In this paper, we consider the L p dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the given data is even. If f ≡ 1, we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

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.

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The immersed boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semi-implicit scheme to remove this stiffness from the immersed boundary method for the Navier–Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semiimplicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the small scale decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semiimplicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the immersed boundary method.

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