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Subrank and optimal reduction of scalar multiplications to generic tensors 将标量乘法简化为通用张量的子级和最优化
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-07-12 DOI: 10.1112/jlms.12963
Harm Derksen, Visu Makam, Jeroen Zuiddam
<p>The subrank of a tensor measures how much a tensor can be diagonalized. We determine this parameter precisely for essentially all (i.e., generic) tensors. Namely, we prove for generic tensors in <span></span><math> <semantics> <mrow> <mi>V</mi> <mo>⊗</mo> <mi>V</mi> <mo>⊗</mo> <mi>V</mi> </mrow> <annotation>$V otimes V otimes V$</annotation> </semantics></math> with <span></span><math> <semantics> <mrow> <mo>dim</mo> <mo>(</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mi>n</mi> </mrow> <annotation>$dim (V) = n$</annotation> </semantics></math> that the subrank is <span></span><math> <semantics> <mrow> <mi>Θ</mi> <mo>(</mo> <msqrt> <mi>n</mi> </msqrt> <mo>)</mo> </mrow> <annotation>$Theta (sqrt {n})$</annotation> </semantics></math>. Our result significantly improves on the previous upper bound from the work of Strassen (1991) and Bürgisser (1990) which was <span></span><math> <semantics> <msup> <mi>n</mi> <mrow> <mn>2</mn> <mo>/</mo> <mn>3</mn> <mo>+</mo> <mi>o</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <annotation>$n^{2/3+o(1)}$</annotation> </semantics></math>. Our result is tight up to an additive constant. Our full result covers not only 3-tensors but also <span></span><math> <semantics> <mi>k</mi> <annotation>$k$</annotation> </semantics></math>-tensors, for which we find that the generic subrank is <span></span><math> <semantics> <mrow> <mi>Θ</mi> <mo>(</mo> <msup> <mi>n</mi> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mrow> <annotation>$Thet
张量的子秩衡量了张量对角化的程度。我们为基本上所有(即泛型)张量精确地确定了这一参数。也就是说,我们证明了对于 dim ( V ) = n $dim (V) = n$ 的 V ⊗ V ⊗ V $V otimes V otimes V$ 中的一般张量,其子秩为 Θ ( n ) $Theta (sqrt {n})$ 。我们的结果大大改进了 Strassen (1991) 和 Bürgisser (1990) 之前的上限,即 n 2 / 3 + o ( 1 ) $n^{2/3+o(1)}$ 。我们的结果在一个可加常数范围内是严密的。我们的完整结果不仅涵盖了 3 张量,还涵盖了 k $k$ 张量,对于这些张量,我们发现其通用子等级为 Θ ( n 1 / ( k - 1 ) ) $Theta (n^{1/(k-1)})$ 。此外,作为应用,我们证明了该子秩在直接相加下不具有可加性。由于我们的结果,我们得到了子秩与张量方法之间的几个大的分离,这些方法最近受到了广泛关注,特别是切片秩(Tao,2016)、解析秩(Gowers-Wolf,2011;Lovett,2018;Bhrushundi-Harsha-Hatami-Kopparty-Kumar,2020)、几何秩(Kopparty-Moshkovitz-Zuiddam,2020)和 G 稳定秩(Derksen,2020)。我们对下界的证明依赖于一个新的技术结果,即把张量空间最优分解为结构子空间,我们认为这可能会引起独立的兴趣。
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引用次数: 0
Intersection matrices for the minimal regular model of X 0 ( N ) ${X}_0(N)$ and applications to the Arakelov canonical sheaf X 0 ( N ) ${X}_0(N)$ 最小正则模型的交集矩阵及其在阿拉克洛夫典范剪辑中的应用
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-07-10 DOI: 10.1112/jlms.12964
Paolo Dolce, Pietro Mercuri
<p>Let <span></span><math> <semantics> <mrow> <mi>N</mi> <mo>></mo> <mn>1</mn> </mrow> <annotation>$N&gt;1$</annotation> </semantics></math> be an integer coprime to 6 such that <span></span><math> <semantics> <mrow> <mi>N</mi> <mo>∉</mo> <mo>{</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>13</mn> <mo>}</mo> </mrow> <annotation>$Nnotin lbrace 5,7,13rbrace$</annotation> </semantics></math> and let <span></span><math> <semantics> <mrow> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <annotation>$g=g(N)$</annotation> </semantics></math> be the genus of the modular curve <span></span><math> <semantics> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> <annotation>$X_0(N)$</annotation> </semantics></math>. We compute the intersection matrices relative to special fibres of the minimal regular model of <span></span><math> <semantics> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> <annotation>$X_0(N)$</annotation> </semantics></math>. Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of <span></span><math> <semantics> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </mrow> <annotation>$X_0(N)$</annotation> </semantics></math> is asymptotic to <span></span><math> <semantics> <mrow> <mn>3</mn> <mi>g</mi> <mi>log</mi> <mi>N</mi> </mrow> <annotation>$3glog N$</annotation> </semantics></math>, for <span></span><math> <semantics> <mrow>
让 N > 1 $N&gt;1$是一个与 6 共乘的整数,使得 N ∉ { 5 , 7 , 13 }。 $Nnotin lbrace 5,7,13rbrace$ 并让 g = g ( N ) $g=g(N)$ 是模态曲线 X 0 ( N ) $X_0(N)$ 的属数。我们计算相对于 X 0 ( N ) $X_0(N)$ 最小正则模型的特殊纤维的交集矩阵。此外,我们还证明了在 N → + ∞ $Nrightarrow +infty$ 时,X 0 ( N ) $X_0(N)$ 的阿拉克洛夫(Arakelov)典范 Sheaf 的自交渐近于 3 g log N $3glog N$ 。
{"title":"Intersection matrices for the minimal regular model of \u0000 \u0000 \u0000 \u0000 X\u0000 0\u0000 \u0000 \u0000 (\u0000 N\u0000 )\u0000 \u0000 \u0000 ${X}_0(N)$\u0000 and applications to the Arakelov canonical sheaf","authors":"Paolo Dolce,&nbsp;Pietro Mercuri","doi":"10.1112/jlms.12964","DOIUrl":"https://doi.org/10.1112/jlms.12964","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N&amp;gt;1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be an integer coprime to 6 such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;∉&lt;/mo&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mn&gt;5&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;7&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;13&lt;/mn&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Nnotin lbrace 5,7,13rbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$g=g(N)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the genus of the modular curve &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$X_0(N)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We compute the intersection matrices relative to special fibres of the minimal regular model of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$X_0(N)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$X_0(N)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is asymptotic to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mi&gt;log&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$3glog N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spectral constant rigidity of warped product metrics 翘曲积度量的谱常数刚性
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-30 DOI: 10.1112/jlms.12958
Xiaoxiang Chai, Juncheol Pyo, Xueyuan Wan

A theorem of Llarull says that if a smooth metric g$g$ on the n$n$-sphere Sn$mathbb {S}^n$ is bounded below by the standard round metric and the scalar curvature Rg$R_g$ of g$g$ is bounded below by n(n1)$n (n - 1)$, then the metric g$g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound Rgn(n1)$R_g geqslant n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature Rg$R_g$. We utilize two methods: spinor and spacetime harmonic function.

拉鲁尔定理指出,如果 n $n$ -球面 S n $mathbb {S}^n$ 上的光滑度量 g $g$ 的下界是标准圆度量,并且 g $g$ 的标量曲率 R g $R_g$ 的下界是 n ( n - 1 ) $n (n - 1)$ ,那么度量 g $g$ 一定是标准圆度量。我们将 R g ⩾ n ( n - 1 ) $R_g geqslant n (n - 1)$ 约束替换为涉及拉普拉卡和标量曲率 R g $R_g$ 的椭圆算子的第一个特征值的下限,从而证明了谱拉鲁尔定理。我们采用两种方法:旋量和时空谐函数。
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引用次数: 0
Spectral large deviations of sparse random matrices 稀疏随机矩阵的谱大偏差
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-30 DOI: 10.1112/jlms.12954
Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam
<p>Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean <span></span><math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math>. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph <span></span><math> <semantics> <msub> <mi>G</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>p</mi> </mrow> </msub> <annotation>$mathcal {G}_{n,p}$</annotation> </semantics></math> equipped with independent and identically distributed (i.i.d.) edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when <span></span><math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math> is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case <span></span><math> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mi>d</mi> <mi>n</mi> </mfrac> </mrow> <annotation>$p = frac{d}{n}$</annotation> </semantics></math>, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph <span></span><math> <semantics> <msub> <mi>G</mi> <mrow> <mi>n</mi> <mo>,</mo> <mfrac> <mi>d</mi> <mi>n</mi> </mfrac> </mrow> </msub> <annotation>$mathcal {G}_{n,frac{d}{n}}$</annotation> </semantics></math> <i>without</i> edge-weights and with <i>Gaussian</i> edge-weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate <span></span><math> <semantics>
维格纳矩阵的特征值一直是研究的主要课题。这类随机矩阵的一个特别重要的子类是所谓的稀疏或稀释随机矩阵,其中维格纳矩阵的每个条目都乘以一个均值为 p $p$ 的独立伯努利随机变量,在许多应用中都非常有用。或者,这样的矩阵可以被看作是厄尔多斯-雷尼图 G n , p $mathcal {G}_{n,p}$ 的邻接矩阵,它配备了独立且同分布(i.i.d.)的边权重。最大特征值是一个特别值得关注的观测值。在本文中,我们将研究此类矩阵最大特征值的大偏差行为,这是多年来备受关注的一个课题。虽然我们已经针对 p $p$ 固定或可能随矩阵大小快速归零的情况设计了某些技术,但我们重点研究 p = d n $p = frac{d}{n}$ 的情况,即平均度恒定的稀疏性机制,由于它与统计力学和其他应用中的许多模型有关,因此是一个核心例子。大多数已知技术都会在这一机制中崩溃,甚至连此类随机矩阵频谱的典型行为都不甚了解。迄今为止,我们只知道 Erdős-Rényi 图 G n , d n $mathcal {G}_{n,frac{d}{n}$ 没有边权重和有高斯边权重的结果。在本文中,我们将考虑一般权重分布的影响。更具体地说,我们考虑的条目分布的尾部概率衰减率为 e - t α $e^{-t^alpha }$,α > 0 $alpha &gt;0$ ,其中 0 < α < 2 $0&lt;alpha &lt; 2$ 和 α > 2 $alpha &gt; 2$ 分别对应于比高斯尾部更重和更轻的尾部。虽然在许多自然环境中,大偏差行为预计会在很大程度上取决于入口分布,但我们建立了一个令人惊讶且罕见的普遍行为,表明当 α > 2 $alpha &gt; 2$ 时情况并非如此。相反,在 α < 2 $alpha &lt; 2$ 的情况下,大偏差率函数不再具有普遍性,而是由一个变异问题的解给出的,对该问题的描述涉及莫茨金-斯特劳斯定理(Motzkin-Straus theorem)的一般化,这是光谱图理论的一个经典结果。作为大偏差结果的副产品,我们还建立了最大特征值的大数定律行为,这似乎也是一个新问题,而且难以用现有方法获得。特别是,我们证明了最大特征值的典型值在 α = 2 $alpha = 2$ 时出现相变,即对应于高斯分布。
{"title":"Spectral large deviations of sparse random matrices","authors":"Shirshendu Ganguly,&nbsp;Ella Hiesmayr,&nbsp;Kyeongsik Nam","doi":"10.1112/jlms.12954","DOIUrl":"https://doi.org/10.1112/jlms.12954","url":null,"abstract":"&lt;p&gt;Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices, useful in many applications, are what are known as sparse or diluted random matrices, where each entry in a Wigner matrix is multiplied by an independent Bernoulli random variable with mean &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Alternatively, such a matrix can be viewed as the adjacency matrix of an Erdős–Rényi graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathcal {G}_{n,p}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; equipped with independent and identically distributed (i.i.d.) edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. While certain techniques have been devised for the case when &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is fixed or perhaps going to zero not too fast with the matrix size, we focus on the case &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$p = frac{d}{n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, that is, constant average degree regime of sparsity, which is a central example due to its connections to many models in statistical mechanics and other applications. Most known techniques break down in this regime and even the typical behavior of the spectrum of such random matrices is not very well understood. So far, results were known only for the Erdős–Rényi graph &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathcal {G}_{n,frac{d}{n}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; &lt;i&gt;without&lt;/i&gt; edge-weights and with &lt;i&gt;Gaussian&lt;/i&gt; edge-weights. In the present article, we consider the effect of general weight distributions. More specifically, we consider entry distributions whose tail probabilities decay at rate &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom 带有 CH 否定和弱强制公理的 C(X) 上的不连续同构
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-30 DOI: 10.1112/jlms.12956
Yushiro Aoki
<p>In this paper, I introduce the properties <span></span><math> <semantics> <msub> <mi>EPC</mi> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </msub> <annotation>$mathrm{EPC}_{aleph _1}$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>ProjCes</mi> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <annotation>$mathrm{ProjCes}(E)$</annotation> </semantics></math> for forcing notions and show that it is consistent that the forcing axiom for <span></span><math> <semantics> <mrow> <msub> <mi>EPC</mi> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </msub> <mo>+</mo> <mi>ProjCes</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <annotation>$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(E)$</annotation> </semantics></math> forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property <span></span><math> <semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> <annotation>$beta _1$</annotation> </semantics></math>. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set <span></span><math> <semantics> <mi>E</mi> <annotation>$E$</annotation> </semantics></math> is an example of an <span></span><math> <semantics> <mrow> <msub> <mi>EPC</mi> <msub> <mi>ℵ</mi> <mn>1</mn> </msub> </msub> <mo>+</mo> <mi>ProjCes</mi> <mrow> <mo>(</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>∖</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <annotation>$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(omega _1 setminus E)$</annotation> </se
在本文中我介绍了强制概念的性质 EPC ℵ 1 $mathrm{EPC}_{aleph _1}$ 和 ProjCes ( E ) $mathrm{ProjCes}(E)$ ,并证明 EPC ℵ 1 + ProjCes ( E ) $mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(E)$ 强制公理成立、连续性假设不成立,并且有元域的超幂有β 1 $beta _1$的性质。这就部分地解决了伍丁(H. Woodin)关于紧凑空间上所有复值连续函数的巴拿赫代数上存在不连续同态的问题。此外,我们还证明了在静态代价集 E $E$ 上梯形系统着色的均匀化是 EPC ℵ 1 + ProjCes ( ω 1 ∖ E ) $mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(omega _1 setminus E)$ 强迫概念的一个例子。作为推论,非自由怀特海群的存在是一致的,连续统假设也是失败的,而且有元域的超幂有β 1 $beta _1$的性质。
{"title":"Discontinuous homomorphisms on C(X) with the negation of CH and a weak forcing axiom","authors":"Yushiro Aoki","doi":"10.1112/jlms.12956","DOIUrl":"https://doi.org/10.1112/jlms.12956","url":null,"abstract":"&lt;p&gt;In this paper, I introduce the properties &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;EPC&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathrm{EPC}_{aleph _1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ProjCes&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathrm{ProjCes}(E)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for forcing notions and show that it is consistent that the forcing axiom for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;EPC&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;ProjCes&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(E)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; forcing notions holds, the continuum hypothesis fails, and an ultrapower of the field of reals has the property &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$beta _1$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. This provides a partial solution to H. Woodin's question concerning the existence of discontinuous homomorphisms on the Banach algebra of all complex-valued continuous functions on a compact space. Furthermore, we prove that the uniformization of a coloring of a ladder system on a stationary–costationary set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;annotation&gt;$E$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an example of an &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;EPC&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ℵ&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;ProjCes&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∖&lt;/mo&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathrm{EPC}_{aleph _1}+ mathrm{ProjCes}(omega _1 setminus E)$&lt;/annotation&gt;\u0000 &lt;/se","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Full Souslin trees at small cardinals 小红雀的全苏树林
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-28 DOI: 10.1112/jlms.12957
Assaf Rinot, Shira Yadai, Zhixing You

A κ$kappa$-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full κ$kappa$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal κ$kappa $. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be 3$aleph _3$ many full 2$aleph _2$-trees such that the product of any countably many of them is an 2$aleph _2$-Souslin tree.

如果κ $kappa$ -树的每个极限层都没有遗漏一个以上的潜在分支,那么这棵树就是完整的。库能(Kunen)问,一棵完整的κ $kappa$ -Souslin 树是否可能一直存在。谢拉给出了一个肯定的答案,即高度强极限马赫洛红心κ $kappa $ 。这里,我们证明了这些树在小红心时可能一直存在。事实上,可以有ℵ 3 $aleph _3$很多棵完整的ℵ 2 $aleph _2$-树,使得其中任意可数棵树的乘积都是一棵ℵ 2 $aleph _2$-苏林树。
{"title":"Full Souslin trees at small cardinals","authors":"Assaf Rinot,&nbsp;Shira Yadai,&nbsp;Zhixing You","doi":"10.1112/jlms.12957","DOIUrl":"https://doi.org/10.1112/jlms.12957","url":null,"abstract":"<p>A <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-tree is <i>full</i> if each of its limit levels omits no more than one potential branch. Kunen asked whether a full <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa $</annotation>\u0000 </semantics></math>. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>3</mn>\u0000 </msub>\u0000 <annotation>$aleph _3$</annotation>\u0000 </semantics></math> many full <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$aleph _2$</annotation>\u0000 </semantics></math>-trees such that the product of any countably many of them is an <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$aleph _2$</annotation>\u0000 </semantics></math>-Souslin tree.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12957","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves 量子计算中的萨尔纳克猜想、循环单元群角和志村曲线
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-25 DOI: 10.1112/jlms.12952
Colin Ingalls, Bruce W. Jordan, Allan Keeton, Adam Logan, Yevgeny Zaytman
<p>Sarnak's conjecture in quantum computing concerns when the groups <span></span><math> <semantics> <msub> <mo>PU</mo> <mn>2</mn> </msub> <annotation>$operatorname{PU}_{2}$</annotation> </semantics></math> and <span></span><math> <semantics> <msub> <mo>PSU</mo> <mn>2</mn> </msub> <annotation>$operatorname{PSU}_{2}$</annotation> </semantics></math> over cyclotomic rings <span></span><math> <semantics> <mrow> <mi>Z</mi> <mo>[</mo> <msub> <mi>ζ</mi> <mi>n</mi> </msub> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>]</mo> </mrow> <annotation>${mathbb {Z}}[zeta _{n}, 1/2]$</annotation> </semantics></math> with <span></span><math> <semantics> <mrow> <msub> <mi>ζ</mi> <mi>n</mi> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo>/</mo> <mi>n</mi> </mrow> </msup> </mrow> <annotation>$zeta _n=e^{2pi i/n}$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mn>4</mn> <mo>|</mo> <mi>n</mi> </mrow> <annotation>$4|n$</annotation> </semantics></math>, are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> has <span></span><math> <semantics> <mrow> <mo>corank</mo> <mi>G</mi> <mo>></mo> <mn>0</mn> </mrow> <annotation>$operatorname{corank}G&gt;0$</annotation> </semantics></math> only if <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary grou
量子计算中的萨纳克猜想涉及到当组 PU 2 $operatorname{PU}_{2}$ 和 PSU 2 $operatorname{PSU}_{2}$ 笼罩在环 Z [ ζ n , 1 / 2 ] 上时。 ${mathbb {Z}}[zeta _{n}, 1/2]$ with ζ n = e 2 π i / n $zeta _n=e^{2pi i/n}$ , 4 | n $4|n$ , 是由克利福德-环原子门集生成的。我们之前利用欧拉-庞加莱特性解决了这个问题。萨纳克猜想的一个推广问题是这些群何时由扭转元素生成。corank 对此提供了一个障碍:只有当 G $G$ 不是由扭转元素生成时,群 G $G$ 才有 corank G &gt; 0 $operatorname{corank}G&amp;gt;0$。在本文中,我们通过让这些单元群作用于布鲁哈特-提茨树(Bruhat-Tits tree),研究了这些单元群在 n = 2 s $n=2^s$ 和 n = 3 - 2 s $n={3cdot 2^s}$ ,n ⩾ 8 $ngeqslant 8$ 族中的角群。这种作用的商是有限图,其第一个贝蒂数是群的角。我们的主要结果是,对于 n = 2 s $n=2^s$ 和 n = 3 - 2 s $n=3cdot 2^s$ 这两个族,当 s →∞ $srightarrow infty$时,corank 在 s $s$ 中以双倍指数增长;而当 n = 8 , 12 , 16 , 24 $n= 8 , 12 , 16 , 24$ 时,corank 恰好为 0。我们用两种不同的方法给出了 corank 的明确下限。第一种方法是通过显式环切来约束树作用中的各向同性子群。第二种是将我们的图与 F n = Q ( ζ n ) 上的志村曲线联系起来。 + $F_n={mathbf {Q}}(zeta _n)^+$ 通过交换局部不变式并应用塞尔伯格和佐格拉夫的一个结果。我们证明循环论证给出了更强的边界。
{"title":"Sarnak's conjecture in quantum computing, cyclotomic unitary group coranks, and Shimura curves","authors":"Colin Ingalls,&nbsp;Bruce W. Jordan,&nbsp;Allan Keeton,&nbsp;Adam Logan,&nbsp;Yevgeny Zaytman","doi":"10.1112/jlms.12952","DOIUrl":"https://doi.org/10.1112/jlms.12952","url":null,"abstract":"&lt;p&gt;Sarnak's conjecture in quantum computing concerns when the groups &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;PU&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$operatorname{PU}_{2}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;PSU&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$operatorname{PSU}_{2}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; over cyclotomic rings &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ζ&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;${mathbb {Z}}[zeta _{n}, 1/2]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ζ&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;e&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mi&gt;π&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$zeta _n=e^{2pi i/n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$4|n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, are generated by the Clifford-cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;corank&lt;/mo&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{corank}G&amp;gt;0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; only if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;annotation&gt;$G$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary grou","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hofer–Zehnder capacity of disc tangent bundles of projective spaces 投影空间圆盘切线束的 Hofer-Zehnder 容量
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-25 DOI: 10.1112/jlms.12948
Johanna Bimmermann

We compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

我们计算任意维度的复投影空间和实投影空间的圆盘切线束的 Hofer-Zehnder 容量。圆盘束是相对于傅比尼-斯图迪(Fubini-Study)和圆度量而言的,但我们可以为任何其他度量获得明确的边界。在复投影空间中,我们还计算了磁扭曲情况下的 Hofer-Zehnder 容量,其中扭曲与 Fubini-Study 形式成正比。对于任意扭曲,我们仍然可以给出明确的上限。
{"title":"Hofer–Zehnder capacity of disc tangent bundles of projective spaces","authors":"Johanna Bimmermann","doi":"10.1112/jlms.12948","DOIUrl":"https://doi.org/10.1112/jlms.12948","url":null,"abstract":"<p>We compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12948","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141488421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Nonfree almost finite actions for locally finite-by-virtually Z ${mathbb {Z}}$ groups 局部有限虚Z ${mathbb {Z}}$ 群的无自由几乎有限作用
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-25 DOI: 10.1112/jlms.12959
Kang Li, Xin Ma

In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let α:GX$alpha:Gcurvearrowright X$ be a minimal action of a locally finite-by-virtually Z${mathbb {Z}}$ group G$G$ on the Cantor set X$X$. We prove that under certain assumptions, the action α$alpha$ is almost finite in measure if and only if α$alpha$ is essentially free. As an application, we obtain that any minimal topologically free action of a virtually Z${mathbb {Z}}$ group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property Γ$Gamma$ and Z$mathcal {Z}$-stability for their crossed product C$C^*$-algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

在本文中,我们将研究非自由作用的几乎有限性和几乎有限性度量。设 α : G ↷ X $alpha:Gcurvearrowright X$ 是局部有限虚数 Z ${mathbb {Z}}$ 群 G $G$ 在康托集 X $X$ 上的最小作用。我们证明,在某些假设条件下,当且仅当 α $alpha$ 本质上是自由的,作用 α $alpha$ 在度量上几乎是有限的。作为一个应用,我们得到,在具有小边界性质的无限紧凑可元空间上,虚Z ${mathbb {Z}}$ 群的任何最小自由拓扑作用都是几乎有限的。这是这个方向上第一个只假定拓扑自由性的一般性结果,这些结果引出了关于其交叉积 C∗ $C^*$ -代数的均匀性质Γ $Gamma$ 和 Z $mathcal {Z}$ -稳定性的新结果。本文提供了一些最小拓扑自由(但非自由)子移动的具体例子。
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引用次数: 0
On K-stability of P 3 $mathbb {P}^3$ blown up along a (2,3) complete intersection 论沿(2,3)完全交点炸开的 P 3 $mathbb {P}^3$ 的 K 稳定性
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-06-24 DOI: 10.1112/jlms.12961
Tiago Duarte Guerreiro, Luca Giovenzana, Nivedita Viswanathan

We prove K-stability of every smooth member of the Fano 3-fold family 2.15 of the Mori, Mukai and Iskovskikh classification.

我们证明了 Mori、Mukai 和 Iskovskikh 分类中法诺 3 折叠族 2.15 的每个光滑成员的 K 稳定性。
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引用次数: 0
期刊
Journal of the London Mathematical Society-Second Series
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