MathOverflow

I was reading an article of matrix completion and met the following lemma The concentration inequality for $\sigma_{\max}$ part is a standard result. However, I didn't find any results like the $\sigma_{\min}$ part. The most similar expression was found...

Let $K$ be a self-similar fractal set with Hausdorff dimension $D \approx 2.4$–$2.7$, equipped with a Laplacian $\Delta$ defined via the standard analysis-on-fractals construction (Dirichlet form / spectral decimation, in the sense of Strichartz, Kigami...

Let $z_1,\ldots,z_n$ be $n\ge 1$ distinct points of $\mathbb R^2$. Define the potential function $U: \mathbb R^2 \to\mathbb R$ by
$$U(x):=\sum_{1\le i\le n} \log(|x-z_i|),$$
where $|\cdot|$ denotes the Euclidean norm. Denote by $F$ be the negative gradient of $U$, i.e....

The general question
It is easy to find on the Wikipedia page for Lebesgue measure that Haar measure is a common generalization that preserves the idea of "invariance under some group action". While wondering about the "most natural" way of defining a measure on lines of...