Given a domain Ω⊂ℂ^{$n$}, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.

We state and prove a Chern–Osserman-type inequality in terms of the volume growth for minimal surfaces$S$which have finite total extrinsic curvature and are properly immersed in a Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0 and such that they are not too curved (on average) with respect to the hyperbolic space with constant sectional curvature given by the upper bound$b$. We also prove the same Chern–Osserman-type inequality for minimal surfaces with finite total extrinsic curvature and properly immersed in an asymptotically hyperbolic Cartan–Hadamard manifold$N$with sectional curvatures bounded from above by a negative quantity$K$_{$N$}≤$b$<0.

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In statistics and machine learning, people are often interested in the eigenvectors (or singular vectors) of certain matrices (eg covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. One usually employs Davis-Kahan \sin theorem to bound the difference between the eigenvectors of a matrix \sin and those of a perturbed matrix \sin , in terms of \sin norm. In this paper, we prove that when \sin is a low-rank and incoherent matrix, the \sin norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of \sin or \sin for left and right vectors, where \sin and \sin are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.

Eigenvalues and capacitors. Let X be a Riemannian manifold and be the Laplace operator on X. It is well-known that if X is compact then the spectrum of is discrete and consists of an increasing sequence {k} k= 1 of the eigenvalues (counted with the multiplicities) where 1= 0 and k as k. Moreover, if n= dim X then Weyls asymptotic formula says that (1.1) k cn