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Persistent homology for functionals 函数的持久同源性
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-12-30 DOI: 10.1142/s0219199723500554
Ulrich Bauer, Anibal M. Medina-Mardones, Maximilian Schmahl

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

我们引入了一大类函数的拓扑条件，确保其相关子级集滤波的持久同调模块承认持久图，这尤其意味着它们满足广义莫尔斯不等式。我们将莫尔斯和汤普金斯给出的不稳定最小面定理的原始证明重铸在一个严谨的现代框架中，以此说明这些结果的适用性。
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引用次数: 0
Fractional anisotropic Calderón problem on complete Riemannian manifolds 完整黎曼流形上的分数各向异性卡尔德龙问题
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-12-09 DOI: 10.1142/s0219199723500578
Mourad Choulli, El Maati Ouhabaz

We prove that the metric tensor g of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator Δg. Our result holds under the condition that the metric tensor g is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].

我们证明，根据与拉普拉斯-贝尔特拉米算子的分数幂相关的局部源到解算子Δg 的知识，完全黎曼流形的度量张量 g 是唯一确定的，直到等度。我们的结果在已知任意小子域中的度量张量 g 的条件下成立。我们还考虑了封闭流形的情况，并对 [A.Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].
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引用次数: 0
Harmonic maps on locally conformal almost cosymplectic manifolds 局部保角近折射流形上的谐波映射
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-17 DOI: 10.1142/s0219199723500529
C. Gherghe, G. Vîlcu
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引用次数: 0
On exponential stability of switched functional differential equations with average dwell-time 论具有平均停留时间的开关函数微分方程的指数稳定性
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-17 DOI: 10.1142/s021919972350058x
P. H. A. Ngoc, Le Trung Hieu, Thai Bao Tran
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引用次数: 0
Convexity of energy functions of harmonic maps homotopic to covering maps of surfaces 与曲面覆盖图同源的谐波图能量函数的凸性
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-17 DOI: 10.1142/s0219199723500542
Inkang Kim, Xueyuan Wan, Genkai Zhang
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引用次数: 0
Boundary restricted Brunn-Minkowski inequalities 边界限制的布伦-闵科夫斯基不等式
IF 1.6 2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-17 DOI: 10.1142/s0219199723500566
S. Artstein-Avidan, Tomer Falah, Boaz A. Slomka
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引用次数: 0
Fast reaction limit of reaction diffusion systems with nonlinear diffusion 非线性扩散反应扩散系统的快速反应极限
2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-10 DOI: 10.1142/s0219199723500426
Elaine Crooks, Yini Du
In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form [Formula: see text] in usual reaction–diffusion equation, which represent linear diffusion, by terms of form [Formula: see text], representing nonlinear diffusion. We prove the convergence in the fast-reaction limit [Formula: see text] that is determined by the unique solution of a certain scalar nonlinear diffusion problem.
本文给出了在无界区域上具有两个反应扩散方程或一个反应扩散方程和一个常微分方程的非线性扩散系统的快速反应极限的一种表征方法。在这里，我们将通常的反应扩散方程中表示线性扩散的形式[公式:见文]的项替换为表示非线性扩散的形式[公式:见文]的项。我们证明了由某标量非线性扩散问题的唯一解决定的快速反应极限[公式:见文]的收敛性。
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引用次数: 1
Properties of gradient maps associated with action of real reductive group 与实约化群作用相关的梯度映射的性质
2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-11-08 DOI: 10.1142/s0219199723500517
Leonardo Biliotti, Joshua O. Windare
Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].
设[公式:见文]是一个Kähler流形，设[公式:见文]是一个具有李代数的紧连通李群[公式:见文]作用于[公式:见文]并保持[公式:见文]。我们假设[公式:见文]-作用全纯地扩展到复化群的一个作用[公式:见文]，并且[公式:见文]上的[公式:见文]-作用是汉密尔顿函数。然后存在一个[公式:见文本]-等变动量图[公式:见文本]。如果[公式:见文]是一个闭合子群，使得Cartan分解[公式:见文]引起Cartan分解[公式:见文]，其中[公式:见文]、[公式:见文]和[公式:见文]是[公式:见文]的李代数，则存在一个相应的梯度映射[公式:见文]。如果[公式:见文]是[公式:见文]的[公式:见文]的一个[公式:见文]不变紧连通实子流形，我们可以把[公式:见文]看作一个映射[公式:见文]给定一个[公式:见文]上的[公式:见文]上的一个[公式:见文]不变标量积，我们得到一个在[公式:见文]上的类莫尔斯函数[公式:见文]。我们指出，在不假设[公式:见文]是一个真正的解析流形的情况下，Lojasiewicz梯度不等式对[公式:见文]成立。因此，[公式:见文]的负梯度流的极限是存在的，并且是唯一的。此外，我们证明了任何[公式:见文]-轨道坍缩为一个[公式:见文]-轨道和[公式:见文]-轨道上的两个临界点属于同一个[公式:见文]-轨道。我们还研究了在阿贝尔情况下梯度映射的凹凸性[公式:见文本]。特别地，我们研究了双轨变化[公式:见文]，并研究了[公式:见文]的拓扑和上同调性质。
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引用次数: 2
The bernstein problem for (X, Y)-lipschitz surfaces in three-dimensional sub-finsler heisenberg groups 三维亚finsler - heisenberg群中(X, Y)-lipschitz曲面的bernstein问题
2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-10-13 DOI: 10.1142/s0219199723500487
Gianmarco Giovannardi, Manuel Ritore
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引用次数: 0
Timelike Ricci bounds for low regularity spacetimes by optimal transport 低正则时空最优输运的类时Ricci界
2区 数学 Q1 Mathematics
Communications in Contemporary Mathematics
Pub Date : 2023-10-13 DOI: 10.1142/s0219199723500499
Mathias Braun, Matteo Calisti
We prove that a globally hyperbolic smooth spacetime endowed with a $smash{mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $smash{mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.
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引用次数: 1
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