带非负权重的有限度量空间的幅度的连续性
Continuity of the magnitude for finite metric spaces with nonnegative weightings
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中文总结 AI 辅助
本文研究带非负权重的有限度量空间的幅度连续性，证明其幅度之比受Gromov-Hausdorff距离指数界约束，得出对数幅度函数Lipschitz连续、幅度函数局部Lipschitz连续的结论。
中文摘要 AI 辅助
对于有限度量空间，幅度是一个关于Gromov-Hausdorff距离处处不连续的不变量。现有文献多聚焦于正定度量空间，本文研究允许非负权重的有限度量空间，该类空间包含已知幅度有特定界的超度量空间。我们将这些界推广，证明对任意两个带非负权重的有限度量空间，其幅度之比受Gromov-Hausdorff距离指数界约束。由此表明，在该类空间中，对数幅度函数是Lipschitz连续的，幅度函数本身是局部Lipschitz连续的，但非Lipschitz连续。
英文摘要
For finite metric spaces, magnitude is an invariant known to be discontinuous everywhere with respect to the Gromov-Hausdorff distance. While much of the literature focuses on positive definite metric spaces, here we consider finite metric spaces admitting a nonnegative weighting. In this paper, we focus on the class of finite metric spaces that admit a nonnegative weighting. This class includes ultrametric spaces, for which specific bounds on magnitude are known. We generalize these bounds by proving that, for any two finite metric spaces with nonnegative weightings, the ratio of their magnitudes is exponentially bounded by their Gromov-Hausdorff distance. Consequently, we show that within this class, the logarithmic magnitude function is Lipschitz continuous, and the magnitude function itself is locally Lipschitz continuous, but not Lipschitz continuous.
发表机构
- The University of Osaka（大阪大学）
机构由 AI 辅助整理，请以论文原文为准。
