In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis-Procesi(DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The $L^2$ stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method.By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the methods.

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main tech- nical tools are local-in-space regularity estimates near the initial time, which are of independent interest.

In this paper, we study robust covariance estimation under the approximate factor model with observed factors. We propose a novel framework to first estimate the initial joint covariance matrix of the observed data and the factors, and then use it to recover the covariance matrix of the observed data. We prove that once the initial matrix estimator is good enough to maintain the element-wise optimal rate, the whole procedure will generate an estimated covariance with desired properties. For data with bounded fourth moments, we propose to use adaptive Huber loss minimization to give the initial joint covariance estimation. This approach is applicable to a much wider class of distributions, beyond sub-Gaussian and elliptical distributions. We also present an asymptotic result for adaptive Hubers M-estimator with a diverging parameter. The conclusions are demonstrated by extensive simulations and real data analysis.

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codimension-two submanifold with constant normalized null expansion (null mean curvature) must lie in a shear-free (umbilical) null hypersurface. These results are generalized for higher order curvature invariants. In particular, the notion of mixed higher order mean curvature is introduced to highlight the special null geometry of the submanifold. Finally, Alexandrov type theorems are established for spacelike submanifolds with constant mixed higher order mean curvature, which are generalizations of hypersurfaces of constant Weingarten curvature in the Euclidean space.

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