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Arithmetic progressions and holomorphic phase retrieval 算术级数和全形相位检索

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-08-13 DOI: 10.1112/blms.13134

Lukas Liehr

We study the determination of a holomorphic function from its absolute value. Given a parameter <math> <semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mi>R</mi> </mrow> <annotation>$theta in mathbb {R}$</annotation> </semantics></math>, we derive the following characterization of uniqueness in terms of rigidity of a set <math> <semantics> <mrow> <mi>Λ</mi> <mo>⊆</mo> <mi>R</mi> </mrow> <annotation>$Lambda subseteq mathbb {R}$</annotation> </semantics></math>: if <math> <semantics> <mi>F</mi> <annotation>$mathcal {F}$</annotation> </semantics></math> is a vector space of entire functions containing all exponentials <math> <semantics> <mrow> <msup> <mi>e</mi> <mrow> <mi>ξ</mi> <mi>z</mi> </mrow> </msup> <mo>,</mo> <mspace></mspace> <mi>ξ</mi> <mo>∈</mo> <mi>C</mi> <mo>∖</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> </mrow> <annotation>$e^{xi z}, , xi in mathbb {C} setminus lbrace 0 rbrace$</annotation> </semantics></math>, then every <math> <semantics> <mrow> <mi>F</mi> <mo>∈</mo> <mi>F</mi> </mrow> <annotation>$F in mathcal {F}$</annotation> </semantics></math> is uniquely determined up to a unimodular phase factor by <math> <semantics> <mrow> <mo>{</mo> <mo>|</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>:</mo> <mi>z</mi> <mo>∈</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>R</mi>

我们研究从全形函数的绝对值确定全形函数的问题。给定一个参数θ ∈ R $theta in mathbb {R}$ ，我们从集合Λ ⊆ R $Lambda subseteq mathbb {R}$ 的刚度方面推导出以下唯一性特征：如果 F $mathcal {F}$ 是包含所有指数 e ξ z , ξ ∈ C ∖ { 0 } 的全函数向量空间 $e^{xi z}, , xi in mathbb {C}setminus lbrace 0 rbrace$, then every F ∈ F $F in mathcal {F}$ is uniquely determined up to a unimodular phase factor by { | F ( z ) | : z ∈ e i θ ( R + i Λ ) } $lbrace |F(z)|: z ∈ e^{itheta }({mathbb {R}}+ i Lambda) rbrace$ 当且仅当 Λ $Lambda$ 不包含在算术级数 a Z + b $amathbb {Z}+b$ 中时。利用这一洞察力，我们为 Gabor 相位检索和保利型唯一性问题确定了一系列结果。例如，对于 L 2 ( R + ) $L^2({mathbb {R}}_+)$ 中的 Gabor 相位检索问题，只要 Z ∼ $tilde{mathbb {Z}}$ 是一个合适的整数扰动，那么 Z × Z ∼ ${mathbb {Z}}/times tilde{mathbb {Z}}$ 就是一个唯一性集合。

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引用次数: 0

On the maximum field of linearity of linear sets 论线性集合的最大线性域

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-08-13 DOI: 10.1112/blms.13133

Bence Csajbók, Giuseppe Marino, Valentina Pepe

Let <math> <semantics> <mi>V</mi> <annotation>$V$</annotation> </semantics></math> denote an <math> <semantics> <mi>r</mi> <annotation>$r$</annotation> </semantics></math>-dimensional <math> <semantics> <msub> <mi>F</mi> <msup> <mi>q</mi> <mi>n</mi> </msup> </msub> <annotation>$mathbb {F}_{q^n}$</annotation> </semantics></math>-vector space. For an <math> <semantics> <mi>m</mi> <annotation>$m$</annotation> </semantics></math>-dimensional <math> <semantics> <msub> <mi>F</mi> <mi>q</mi> </msub> <annotation>$mathbb {F}_q$</annotation> </semantics></math>-subspace <math> <semantics> <mi>U</mi> <annotation>$U$</annotation> </semantics></math> of <math> <semantics> <mi>V</mi> <annotation>$V$</annotation> </semantics></math>, assume that <math> <semantics> <mrow> <msub> <mo>dim</mo> <mi>q</mi> </msub> <mfenced> <msub> <mrow> <mo>⟨</mo> <mi>v</mi> <mo>⟩</mo> </mrow> <msub> <mi>F</mi> <msup> <mi>q</mi> <mi>n</mi> </msup> </msub> </msub> <mo>∩</mo> <mi>U</mi> </mfenced> <mo>⩾</mo> <mn>2</mn> </mrow> <annotation>$dim _q left(langle {bf v}rangle _{mathbb {F}_{q^n}} cap Uright) geqslant 2$</annotation> </semantics></math> for each nonzero vector <math> <semantics> <mrow> <mi>v</mi> <mo>∈</mo> <mi>U</mi> </mrow> <annotation>${bf v}in U$</annotation> </semantics></math>. If <math> <semantics> <mrow> <

让 V $V$ 表示一个 r $r$ -dimensional F q n $mathbb {F}_{q^n}$ -vector 空间。对于 V $V$ 的一个 m $m$ -dimensional F q $mathbb {F}_q$ -subspace U $U$ , 假设对于每个非零向量 v ∈ U ${bf v}rangle _mathbb {F}_{q^n}} cap Uright) geqslant 2$ 。如果 n ⩽ q $nleqslant q$ ，那么我们证明存在一个整数 1 < d ∣ n $1&lt;d mid n$，使得由 U $U$ 的非零向量生成的一维 F q n $mathbb {F}_{q^n}$ 子空间的集合与由⟨ U ⟩ F q d $langle Urangle _{mathbb {F}_{q^d}}$ 的非零向量生成的一维 F q n $mathbb {F}_{q^n}$ 子空间的集合相同。如果我们把 U $U$ 看作 AG ( r , q n ) ${mathrm{AG}},(r,q^n)$ 的点集，这意味着 U $U$ 和 ⟨ U ⟩ F q d $langle U rangle _{mathbb {F}_{q^d}}$ 决定了同一组方向。当 n ∣ m $nmid m$ 时，我们会证明一个更有力的声明。

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引用次数: 0

Autoequivalences of blow-ups of minimal surfaces 极小曲面炸开的自等价性

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-08-08 DOI: 10.1112/blms.13131

Xianyu Hu, Johannes Krah

Let $� � X � �$ be the blow-up of $� � �_{P}^{�} � �$ in a finite set of very general points. We deduce from the work of Uehara [Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529–3547] that $� � X � �$ has only standard autoequivalences, no non-trivial Fourier–Mukai partners, and admits no spherical objects. If $� � X � �$ is the blow-up of $� � �_{P}^{�} � �$ in 9 very general points, we provide an alternate and direct proof of the corresponding statement. Further, we show that the same result holds if $� � X � �$ is a blow-up of finitely many points in a minimal surface of non-negative Kodaira dimension which contains no $� � � (� - � 2 �) � � �$ -curves. Independently, we characterize spherical objects on blow-ups of minimal surfaces of positive Kodaira dimension.

让 X $X$ 是 P C 2 $mathbb {P}^2_mathbb {C}$ 在一个有限的非常一般的点集中的炸开。我们从上原的研究[Trans. Amer. Math. Soc. 371 (2019), no. 5, 3529-3547]中推导出，X $X$只有标准的自等价性，没有非三维的傅立叶-穆凯伙伴，也不允许球面对象。如果 X $X$ 是 P C 2 $mathbb {P}^2_mathbb {C}$ 在 9 个非常一般的点上的炸开，我们提供了相应声明的另一种直接证明。此外，我们还证明了如果 X $X$ 是不包含 ( - 2 ) $(-2)$ 曲线的非负柯达伊拉维度的最小曲面中有限多个点的炸开，则同样的结果成立。另外，我们还描述了科代拉维度为正的极小曲面炸开后的球面对象的特征。

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引用次数: 0

Galois invariants of finite abelian descent and Brauer sets 有限阿贝尔后裔和布劳尔集的伽罗瓦不变式

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-08-01 DOI: 10.1112/blms.13130

Brendan Creutz, Jesse Pajwani, José Felipe Voloch

For a variety over a global field, one can consider subsets of the set of adelic points of the variety cut out by finite abelian descent or Brauer–Manin obstructions. Given a Galois extension of the ground field, one can consider similar sets over the extension and take Galois invariants. In this paper, we study under which circumstances the Galois invariants recover the obstruction sets over the ground field. As an application of our results, we study finite abelian descent and Brauer–Manin obstructions for isotrivial curves over function fields and extend results obtained by the first and last authors for constant curves to the isotrivial case.

对于全域上的一个综，我们可以考虑由有限阿贝尔后裔或布劳尔-马宁障碍切出的综的阿贝尔点集合的子集。给定基域的伽罗瓦扩展，我们可以考虑扩展上的类似集合，并求取伽罗瓦不变式。在本文中，我们将研究在什么情况下，伽罗瓦不变式可以恢复地域上的障碍集。作为我们结果的应用，我们研究了函数域上等差数列曲线的有限非比利亚后裔和布劳尔-马宁障碍，并将第一位和最后一位作者获得的常数曲线结果扩展到等差数列情况。

引用次数: 0

Local mirror symmetry via SYZ 通过 SYZ 实现局部镜像对称

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-08-01 DOI: 10.1112/blms.13126

Benjamin Gammage

In this note, we explain how mirror symmetry for basic local models in the Gross–Siebert program can be understood through the nontoric blowup construction described by Gross–Hacking–Keel. This is part of a program to understand the symplectic geometry of affine cluster varieties through their SYZ fibrations.

在本注释中，我们将解释如何通过格罗斯-哈金-基尔（Gross-Hacking-Keel）描述的非oric炸开构造来理解格罗斯-西伯特（Gross-Siebert）计划中基本局部模型的镜像对称性。这是通过仿射簇变体的 SYZ 纤维来理解其交映几何的计划的一部分。

引用次数: 0

The inertia bound is far from tight 惯性约束远远不够

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-07-30 DOI: 10.1112/blms.13127

Matthew Kwan, Yuval Wigderson

The inertia bound and ratio bound (also known as the Cvetković bound and Hoffman bound) are two fundamental inequalities in spectral graph theory, giving upper bounds on the independence number <math> <semantics> <mrow> <mi>α</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <annotation>$alpha (G)$</annotation> </semantics></math> of a graph <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> in terms of spectral information about a weighted adjacency matrix of <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math>. For both inequalities, given a graph <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math>, one needs to make a judicious choice of weighted adjacency matrix to obtain as strong a bound as possible. While there is a well-established theory surrounding the ratio bound, the inertia bound is much more mysterious, and its limits are rather unclear. In fact, only recently did Sinkovic find the first example of a graph for which the inertia bound is not tight (for any weighted adjacency matrix), answering a longstanding question of Godsil. We show that the inertia bound can be extremely far from tight, and in fact can significantly underperform the ratio bound: for example, one of our results is that for infinitely many <math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math>, there is an <math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math>-vertex graph for which even the unweighted ratio bound can prove <math> <semantics> <mrow> <mi>α</mi> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>⩽</mo> <mn>4</mn> <msup> <mi>n</mi> <mrow> <mn>3</mn> <mo>/</mo> <mn>4</mn> </mrow> </msup> </mrow> <annotation>$alpha (G)leqslant 4n^{3/4}$</annotation> </semantics></math>, but the inertia bound is always at least <math> <semantics> <mrow>

惯性约束和比值约束（又称 Cvetković 约束和 Hoffman 约束）是谱图理论中的两个基本不等式，根据 G $G$ 的加权邻接矩阵的谱信息给出了图 G $G$ 的独立性数 α ( G ) $alpha (G)$ 的上限。对于这两个不等式，给定一个图 G $G$ 时，需要明智地选择加权邻接矩阵，以获得尽可能强的约束。虽然围绕比值约束有一套成熟的理论，但惯性约束要神秘得多，其极限也相当不明确。事实上，直到最近，辛科维奇才找到了第一个惯性约束不严格（对于任何加权邻接矩阵）的图的例子，回答了戈德希尔长期以来提出的一个问题。我们的研究表明，惯性约束离严密可能相去甚远，而且事实上可能大大低于比率约束：例如，我们的结果之一是，对于无限多的 n $n$ ，存在一个 n $n$ -顶点图，对于该图，即使无权比率约束也能证明 α ( G ) ⩽ 4 n 3 / 4 $alpha (G)leqslant 4n^{3/4}$ ，但惯性约束总是至少 n / 4 $n/4$ 。这些结果特别解决了鲁尼、辛科维奇和沃奇扬-埃尔菲克-阿比阿德的问题。

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引用次数: 0

The sharp doubling threshold for approximate convexity 近似凸性的急剧加倍阈值

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-07-25 DOI: 10.1112/blms.13129

Peter van Hintum, Peter Keevash

We show for <math> <semantics> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊂</mo> <msup> <mi>R</mi> <mi>d</mi> </msup> </mrow> <annotation>$A,Bsubset mathbb {R}^d$</annotation> </semantics></math> of equal volume and <math> <semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>]</mo> </mrow> <annotation>$tin (0,1/2]$</annotation> </semantics></math> that if <math> <semantics> <mrow> <mrow> <mo>|</mo> <mi>t</mi> <mi>A</mi> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>B</mi> <mo>|</mo> <mo><</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mi>d</mi> </msup> <mo>)</mo> </mrow> <mrow> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> </mrow> <annotation>$|tA+(1-t)B|< (1+t^d)|A|$</annotation> </semantics></math>, then (up to translation) <math> <semantics> <mrow> <mo>|</mo> <mo>co</mo> <mo>(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo>)</mo> <mo>|</mo> <mo>/</mo> <mo>|</mo> <mi>A</mi> <mo>|</mo> </mrow> <annotation>$|operatorname{co}(Acup B)|/|A|$</annotation> </semantics></math> is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by

我们证明，对于 A , B ⊂ R d $A,Bsubset mathbb {R}^d$ 体积相等且 t∈ ( 0 , 1 / 2 ]$，如果 | t A + ( 1 - t ) 在 (0,1/2]$ 中，则 | t A + ( 1 - t ) = $。 $tin (0,1/2]$ that if | t A + ( 1 - t ) B | < ( 1 + t d ) | A | $|tA+(1-t)B|< (1+t^d)|A|$ ，则（直至平移） | co ( A ∪ B ) | / | A | $|operatorname{co}(Acup B)|/|A|$ 是有界的。这就确立了菲加里、范欣图姆和蒂巴最近建立的布鲁恩-明考斯基不等式定量稳定性的尖锐阈值，其证明使用了我们当前的结果。我们还为迭代和集建立了类似的尖锐临界值。

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引用次数: 0

Unstability problem of real analytic maps 实解析映射的不稳定性问题

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-07-22 DOI: 10.1112/blms.13124

Karim Bekka, Satoshi Koike, Toru Ohmoto, Masahiro Shiota, Masato Tanabe

As well known, the $� � �^{C � \infty �} � �$ stability of proper $� � �^{C � \infty �} � �$ maps is characterized by the infinitesimal $� � �^{C � \infty �} � �$ stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal $� � �^{C � ω �} � �$ stability does not imply $� � �^{C � ω �} � �$ stability; for instance, a Whitney umbrella $� � � �^{R � 2 �} � \to � �^{R � 3 �} � � �$ is not $� � �^{C � ω �} � �$ stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

众所周知，适当 C ∞ $C^infty$ 映射的 C ∞ $C^infty$ 稳定性是以无穷小 C ∞ $C^infty$ 稳定性为特征的。在本文中，我们研究了实解析背景下的对应关系。特别是，我们证明了无穷小 C ω $C^omega$ 稳定性并不意味着 C ω $C^omega$ 稳定性；例如，惠特尼伞 R 2 → R 3 $mathbb {R}^2 rightarrow mathbb {R}^3$ 不是 C ω $C^omega$ 稳定性。证明的一个主要工具是惠特尼解析近似定理的一个相对版本，它是通过 H. Cartan 的定理 A 和 B 来证明的。

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引用次数: 0

Solubility of additive forms of twice odd degree over totally ramified extensions of Q 2 $mathbb {Q}_2$ 在 Q2$mathbb {Q}_2$ 的完全夯实扩展上的两倍奇数度加法形式的可溶性

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-07-22 DOI: 10.1112/blms.13120

Drew Duncan

We prove that an additive form of degree $� � � d � = � 2 � m � � �$ , $� � m � �$ odd over any totally ramified extension of $� � �_{Q � 2 �} � �$ has a nontrivial zero if the number of variables $� � s � �$ satisfies $� � � s � ⩾ � \frac{�}{�^{d � 2 �}} � + � 3 � d � + � 1 � � �$ .

我们证明，如果变量数满足.

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引用次数: 0

The uniqueness of the Wiener–Hopf factorisation of Lévy processes and random walks 莱维过程和随机游走的维纳-霍普夫因式分解的唯一性

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society

Pub Date : 2024-07-17 DOI: 10.1112/blms.13112

Leif Döring, Mladen Savov, Lukas Trottner, Alexander R. Watson

We prove that the spatial Wiener–Hopf factorisation of a Lévy process or random walk without killing is unique.

我们证明，无杀的莱维过程或随机漫步的空间维纳-霍普夫因子化是唯一的。

引用次数: 0

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