Advances in Mathematics最新文献

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Integral p-adic non-abelian Hodge theory for small representations 小表征的积分 p-adic 非阿贝尔霍奇理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.aim.2024.109950

Yu Min , Yupeng Wang

Let $X$ be a smooth p-adic formal scheme over $O_{C}$ with rigid generic fiber X. In this paper, we construct a new integral period sheaf $O {\hat{C}}_{pd}^{+}$ on $X_{pro \overset{´}{e} t}$ and use it to establish an integral p-adic Simpson correspondence for small ${\hat{O}}_{X}^{+}$ -representations on $X_{pro \overset{´}{e} t}$ and small Higgs bundles on $X_{\overset{´}{e} t}$ , which recovers rational p-adic Simpson correspondence for small coefficients after inverting p (at least in the good reduction case). Moreover, for a small ${\hat{O}}_{X}^{+}$ -representation $L$ with induced Higgs bundle $(H, θ_{H})$ , we provide a natural morphism $HIG (H, θ_{H}) \to R ν_{⁎} L$ with a bounded $p^{\infty}$ -torsion cofiber. Finally, we shall use this natural map to study an analogue of Deligne–Illusie decomposition with coefficients in small ${\hat{O}}_{X}^{+}$ -representations.

在本文中，我们在 Xproe´t 上构造了一个新的积分周期舍弗 OCˆpd+，并用它为 Xproe´t 上的小 OˆX+ 表示和 Xe´t 上的小希格斯束建立了一个积分 p-adic Simpson 对应关系，在反转 p 之后（至少在良好的还原情况下）恢复了小系数的理性 p-adic Simpson 对应关系。此外，对于具有诱导希格斯束（H,θH）的小 OˆX+ 表示 L，我们提供了一个具有有界 p∞ 扭转共纤的自然态射 HIG(H,θH)→Rν⁎L。最后，我们将利用这一自然映射来研究德利涅-伊卢西分解的一个类似方法，其系数为小 OˆX+ 表示。

引用次数: 0

Goerss–Hopkins obstruction theory for ∞-categories ∞类的戈尔斯-霍普金斯阻塞理论

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.aim.2024.109951

Aaron Mazel-Gee

Goerss–Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model ∞-categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable ∞-category (such as that of equivariant spectra or of motivic spectra).

戈尔斯-霍普金斯阻塞理论是从纯代数数据构建结构化环谱的有力工具。利用模型∞范畴的形式主义，我们提供了一种适用于任意现对称单稳态∞范畴（如等变谱或动机谱范畴）的广义方法。

引用次数: 0

Weighted anisotropic isoperimetric inequalities and existence of extremals for singular anisotropic Trudinger-Moser inequalities 加权各向异性等周不等式和奇异各向异性特鲁丁格-莫泽不等式的极值存在性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.aim.2024.109949

Guozhen Lu , Yansheng Shen , Jianwei Xue , Maochun Zhu

<div>In this paper, we establish a class of isoperimetric inequalities on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> with respect to weights which are negative powers of the distance to the origin associated with the Finsler metric. (See Theorem 1.1.) Based on these weighted anisotropic isoperimetric inequalities, we can classify a class of singular Liouville's equation associated with the n-Finsler-Laplacian (1.10) and construct a blow-up sequence to show the existence of extremals for the singular Trudinger-Moser inequality involving the anisotropic Dirichlet norm:<math><munder><mi>sup</mi><mrow><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>,</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>F</mi><msup><mrow><mo>(</mo><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>≤</mo><mn>1</mn></mrow></munder><mo>⁡</mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></msup><msup><mrow><mi>F</mi></mrow><mrow><mo>∘</mo></mrow></msup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>β</mi></mrow></msup><mi>d</mi><mi>x</mi><mo><</mo><mo>∞</mo></math> for any <math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></math>, where <math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> is a smooth and bounded domain containing the origin, and <math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mi>β</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup></math>. Here F is a convex function, which is even and positively homogeneous of degree 1, and its polar <math><msup><mrow><mi>F</mi></mrow><mrow><mo>∘</mo></mrow></msup></math> represents a Finsler metric on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is the Lebesgue measure of the unit Wulff ball.The presence of the weight in Theorem 1.1 adds significant difficulties because

在本文中，我们在 Rn 上建立了一类等周不等式，其权重是与芬斯勒度量相关的到原点距离的负幂次。(见定理 1.1。）基于这些加权各向异性等周不等式，我们可以划分出一类与 n-Finsler 拉普拉斯相关的奇异 Liouville 方程 (1. 10)，并构造出一个吹胀序列。10），并构造一个吹胀序列来证明涉及各向异性狄利克特规范的奇异特鲁丁格-莫泽不等式的极值存在：supu∈W01,n(Ω),∫ΩF(∇u)ndx≤1∫Ωeλn,β|u|nn-1F∘(x)-βdx<∞ 对于任意 β∈(0,n)，其中 Ω⊂Rn 是包含原点的光滑有界域，且 λn,β:=(n-βn)nnn-1κn1n-1。这里 F 是一个凸函数，它是阶数为 1 的偶次正均质函数，其极点 F∘ 表示 Rn 上的 Finsler 度量，κn 是单位 Wulff 球的 Lebesgue 度量。由于缺乏与权重函数适当的对称性原理，定理 1.1 中权重的存在增加了很大的困难。为此，我们采用了一种与芬斯勒度量相关的新型准共形映射来处理这个权重。定理 1.1 对于划分与 n-Finsler 拉普拉奇（1.10）相关的一类奇异 Liouville 方程的解至关重要。(见定理 4.1。）这一分类在建立奇异各向异性特鲁丁格-莫泽不等式的极值函数的存在性方面起着重要作用。(见定理 1.2）。

引用次数: 0

A descent basis for the Garsia-Procesi module 加西亚-普罗切西模块的下降基础

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-17 DOI: 10.1016/j.aim.2024.109945

Erik Carlsson, Raymond Chou

We assign to each Young diagram λ a subset $B_{λ^{'}}^{maj}$ of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module $R_{λ}$ , whose graded character is the Hall-Littlewood polynomial ${\tilde{H}}_{λ} [X; t]$ [14], [10], [29]. This basis is a major index analogue of the basis $B_{λ} \subset R_{λ}$ defined by certain recursions, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra $R_{n}$ , which in fact corresponds to the case when $λ = 1^{n}$ . By anti-symmetrizing a subset of this basis with respect to the corresponding Young subgroup under the Springer action, we obtain a basis in the parabolic case, as well as a corresponding formula for the expansion of ${\tilde{H}}_{λ} [X; t]$ . Despite a similar appearance, it does not appear obvious how to connect the formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q = 0$ .

我们为每个杨图 λ 指定了一个加西亚-斯坦顿下降单项式集合的子集 Bλ′maj ，并证明它决定了加西亚-普罗切西模块 Rλ 的一个基，而 Rλ 的级数特征是霍尔-利特尔伍德多项式 H˜λ[X;t] [14], [10], [29]。这个基是某些递归定义的基 Bλ⊂Rλ 的大指数类似物，就像下降基与共变代数 Rn 的阿廷基的关系一样，实际上对应于 λ=1n 的情况。通过反对称该基的一个子集与斯普林格作用下的相应杨子群，我们得到了抛物线情况下的基，以及 H˜λ[X;t]的相应展开式。尽管表面相似，但如何将这些公式与哈格伦德、海曼和卢尔在 q=0 时的修正麦克唐纳公式的特殊化联系起来，似乎并不明显。

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引用次数: 0

Virtual linearity for KPP reaction-diffusion equations KPP 反应扩散方程的虚拟线性关系

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-17 DOI: 10.1016/j.aim.2024.109948

Andrej Zlatoš

We show that long time solution dynamic for general reaction-advection-diffusion equations with KPP reactions is virtually linear in the following sense. Its leading order depends on the non-linear reaction only through its linearization at $u = 0$ , and it can also be recovered for general initial data by instead solving the PDE for restrictions of the initial condition to unit cubes on $R^{d}$ (the latter means that non-linear interaction of these restricted solutions has only lower order effects on the overall solution dynamic). The result holds under a uniform bound on the advection coefficient, which we show to be sharp. We also extend it to models with non-local diffusion and KPP reactions.

我们的研究表明，具有 KPP 反应的一般反应-平流-扩散方程的长时间动态解在以下意义上几乎是线性的。其前导阶仅通过 u=0 处的线性化而依赖于非线性反应，而且对于一般初始数据，也可以通过求解初始条件对 Rd 上单位立方体的限制的 PDE 来恢复（后者意味着这些限制解的非线性相互作用对整体解动态仅有低阶影响）。这一结果在平流系数的统一约束下成立，我们证明这一约束是尖锐的。我们还将其扩展到具有非局部扩散和 KPP 反应的模型。

引用次数: 0

The non-resonant bilinear Hilbert-Carleson operator 非共振双线性希尔伯特-卡列松算子

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-16 DOI: 10.1016/j.aim.2024.109939

Cristina Benea , Frédéric Bernicot , Victor Lie , Marco Vitturi

<div>In this paper we introduce the class of bilinear Hilbert-Carleson operators <math><msub><mrow><mo>{</mo><mi>B</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msup><mo>}</mo></mrow><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></msub></math> defined by<math><mi>B</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msup><mo>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mi>sup</mi><mrow><mi>λ</mi><mo>∈</mo><mi>R</mi></mrow></munder><mo>⁡</mo><mo>|</mo><mo>∫</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi>t</mi><mo>)</mo><mspace></mspace><mi>g</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>t</mi><mo>)</mo><mspace></mspace><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mi>a</mi></mrow></msup></mrow></msup><mspace></mspace><mfrac><mrow><mi>d</mi><mi>t</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>|</mo></math> and show that in the non-resonant case <math><mi>a</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo><mo>∖</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></math> the operator <math><mi>B</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msup></math> extends continuously from <math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo><mo>×</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math> into <math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math> whenever <math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>q</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac></math> with <math><mn>1</mn><mo><</mo><mi>p</mi><mo>,</mo><mspace></mspace><mi>q</mi><mo>≤</mo><mo>∞</mo></math> and <math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo><</mo><mi>r</mi><mo><</mo><mo>∞</mo></math>.A key novel feature of these operators is that – in the non-resonant case – <math><mi>B</mi><msup><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msup></math> has a hybrid nature enjoying both<ul><li>(I)zero curvature features inherited from the modulation invariance property of the classical bilinear Hilbert transform (BHT), and</li><li>(II)non-zero curvature features arising from the Carleson-type operator with nonlinear phase <math><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mi>a</mi></mrow></msup></math>.</li></ul> In order to simultaneously control these two competing facets of our operator we develop a two-resolution approach:<ul><li>•A low resolution, multi-scale an

本文介绍了双线性希尔伯特-卡莱森算子 {BCa}a>0 类，其定义为：BCa(f,g)(x)：=supλ∈R|∫f(x-t)g(x+t)eiλtadtt|，并证明在非共振情况下 a∈(0,∞)∖{1,2}，只要 1p+1q=1r 且 1<p,q≤∞ 和 23<r<∞，算子 BCa 就从 Lp(R)×Lq(R) 连续扩展到 Lr(R)。这些算子的一个关键新特征是--在非共振情况下--BCa 具有混合性质，同时享有（I）经典双线性希尔伯特变换（BHT）的调制不变性所继承的零曲率特征和（II）具有非线性相位 λta 的卡莱森型算子所产生的非零曲率特征。为了同时控制算子的这两个相互竞争的方面，我们开发了一种双分辨率方法：-针对(I)的低分辨率多尺度分析，依赖于将 BCa 的时频离散化为 "扩张的 "相空间 BHT 类肖像的合适版本。由此产生的分解将产生秩一的三方格{Pm}m族，使得任何此类三方格的分量不再具有区域一海森堡定位。对这些族的控制将通过[35]和[36]中介绍的时频方法的改进来实现。-针对（II）的高分辨率、单一尺度分析，依赖于将每个三面体 P∈Pm 进一步离散化为四参数的三面体 S(P)族，由此产生的每个三面体 s∈S(P) 现在都服从区域一海森堡定位。后面这些族的设计以及每个给定 P 内乘法器相位非零曲率中编码的消除提取，都依赖于 [41] 中介绍的 LGC 方法。我们工作的另一个有趣之处在于，高分辨率分析本身涉及两种类型的分解，以捕捉算子的局部（单一尺度）行为：-连续相位线性化空间模型，作为从乘法器相位中提取抵消的载体。后者通过 TT⁎参数、数论工具（Weyl 和）和利用时频相关性的相位水平集分析来实现。-离散相位线性化波包模型，采用刚刚捕获的相位抵消，并将其输入低分辨率分析，以实现对 BCa 的全局控制。

引用次数: 0

Strong law of large numbers for generalized operator means 广义算子手段的强大数定律

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.aim.2024.109933

Zoltán Léka , Miklós Pálfia

Sturm's strong law of large numbers in $CAT (0)$ spaces and in the Thompson metric space of positive invertible operators is not only an important theoretical generalization of the classical strong law but also serves as a root-finding algorithm in the spirit of a proximal point method with splitting. It provides an easily computable stochastic approximation based on inductive means. The purpose of this paper is to extend Sturm's strong law and its deterministic counterpart, known as the “nodice” version, to unique solutions of nonlinear operator equations that generate exponentially contracting ODE flows in the Thompson metric. This includes a broad family of so-called generalized (Karcher) operator means introduced by Pálfia in 2016. The setting of the paper also covers the framework of order-preserving flows on Thompson metric spaces, as investigated by Gaubert and Qu in 2014, and provides a generally applicable resolvent theory for this setting.

斯特姆在 CAT(0) 空间和正可逆算子的汤普森度量空间中的强大数定律不仅是对经典强定律的重要理论概括，而且还是一种具有分裂精神的近点法的寻根算法。它基于归纳法提供了一种易于计算的随机近似方法。本文的目的是将 Sturm 强定律及其被称为 "nodice "版本的确定性对应定律扩展到在汤普森度量中产生指数收缩 ODE 流的非线性算子方程的唯一解。这包括帕尔菲亚在 2016 年提出的所谓广义（卡尔希尔）算子手段的广泛系列。论文的设定还涵盖了汤普森度量空间上的保阶流框架，正如高伯特和瞿秋白在 2014 年所研究的那样，并为这一设定提供了普遍适用的解析理论。

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引用次数: 0

Subspace concentration of zonoids and a sharp Minkowski mixed volume inequality zonoids 的子空间集中和尖锐的 Minkowski 混合体积不等式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.aim.2024.109934

Qiang Sun, Ge Xiong

The mixed volume measure $V_{K_{1}, \dots, K_{n}}$ of compact convex sets $K_{1}, \dots, K_{n}$ in $R^{n}$ is the localization of the classic Minkowski mixed volume $V (K_{1}, \dots, K_{n})$ and is one of the generalizations of the important cone-volume measure. When $K_{1}, \dots, K_{n - 1}$ are zonotopes and $K_{n}$ is a convex body or $K_{1}, \dots, K_{n - 1}, K_{n}$ are zonoids in $R^{n}$ , the subspace concentration of $V_{K_{1}, \dots, K_{n}}$ is proved. As applications, a subspace concentration phenomenon for quermassintegrals is revealed and a sharp affine isoperimetric inequality for the Minkowski mixed volume is established.

Rn中紧凑凸集K1,...,Kn的混合体积度量VK1,...,Kn是经典的闵科夫斯基混合体积V(K1,...,Kn)的局部化，是重要的锥体积度量的广义化之一。当 K1，...，Kn-1 是众凸体且 Kn 是凸体或 K1，...，Kn-1，Kn 是 Rn 中的众凸体时，证明了 VK1，...，Kn 的子空间集中。作为应用，揭示了量子整数的子空间集中现象，并建立了明考斯基混合体积的尖锐仿射等周不等式。

引用次数: 0

Density of exponentials and Perron-Frobenius operators 指数密度和佩伦-弗罗贝尼斯算子

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.aim.2024.109932

Somnath Ghosh , Debkumar Giri

<div>In this article, we study the weak-star density of the linear span of the trigonometric functions<math><mrow><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>π</mi><mi>i</mi><mo>(</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>n</mi><mi>y</mi><mo>)</mo></mrow></msup><mo>,</mo><mspace></mspace><msubsup><mrow><mi>e</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow><mrow><mo><</mo><mi>β</mi><mo>></mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>π</mi><mi>i</mi><mi>β</mi><mo>(</mo><mi>m</mi><mo>/</mo><mi>x</mi><mo>+</mo><mi>n</mi><mo>/</mo><mi>y</mi><mo>)</mo></mrow></msup><mo>;</mo><mspace></mspace></mrow><mspace></mspace><mrow><mspace></mspace><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mo>}</mo></mrow></math> for a positive real β. We aim to extend the results of Hedenmalm and Montes-Rodríguez (2011) [18] and Canto-Martín, Hedenmalm, and Montes-Rodríguez (2014) [8] in the plane. They have extensively studied the weak-star completeness of the hyperbolic trigonometric system in <math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math>. This is the dual formulation of the Heisenberg uniqueness pair for (hyperbola, certain lattice-cross).As in their work, <math><mi>β</mi><mo>=</mo><mn>1</mn></math> turns out to be the critical value. In particular, one of our main results asserts that the space spanned by the aforesaid trigonometric functions is weak-star dense in <math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math> of the set <math><msub><mrow><mi>Θ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>β</mi></mrow></msub><mo>=</mo><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>∪</mo><msup><mrow><mo>(</mo><mi>R</mi><mo>∖</mo><mo>(</mo><mo>−</mo><mi>β</mi><mo>,</mo><mi>β</mi><mo>]</mo><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> if and only if <math><mn>0</mn><mo><</mo><mi>β</mi><mo>≤</mo><mn>1</mn></math>, and the corresponding pre-annihilator space has finite dimension whenever <math><mi>β</mi><mo>></mo><mn>1</mn></math>. However, for <math><mi>β</mi><mo>></mo><mn>1</mn></math>, the pre-annihilator space can be made infinite-dimensional by allowing functions with slightly bigger support than <math><msub><mrow><mi>Θ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>β</mi></mrow></msub></math>. To be precise, let <math><msubsup><mrow><mi>Θ</mi></mrow><mrow><mi>β</mi></mrow><mrow><mo>″</mo></mrow></msubsup><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo

本文研究正实数 β 时三角函数{em,n(x,y)=eπi(mx+ny),em,n<β>(x,y)=eπiβ(m/x+n/y);m,n∈Z}线性跨度的弱星密度。我们的目标是在平面上扩展 Hedenmalm 和 Montes-Rodríguez (2011) [18] 以及 Canto-Martín、Hedenmalm 和 Montes-Rodríguez (2014) [8] 的成果。他们广泛研究了 L∞(R)中双曲三角系统的弱星完备性。与他们的研究一样，β=1 被证明是临界值。特别是，我们的主要结果之一断言，当且仅当 0<β≤1 时，上述三角函数所跨越的空间在集合 Θ1,β=(-1,1]2∪(R∖(-β,β])2 的 L∞ 中是弱星密集的，并且当 β>1 时，相应的前平稳器空间具有有限维。然而，对于 β>1，可以通过允许支持度比Θ1,β 稍大的函数来使前咝声空间无限维。确切地说，让 Θβ″⊆R2∖Θ1,β 使得 (-β,β]2∩Θβ″ 具有正的 Lebesgue 度量。我们证明，当 m,n 在 Z 上变化时，当 β>1 时，em,n 和 em,n<β> 的线性跨度的弱星闭包在 L∞(Θ1,β∪Θβ″)中具有无限的编码维度。我们的证明是通过分析二维高斯型映射及其相应的 Perron-Frobenius 算子进行的。

引用次数: 0

Regularizable collinear periodic solutions in the n-body problem with arbitrary masses 具有任意质量的 n 体问题中可规整的共线周期解

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.aim.2024.109947

Guowei Yu

For n-body problem with arbitrary positive masses, we prove there are regularizable collinear periodic solutions for any ordering of the masses, going from a simultaneous binary collision to another in half of a period with half of the masses moving monotonically to the right and the other half monotonically to the left. When the masses satisfy certain equality condition, the solutions have extra symmetry. This also gives a new proof of the existence of Schubart orbit, when $n = 3$ .

对于具有任意正质量的 n 体问题，我们证明了对于质量的任意排序都存在可正则化的碰撞周期解，从同时发生的二元碰撞到另一组碰撞只需半个周期，其中一半质量单调地向右移动，另一半质量单调地向左移动。当质量满足某些相等条件时，解具有额外的对称性。这也给出了 n=3 时舒巴特轨道存在的新证明。

引用次数: 0

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