Advances in Mathematics最新文献

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Approximation properties for dynamical W⁎-correspondences 动态 W⁎ 对应关系的近似特性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-30 DOI: 10.1016/j.aim.2024.109958

K. De Commer, J. De Ro

Let

G

be a locally compact quantum group, and

A, B

von Neumann algebras on which

G

acts. We refer to these as

G

-dynamical W^⁎-algebras. We make a study of

G

-equivariant A-B-correspondences, that is, Hilbert spaces

H

with an A-B-bimodule structure by ⁎-preserving normal maps, and equipped with a unitary representation of

G

which is equivariant with respect to the above bimodule structure. Such structures are a Hilbert space version of the theory of

G

-equivariant Hilbert C^⁎-bimodules. We show that there is a well-defined Fell topology on equivariant correspondences, and use this to formulate approximation properties for them. Within this formalism, we then characterize amenability of the action of a locally compact group on a von Neumann algebra, using recent results due to Bearden and Crann. We further consider natural operations on equivariant correspondences such as taking opposites, composites and crossed products, and examine the continuity of these operations with respect to the Fell topology.

让 G 是一个局部紧密的量子群，A,B 是 G 作用于其上的 von Neumann 对象。我们称这些为 G-dynamical W⁎-gebras。我们研究了 G 的等变 A-B 对应，即通过⁎保留的法映射具有 A-B 双模块结构的希尔伯特空间 H，并配备了 G 的单元表示，该表示与上述双模块结构有关。这种结构是 G 等变希尔伯特 C⁎-双模理论的希尔伯特空间版本。我们证明在等变对应关系上存在定义明确的费尔拓扑，并以此为基础提出了等变对应关系的近似性质。在这一形式中，我们利用贝登和克兰恩的最新成果，描述了冯-诺依曼代数上局部紧凑群作用的可近似性。我们进一步考虑了等价对应的自然运算，如取对立面、复合和交叉积，并研究了这些运算在费尔拓扑学方面的连续性。

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引用次数: 0

Curvature bound for Lp Minkowski problem Lp Minkowski 问题的曲率约束

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-27 DOI: 10.1016/j.aim.2024.109959

Kyeongsu Choi , Minhyun Kim , Taehun Lee

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f, any solution to the

L_{p}

Minkowski problem in

R^{n + 1}

with

p \leq - n + 2

is a hypersurface of class

C^{1, 1}

. This is a sharp result because for each

p \in [- n + 2, 1)

there exists a convex hypersurface of class

C^{1, \frac{1}{n + p - 1}}

which is a solution to the

L_{p}

Minkowski problem for a positive smooth density f. In particular, the

C^{1, 1}

regularity is optimal in the case

p = - n + 2

which includes the logarithmic Minkowski problem in

R^{3}

我们建立了各向异性高斯曲率流的曲率估计。利用这一点，我们证明了给定一个具有正光滑密度 f 的度量 μ，Rn+1 中 p≤-n+2 的 Lp Minkowski 问题的任何解都是类 C1,1 的超曲面。这是一个尖锐的结果，因为对于每个 p∈[-n+2,1)，都存在一个 C1,1n+p-1 类的凸超曲面，它是正光滑密度 f 的 Lp Minkowski 问题的解。

引用次数: 0

Arithmetic Demailly approximation theorem 戴梅利算术近似定理

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-27 DOI: 10.1016/j.aim.2024.109961

Binggang Qu , Hang Yin

We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.

As an application, let

X / Q

be an integral projective variety and

\overline{N}

be an adelic line bundle on X. We prove that

ess (\overline{N}) \geq 0

⟹

\overline{N}

pseudo-effective. This was proved in [1], assuming

\overline{N}

relatively semipositive.

We show in the appendix that the above assertion is also true for adelic line bundles on quasi-projective varieties, under the framework of [17].

作为应用，假设 X/Q 是一个积分射影变项，N‾是 X 上的一个自立线束。我们在附录中证明，在[17]的框架下，上述论断对于准投影变体上的自立线束也是成立的。

引用次数: 0

Global rigidity of triangulated manifolds 三角流形的全局刚性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.aim.2024.109953

James Cruickshank , Bill Jackson , Shin-ichi Tanigawa

We prove that if G is the graph of a connected triangulated

(d - 1)

-manifold, for

d \geq 3

, then G is generically globally rigid in

R^{d}

if and only if it is

(d + 1)

-connected and, if

d = 3

, G is not planar. The special case

d = 3

verifies a conjecture of Connelly. Our results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. We also give two significant applications of our main theorems. We show that the characterisation of pseudomanifolds with extremal edge numbers given by the Lower Bound Theorem extends to circuits of the simplicial matroid. We also prove the generic case of a conjecture of Kalai concerning the reconstructability of a polytope from its space of stresses. The proofs of our main results adapt earlier ideas of Fogelsanger and Whiteley to the setting of global rigidity. In particular we verify a special case of Whiteley's vertex splitting conjecture for global rigidity.

我们证明，如果 G 是连通的三角形 (d-1)-manifold 的图，对于 d≥3，那么当且仅当 G 是 (d+1)-connected 时，G 在 Rd 中一般是全局刚性的，并且当 d=3 时，G 不是平面的。d=3 的特殊情况验证了康奈利的猜想。我们的结果实际上适用于更大的一类简单复数，即简单矩阵的回路。我们还给出了主要定理的两个重要应用。我们证明了下界定理给出的具有极值边数的伪曼折线的特征可以扩展到单纯 matroid 的回路。我们还证明了 Kalai 关于从应力空间重构多面体的猜想的一般情况。我们主要结果的证明将福格尔桑格和怀特利早先的想法改编成了全局刚性。我们特别验证了怀特利顶点分裂猜想在全局刚性方面的一个特例。

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引用次数: 0

Generalized Frank characterizations of Muckenhoupt weights and homogeneous ball Banach Sobolev spaces 穆肯霍普特权重和同质球巴纳赫索波列夫空间的广义弗兰克特性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.aim.2024.109957

Yirui Zhao, Yinqin Li, Dachun Yang, Wen Yuan, Yangyang Zhang

In this article, the authors first establish a new characterization of Muckenhoupt weights in terms of oscillations. As an application, the authors give a new characterization of homogeneous ball Banach Sobolev spaces, which extends the elegant characterization of Sobolev spaces obtained by R. L. Frank in 2024 and is a variant of the famous formula obtained by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung in 2024 with difference quotients replaced by oscillations. Moreover, the authors also obtain new representation formulae of gradients in terms of oscillations in ball Banach function spaces, which even include the critical case where Frank did not consider. Furthermore, via some counterexamples, we prove that all the main results are sharp. Applying these results, the authors further reveal the mutual equivalences among Muckenhoupt weights, the weighted upper estimate of the characterization of Frank, and the weighted upper estimate of the formula of Brezis et al.

在本文中，作者首先从振荡的角度建立了穆肯霍普特权重的新表征。作为应用，作者给出了同质球巴纳赫索波列夫空间的新表征，它扩展了弗兰克（R. L. Frank）在 2024 年获得的索波列夫空间的优雅表征，是布雷齐斯（H. Brezis）、西格（A. Seeger）、范沙夫廷根（J. Van Schaftingen）和容永（P.-L. Yung）在 2024 年获得的著名公式的变体，用振荡代替了差商。此外，作者还获得了球巴纳赫函数空间中以振荡表示的梯度的新表示公式，其中甚至包括弗兰克没有考虑的临界情况。此外，通过一些反例，我们证明了所有主要结果都是尖锐的。应用这些结果，作者进一步揭示了穆肯霍普特权重、弗兰克表征的加权上估计值和布雷齐斯等人公式的加权上估计值之间的相互等价性。

引用次数: 0

Correlation inequalities for linear extensions 线性扩展的相关不等式

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.aim.2024.109954

Swee Hong Chan , Igor Pak

We employ the combinatorial atlas technology to prove new correlation inequalities for the number of linear extensions of finite posets. These include the approximate independence of probabilities and expectations of values of random linear extensions, closely related to Stanley's inequality. We also give applications to the numbers of standard Young tableaux and to Euler numbers.

我们利用组合图集技术证明了有限正集线性扩展数的新相关不等式。这些不等式包括随机线性扩展的概率和期望值的近似独立性，与斯坦利不等式密切相关。我们还给出了标准扬台数和欧拉数的应用。

引用次数: 0

Deformations of Lagrangian NQ-submanifolds 拉格朗日NQ子曼形体的变形

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.aim.2024.109952

Miquel Cueca , Jonas Schnitzer

In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian NQ-submanifolds of degree n symplectic NQ-manifolds. Using Weinstein's Lagrangian tubular neighbourhood Theorem, we attach to every Lagrangian NQ-submanifold an

L_{\infty}

-algebra, which controls its deformation theory. The main examples are coisotropic submanifolds of Poisson manifolds and (higher) Dirac structures with support in (higher) Courant algebroids.

在本文中，我们证明了达布定理和温斯坦拉格朗日管状邻域定理的分级版本，以研究 n 度交映 NQ-manifolds的拉格朗日 NQ-submanifolds的变形理论。利用韦恩斯坦拉格朗日管状邻域定理，我们给每个拉格朗日 NQ 子曼形体附加了一个 L∞ 代数，这个代数控制着它的变形理论。主要的例子是泊松流形的各向同性子流形，以及在（高）库朗特实体中具有支持的（高）狄拉克结构。

引用次数: 0

Mating quadratic maps with the modular group III: The modular Mandelbrot set 将二次方程图与模态群结合起来 III：模态曼德布罗特集

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.aim.2024.109956

Shaun Bullett , Luna Lomonaco

We prove that there exists a homeomorphism χ between the connectedness locus

M_{Γ}

for the family

F_{a}

(2 : 2)

holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set

M_{1}

. The homeomorphism χ is dynamical (

F_{a}

is a mating between

P S L (2, Z)

and

P_{χ (a)}

), it is conformal on the interior of

M_{Γ}

, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces.

Following the recent proof by Petersen and Roesch that

M_{1}

is homeomorphic to the classical Mandelbrot set

M

, we deduce that

M_{Γ}

is homeomorphic to

M

我们证明了布尔利特和彭罗斯引入的 (2:2) 全形对应系 Fa 的连通性位置 MΓ 与抛物线曼德尔布罗特集 M1 之间存在同构关系 χ。同构 χ 是动态的（Fa 是 PSL(2,Z) 和 Pχ(a) 之间的配位），它在 MΓ 的内部是保角的，并扩展为各自一参数模空间中适当定义的邻域之间的同构。

引用次数: 0

The homological slice spectral sequence in motivic and Real bordism motivic 和 Real bordism 中的同调切片谱序列

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.aim.2024.109955

Christian Carrick , Michael A. Hill , Douglas C. Ravenel

<div>For a motivic spectrum <math><mi>E</mi><mo>∈</mo><mrow><mi>SH</mi></mrow><mo>(</mo><mi>k</mi><mo>)</mo></math>, let <math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math> denote the global sections spectrum, where E is viewed as a sheaf of spectra on <math><msub><mrow><mi>Sm</mi></mrow><mrow><mi>k</mi></mrow></msub></math>. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of <math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math>. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of <math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math> and study the case <math><mi>E</mi><mo>=</mo><mi>B</mi><mi>P</mi><mi>G</mi><mi>L</mi><mo>〈</mo><mi>m</mi><mo>〉</mo></math> for <math><mi>k</mi><mo>=</mo><mi>R</mi></math> in detail. We show that this spectral sequence contains the <math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math>-comodule algebra <math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> as permanent cycles, and we determine a family of differentials interpolating between <math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. Using this, we compute the spectral sequence completely for <math><mi>m</mi><mo>≤</mo><mn>3</mn></math>.In the height 2 case, the Betti realization of <math><mi>B</mi><mi>P</mi><mi>G</mi><mi>L</mi><mo>〈</mo><mn>2</mn><mo>〉</mo></math> is the <math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-spectrum <math><mi>B</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>〈</mo><mn>2</mn><mo>〉</mo></math>, a form of which was shown by Hill and Meier to be an equivariant model for <math><msub><mrow><mi>tmf</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math>. Our spectral sequence therefore gives a computation of the comodule algebra <math><msub><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mi>tmf</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math>. As a consequence, we deduce a new (2-local) Wood-t

对于一个动机谱 E∈SH(k)，让Γ(E) 表示全局剖面谱，其中 E 被视为 Smk 上的一个谱片。Voevodsky 的切片滤波决定了收敛于 Γ(E) 同调群的谱序列。在本文中，我们引入了收敛于 Γ(E) 的 mod 2 同调的谱序列，并详细研究了 k=R 时 E=BPGL〈m〉的情况。我们证明这个谱序列包含作为永久循环的 A⁎-omodule 代数 A⁎□A(m)⁎F2，并确定了介于 A⁎□A(0)⁎F2 和 A⁎□A(m)⁎F2 之间的微分族。在高度 2 的情况下，BPGL〈2〉的贝蒂实现是 C2 谱 BPR〈2〉，希尔和迈尔证明了它的一种形式是 tmf1(3) 的等变模型。因此，我们的谱序列给出了逗点代数 H⁎tmf0(3)的计算结果。因此，我们推导出了戴维斯和马霍瓦尔德预测的 tmf 模块的新的（2-局部）伍德型分裂 tmf∧X≃tmf0(3) ，X 是某个 10 单元复数。

引用次数: 0

Growth of polynomials on arcs in the complex plane 复平面内弧上多项式的增长

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.aim.2024.109940

Annie R. Wei

We prove that the growth rate of polynomials on an arc in the complex plane is exponential in its degree and can be computed by a linear program.

我们证明，复平面内弧上多项式的增长率与弧度成指数关系，并可通过线性程序计算。

引用次数: 0

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