆曲线 p 塞尔默群的伽罗瓦后裔失效

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-09-19 DOI: 10.1017/s0305004124000197

ROSS PATERSON

We show that if F is <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline1.png">$mathbb{Q}$</img> or a multiquadratic number field, <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline2.png">$pinleft{{2,3,5}right}$</img>, and <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline3.png">$K/F$</img> is a Galois extension of degree a power of p, then for elliptic curves <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline4.png">$E/mathbb{Q}$</img> ordered by height, the average dimension of the p-Selmer groups of <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline5.png">$E/K$</img> is bounded. In particular, this provides a bound for the average K-rank of elliptic curves <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline6.png">$E/mathbb{Q}$</img> for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F.The central result is that: for each finite Galois extension <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline7.png">$K/F$</img> of number fields and prime number p, as <img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918140027897-0543:S0305004124000197:S0305004124000197_inline8.png"/>$E/mathbb{Q}$ varies, the difference in dimension betwe

我们证明，如果 F 是 $mathbb{Q}$ 或一个多二次数域，$pinleft{2,3,5}right}$ 并且 $K/F$ 是 p 的幂级数的伽罗瓦扩展，那么对于按高度排序的椭圆曲线 $E/mathbb{Q}$，$E/K$ 的 p-Selmer 群的平均维度是有界的。此外，我们还给出了这种 F 的伽罗瓦扩展上的莫德尔-韦尔群的某些表示论不变式的边界。核心结果是：对于数域和素数 p 的每个有限伽罗瓦扩展 $K/F$，随着 $E/mathbb{Q}$ 的变化，$E/K$ 的 p 塞尔默群和 $E/F$ 的 p 塞尔默群的伽罗瓦固定空间维数之差具有有界平均数。

引用次数: 0

Differential forms on universal K3 surfaces 通用 K3 表面上的微分形式

IF 0.8 3区数学 Q3 MATHEMATICS

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-07-12 DOI: 10.1017/s0305004124000100

SHOUHEI MA

We give a vanishing and classification result for holomorphic differential forms on smooth projective models of the moduli spaces of pointed K3 surfaces. We prove that there is no nonzero holomorphic k-form for

$0<k<10$

and for even

$k>19$

. In the remaining cases, we give an isomorphism between the space of holomorphic k-forms with that of vector-valued modular forms (

$10leq k leq 18$

) or scalar-valued cusp forms (odd

$kgeq 19$

) for the modular group. These results are in fact proved in the generality of lattice-polarisation.

我们给出了尖 K3 曲面模空间光滑投影模型上的全形微分形式的消失和分类结果。我们证明，在 $0<k<10$ 和偶数 $k>19$ 时，不存在非零的全形 k 形式。在其余情况下，我们给出了全形 k 形式空间与模数群的矢量值模数形式（$10leq k leq 18$）或标量值尖顶形式（奇$kgeq 19$）空间之间的同构关系。这些结果实际上是在格极化的一般性中证明的。

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引用次数: 0

A better than exponent for iterated sums and products over 迭代求和与乘积的指数，比

IF 0.8 3区数学 Q2 Mathematics

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-05-10 DOI: 10.1017/s0305004124000112

OLIVER ROCHE–NEWTON

In this paper, we prove that the bound

begin{equation*}max { |8A-7A|,|5f(A)-4f(A)| } gg |A|^{frac{3}{2} + frac{1}{54}}end{equation*}

holds for all

$A subset mathbb R$

, and for all convex functions f which satisfy an additional technical condition. This technical condition is satisfied by the logarithmic function, and this fact can be used to deduce a sum-product estimate

begin{equation*}max { |16A|, |A^{(16)}| } gg |A|^{frac{3}{2} + c},end{equation*}

for some

$cgt 0$

. Previously, no sum-product estimate over

$mathbb R$

with exponent strictly greater than

$3/2$

was known for any number of variables. Moreover, the technical condition on f seems to be satisfied for most interesting cases, and we give some further applications. In particular, we show that

begin{equation*}|AA| leq K|A| implies ,forall d in mathbb R setminus {0 }, ,, |{(a,b) in A times A : a-b=d }| ll K^C |A|^{frac{2}{3}-c^{prime}},end{equation*}

where

$c,C gt 0$

在本文中，我们证明了束缚（begin{equation*}max { |8A-7A|,|5f(A)-4f(A)| }|gg |A|^{frac{3}{2}+ frac{1}{54}}end{equation*} 对于所有 $A subset mathbb R$ 以及所有满足附加技术条件的凸函数 f 都成立。对数函数满足这个技术条件，这个事实可以用来推导出一个和积估计值。|A|^{frac{3}{2}.+ c},end{equation*} for some $cgt 0$ .在此之前，对于任意数量的变量，都不知道在 $mathbb R$ 上有指数严格大于 3/2$ 的和积估计。此外，关于 f 的技术条件似乎在大多数有趣的情况下都能满足，我们给出了一些进一步的应用。特别是，我们证明了begin{equation*}|AA|leq K|A| implies ,forall d in mathbb R setminus {0 }，,,||{(a,b)in A times A ：a-b=d }| |ll K^C |A|^{frac{2}{3}-c^{prime}},end{equation*} 其中 $c,C gt 0$ 是绝对常数。

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引用次数: 0

The Ulam–Hammersley problem for multiset permutations 多集排列的乌兰-哈默斯利问题

IF 0.8 3区数学 Q2 Mathematics

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-05-09 DOI: 10.1017/s0305004124000124

LUCAS GERIN

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in

${1,dots,n}$

occurs k times, where k may depend on n. This generalises the famous Ulam–Hammersley problem of the case

$k=1$

. The proof relies on poissonisation and on a careful non-asymptotic analysis of variants of the Hammersley–Aldous–Diaconis particle system.

我们得到了随机均匀多集排列中最长递增/非递减子序列的渐近行为，其中 ${1,dots,n}$ 中的每个元素都出现了 k 次，其中 k 可能取决于 n。证明依赖于泊松化和对哈默斯利-阿尔都斯-迪亚科尼斯粒子系统变体的仔细非渐进分析。

引用次数: 0

Non-invertible planar self-affine sets 非可逆平面自阿芬集合

IF 0.8 3区数学 Q2 Mathematics

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-05-08 DOI: 10.1017/s0305004124000136

ANTTI KÄENMÄKI, PETTERI NISSINEN

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.

我们比较了不可反转自证集的维度和相应的可反转自证集的维度。特别是，对于一般的平面自阿芬集合，我们证明，当它们的维数较大时，它们的维数是重合的，而当它们的维数较小时，它们的维数是不同的。我们的研究依赖于热力学形式主义，对于主矩阵和不可还原矩阵，我们可以完全描述压力的行为特征。

引用次数: 0

PSP volume 176 issue 3 Cover and Back matter PSP 第 176 卷第 3 期封面和封底

IF 0.8 3区数学 Q2 Mathematics

Mathematical Proceedings of the Cambridge Philosophical Society

Pub Date : 2024-04-12 DOI: 10.1017/s0305004124000094

引用次数: 0

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