In statistics and machine learning, we are 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. The Davis-Kahan sin theorem is often used to bound the difference between the eigenvectors of a matrix A and those of a perturbed matrix A= A+ E, in terms of l2 norm. In this paper, we prove that when A is a low-rank and incoherent matrix, the l norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of

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The nth power of the volume functional Vnof polytopes P in R^n, according to dimensions of the spaces spanned by any nunit outer normal vectors of P, is decomposed into nhomogeneous polynomials of degree n. A set of new sharp affine isoperimetric inequalities for these volume decomposition functionals in R^3 are established, which essentially characterize the geometric and algebraic structures of polytopes.

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In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.