Metric Geometry

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New submissions (showing 1 of 1 entries )

[1] arXiv:2509.21025 [cn-pdf, pdf, html, other]

Title: A Reshetnyak type theorem for quasiregular values on Carnot groups of $H$-type Show Chinese title

Title: 一个关于Carnot群上拟正则值的Reshetnyak型定理$H$-类型

Deguang Zhong

Subjects: Metric Geometry (math.MG) ; Complex Variables (math.CV)

In this paper, we establish a Reshetnyak type theorem for quasiregular values on the setting of Carnot group of $H$-type. Show Chinese abstract

在本文中，我们在 Carnot 群的$H$类型的背景下，建立了关于拟正则值的 Reshetnyak 型定理。

Replacement submissions (showing 1 of 1 entries )

[2] arXiv:2509.21066 (replaced) [cn-pdf, pdf, html, other]

Title: A Riemannian Variational and Spectral Framework for High-Dimensional Sphere Packing: Barrier-Dynamics Reconciliation, Periodic Rigidity, and Discrete-Time Guarantees Show Chinese title

Title: 一种高维球体填充的黎曼变分与谱框架：障碍物动力学协调、周期刚性与离散时间保证

Faruk Alpay, Hamdi Alakkad

Comments: 9 pages

Subjects: Optimization and Control (math.OC) ; Metric Geometry (math.MG)

We develop a unified framework that reconciles a barrier based geometric model of periodic sphere packings with a provably convergent discrete time dynamics. First, we introduce a C2 interior barrier U_nu that is compatible with a strict feasibility safeguard and has a Lipschitz gradient on the iterates domain. Second, we correct and formalize the discrete update and give explicit step size and damping rules. Third, we prove barrier to KKT consistency and state an interior variant clarifying the role of the quadratic term. Fourth, we show that strict prestress stability implies periodic infinitesimal rigidity of the contact framework. Fifth, we establish a Lyapunov energy descent principle, an energy nonexpansive feasibility projection (including a joint x and lattice basis B variant), and local linear convergence for the Spectral Projected Interior Trajectory (Spit) method. We also provide practical Hessian vector formulas to estimate smoothness and curvature, minimal schematic illustrations, and a short reproducibility stub. The emphasis is on rigorous assumptions and proofs; empirical evaluation is deferred. Show Chinese abstract

我们开发了一个统一的框架，将基于障碍的周期性球体包装的几何模型与可证明收敛的离散时间动力学相结合。 首先，我们引入一个C2内部障碍U_nu，该障碍与严格的可行性保障兼容，并在迭代域上具有Lipschitz梯度。 其次，我们修正并形式化了离散更新，并给出了显式的步长和阻尼规则。 第三，我们证明了障碍到KKT的一致性，并提出了一个内部变体，阐明了二次项的作用。 第四，我们证明了严格预应力稳定性意味着接触框架的周期性无限小刚性。 第五，我们建立了Lyapunov能量下降原理、能量非扩张可行性投影（包括联合x和格子基B的变体），以及Spectral Projected Interior Trajectory（Spit）方法的局部线性收敛性。 我们还提供了实用的Hessian向量公式来估计平滑性和曲率，最小的示意图，以及一个简短的可重复性存根。 重点在于严格的假设和证明；经验评估被推迟。