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Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics 轴对称不可压缩粘性等离子体：全局拟然性和渐近性

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-10 DOI: 10.1017/fms.2024.60

Diogo Arsénio, Zineb Hassainia, Haroune Houamed

This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier–Stokes–Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light

$cin (c_0, infty )$

, for some threshold

$c_0>0$

depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime

$crightarrow infty $

and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting. The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler–Maxwell system to exploit the dissipative–dispersive structure of Maxwell’s system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime

$cto infty $

allows us to derive a robust nonlinear energy estimate which holds uniformly in c. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system. This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as

$cto infty $

本文致力于三维轴对称纳维-斯托克斯-麦克斯韦方程的全局分析。更确切地说，我们能够证明，对于光速的大值 $cin (c_0, infty )$ ，对于仅取决于初始数据的某个阈值 $c_0>0$，相关系统具有唯一的全局解。随之而来的解的边界与光速是一致的，这使得我们可以研究奇异机制 $crightarrow infty $，并严格推导出轴对称环境下的极限粘性磁流体动力学（MHD）系统。我们的证明策略借鉴了最近关于二维不可压缩的欧拉-麦克斯韦系统的结果，利用了麦克斯韦系统在轴对称环境下的耗散-分散结构。此外，通过对渐近机制 $cto infty $ 的详细分析，我们得出了在 c 中均匀成立的稳健非线性能量估计。这一系列结果似乎首次建立了三维粘性等离子体的全局好拟性，其中电场和磁场由完整的麦克斯韦方程支配，对于$cto infty $ 的大初始数据。

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引用次数: 0

Stability in the category of smooth mod-p representations of 光滑模-p 表示类别中的稳定性

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-09 DOI: 10.1017/fms.2024.37

Konstantin Ardakov, Peter Schneider

Let $p geq 5$ be a prime number, and let $G = {mathrm {SL}}_2(mathbb {Q}_p)$ . Let $Xi = {mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $mathcal {R} subset Xi times Xi $ denote the support of the pro-p Iwahori ${mathrm {Ext}}$ -algebra of G, viewed as a $(Z,Z)$ -bimodule. We show that the locally ringed space $Xi /mathcal {R}$ is a projective algebraic curve over ${mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $X

让 $p geq 5$ 是一个素数，让 $G = {mathrm {SL}}_2(mathbb {Q}_p)$ .让 $Xi = {mathrm {Spec}}(Z)$ 表示 G 的亲 p 岩崛-赫克代数的中心 Z 的谱，其系数在特征 p 的域 k 中。让 $mathcal {R} subset Xi times Xi $ 表示 G 的 pro-p Iwahori ${mathrm {Ext}}$ 代数的支持，看作 $(Z,Z)$ 双模块。我们证明了局部环形空间 $Xi /mathcal {R}$ 是一条在 ${mathrm {Spec}}(k)$ 上的投影代数曲线，它有两个连通分量，并且每个连通分量都是一条投影线链。对于 $Xi /mathcal {R}$ 的每个扎里斯基开放子集 U，我们都会为 G 的光滑 k 线性表示范畴构建一个稳定的局部化子范畴 $mathcal {L}_U$ 。

引用次数: 0

Bounds on multiplicities of symmetric pairs of finite groups 有限群对称对的乘数界值

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-09 DOI: 10.1017/fms.2024.58

Avraham Aizenbud, Nir Avni

Let $Gamma $ be a finite group, let $theta $ be an involution of $Gamma $ and let $rho $ be an irreducible complex representation of $Gamma $ . We bound ${operatorname {dim}} rho ^{Gamma ^{theta }}$ in terms of the smallest dimension of a faithful $mathbb {F}_p$ -representation of $Gamma /operatorname {mathrm {Rad}}_p(Gamma )$ , where p is any odd prime and $operatorname {mathrm {Rad}}_p(Gamma )$ is the maximal normal p-subgroup of $Gamma $ . This implies, in particular, tha

让 $Gamma $ 是一个有限群，让 $theta $ 是 $Gamma $ 的一个反卷，让 $rho $ 是 $Gamma $ 的一个不可还原复代表。我们将 ${operatorname {dim}的最小维度，其中 p 是任意奇素数，$operatorname {mathrm {Rad}}_p(Gamma )$ 是 $Gamma $ 的最大法向 p 子群。这就意味着，如果 $mathbf {G}$ 是一个在 $mathbb {Z}$ 上的群方案，并且 $theta $ 是 $mathbf {G}$ 的一个内卷，那么在 $C^infty left( mathbf {G}(mathbb {Z}_p)/ mathbf {G} 中的任何不可还原表征的多重性就是 $C^infty left( mathbf {G}(mathbb {Z}_p)/ mathbf {G})^{theta }(mathbb {Z}_p) right)$ 是有界的，在 p 中均匀分布。

引用次数: 0

Pressure of a dilute spin-polarized Fermi gas: Lower bound 稀释自旋极化费米气体的压力：下限

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-09 DOI: 10.1017/fms.2024.56

Asbjørn Bækgaard Lauritsen, Robert Seiringer

We consider a dilute fully spin-polarized Fermi gas at positive temperature in dimensions

$din {1,2,3}$

. We show that the pressure of the interacting gas is bounded from below by that of the free gas plus, to leading order, an explicit term of order

$a^drho ^{2+2/d}$

, where a is the p-wave scattering length of the repulsive interaction and

$rho $

is the particle density. The results are valid for a wide range of repulsive interactions, including that of a hard core, and uniform in temperatures at most of the order of the Fermi temperature. A central ingredient in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), pp. 237–260).

我们考虑了在正温度下维数为 $din {1,2,3}$的稀释全自旋极化费米气体。我们证明，相互作用气体的压力自下而上由自由气体的压力加上一个前导阶为 $a^drho ^{2+2/d}$ 的显式项限定，其中 a 是斥性相互作用的 p 波散射长度，$rho $ 是粒子密度。这些结果适用于广泛的斥力相互作用，包括硬核的斥力相互作用，并且在费米温度数量级的温度下是一致的。证明的一个核心要素是对高丁、吉莱斯皮和里普卡（Nucl.A，176.2 (1971)，第 237-260 页）。

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引用次数: 0

A proof of the Elliott–Rödl conjecture on hypertrees in Steiner triple systems 关于斯坦纳三重系统中肥大树的埃利奥特-罗德尔猜想的证明

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-06 DOI: 10.1017/fms.2024.34

Seonghyuk Im, Jaehoon Kim, Joonkyung Lee, Abhishek Methuku

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given

$mu>0$

, there exists

$n_0$

such that the following holds. Every n-vertex Steiner triple system contains all hypertrees with at most

$(1-mu )n$

vertices whenever

$ngeq n_0$

. We prove this conjecture.

超树是线性超图，其中每两个顶点都由一条唯一的路径连接。埃利奥特和罗德尔猜想，对于任何给定的 $mu>0$ ，都存在 $n_0$ ，使得以下条件成立。当 $ngeq n_0$ 时，每一个 n 个顶点的斯坦纳三重系统都包含最多有 $(1-mu)n$ 顶点的所有高树。我们将证明这一猜想。

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引用次数: 0

On the local -Bound of the Eisenstein series 关于爱森斯坦数列的局部束缚

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-06 DOI: 10.1017/fms.2024.59

Subhajit Jana, Amitay Kamber

We study the growth of the local

$L^2$

-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a poly-logarithmic bound on an average, for a large class of reductive groups. The method is based on Arthur’s development of the spectral side of the trace formula, and ideas of Finis, Lapid and Müller. As applications of our method, we prove the optimal lifting property for

$mathrm {SL}_n(mathbb {Z}/qmathbb {Z})$

for square-free q, as well as the Sarnak–Xue [52] counting property for the principal congruence subgroup of

$mathrm {SL}_n(mathbb {Z})$

of square-free level. This makes the recent results of Assing–Blomer [8] unconditional.

我们研究了数域上还原群的单元式爱森斯坦数列的局部 $L^2$ -项的增长，以其参数为基础。我们得出了一大类还原群的多对数平均约束。该方法基于阿瑟对迹线公式谱侧的发展，以及菲尼斯、拉皮德和缪勒的观点。作为我们方法的应用，我们证明了无平方q的$mathrm {SL}_n(mathbb {Z}/qmathbb {Z})$的最优提升性质，以及无平方级的$mathrm {SL}_n(mathbb {Z})$的主全等子群的萨尔纳克-薛[52]计数性质。这使得阿辛-布鲁默[8]的最新结果成为无条件的。

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引用次数: 0

Flags of sheaves, quivers and symmetric polynomials 剪切、四元组和对称多项式的标志

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-09-06 DOI: 10.1017/fms.2024.43

Giulio Bonelli, Nadir Fasola, Alessandro Tanzini

We study a quiver description of the nested Hilbert scheme of points on the affine plane and its higher rank generalization – that is, the moduli space of flags of framed torsion-free sheaves on the projective plane. We show that stable representations of the quiver provide an ADHM-like construction for such moduli spaces. We introduce a natural torus action and use equivariant localization to compute some of their (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials. From the physics viewpoint, the quivers we study describe a class of surface defects in four-dimensional supersymmetric gauge theories in terms of nested instantons.

我们研究了仿射平面上点的嵌套希尔伯特方案的簇描述及其高阶广义--即投影平面上有框无扭剪的旗的模空间。我们证明，簇的稳定表示为这类模量空间提供了类似于 ADHM 的构造。我们引入了一个自然的环作用，并使用等变局部化来计算它们的一些（虚拟）拓扑不变式，包括紧凑环曲面的情况。我们猜想，秩一的全形欧拉特征的生成函数是由等变权重多项式给出的，对于特定的数值类型，这些多项式与（修正的）麦克唐纳多项式重合。从物理学的角度来看，我们研究的震颤子以嵌套瞬子的形式描述了四维超对称规理论中的一类表面缺陷。

引用次数: 0

Persistent transcendental Bézout theorems 持久超越贝祖特定理

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-08-27 DOI: 10.1017/fms.2024.49

Lev Buhovsky, Iosif Polterovich, Leonid Polterovich, Egor Shelukhin, Vukašin Stojisavljević

An example of Cornalba and Shiffman from 1972 disproves in dimension two or higher a classical prediction that the count of zeros of holomorphic self-mappings of the complex linear space should be controlled by the maximum modulus function. We prove that such a bound holds for a modified coarse count inspired by the theory of persistence modules originating in topological data analysis.

科纳尔巴和希夫曼在 1972 年提出的一个例子推翻了在二维或更高维度上的一个经典预言，即复数线性空间全形自映射的零点计数应受最大模函数的控制。我们证明，在拓扑数据分析的持久性模块理论的启发下，修正后的粗计数也存在这样的约束。

引用次数: 0

Sums of squares, Hankel index and almost real rank 平方和、汉克尔指数和几乎实等级

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-05-30 DOI: 10.1017/fms.2024.45

Grigoriy Blekherman, Justin Chen, Jaewoo Jung

The Hankel index of a real variety X is an invariant that quantifies the difference between nonnegative quadrics and sums of squares on X. In [5], the authors proved an intriguing bound on the Hankel index in terms of the Green–Lazarsfeld index, which measures the ‘linearity’ of the minimal free resolution of the ideal of X. In all previously known cases, this bound was tight. We provide the first class of examples where the bound is not tight; in fact, the difference between Hankel index and Green–Lazarsfeld index can be arbitrarily large. Our examples are outer projections of rational normal curves, where we identify the center of projection with a binary form F. The Green–Lazarsfeld index of the projected curve is given by the complex Waring border rank of F [16]. We show that the Hankel index is given by the almost real rank of F, which is a new notion that comes from decomposing F as a sum of powers of almost real forms. Finally, we determine the range of possible and typical almost real ranks for binary forms.

在[5]中，作者用格林-拉扎斯菲尔德指数（Green-Lazarsfeld index）证明了关于汉克尔指数的一个有趣的约束，格林-拉扎斯菲尔德指数衡量的是 X 理想的最小自由解的 "线性度"。事实上，汉克尔指数和格林-拉扎斯菲尔德指数之间的差异可以是任意大的。我们的例子是有理法线曲线的外投影，我们用二元形式 F 确定投影中心。投影曲线的格林-拉扎斯菲尔德指数由 F 的复瓦林边界秩给出 [16]。我们证明，汉克尔指数由 F 的近实阶给出，这是将 F 分解为近实形式的幂和后得到的新概念。最后，我们确定了二元形式可能的和典型的近实阶范围。

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引用次数: 0

L-invariants for cohomological representations of PGL(2) over arbitrary number fields 任意数域上 PGL(2) 同调表示的 L 不变式

IF 1.7 2区数学 Q1 MATHEMATICS

Forum of Mathematics Sigma

Pub Date : 2024-05-30 DOI: 10.1017/fms.2024.51

Lennart Gehrmann, Maria Rosaria Pati

Let $pi $ be a cuspidal, cohomological automorphic representation of an inner form G of $operatorname {{PGL}}_2$ over a number field F of arbitrary signature. Further, let $mathfrak {p}$ be a prime of F such that G is split at $mathfrak {p}$ and the local component $pi _{mathfrak {p}}$ of $pi $ at $mathfrak {p}$ is the Steinberg representation. Assuming that the representation is noncritical at $mathfrak {p}$ , we construct automorphic $mathcal {L}$ -invariants for the representation <jats:tex-m

设 $pi $ 是任意签名数域 F 上 $operatorname {{PGL}}_2$ 的内形式 G 的一个尖顶同调自形表示。此外，让 $mathfrak {p}$ 是 F 的一个素数，使得 G 在 $mathfrak {p}$ 处分裂，并且 $pi $ 在 $mathfrak {p}$ 处的局部成分 $pi _{mathfrak {p}}$ 是 Steinberg 表示。假定这个表示在 $mathfrak {p}$ 处是非临界的，我们就为这个表示 $pi $ 构造自变$mathcal {L}$ -变量。如果数域 F 是全实数，我们证明这些自变$mathcal {L}$ -不变式与相关 p-adic 伽罗瓦表示的 Fontaine-Mazur $mathcal {L}$ -不变式是一致的。这将斯皮埃斯（Spieß）、罗索（Spieß respectively Rosso）和第一作者的最新成果从平行权重 2$ 的情况推广到了任意同调权重。

引用次数: 0

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