In this paper, we investigate the vanishing viscosity limit problem for the 3-dimensional (3D) incompressible Navier–Stokes equations in a general bounded smooth domain of R ³ with the generalized Navier-slip boundary conditions $u^{\varepsilon}\cdot n = 0,\ n\times(\omega^{\varepsilon}) = [B u^{\varepsilon}]_{\tau}\ {\rm on} \ \partial\varOmega$. Some uniform estimates on rates of convergence in C([0,T],L ²(Ω)) and C([0,T],H ¹(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.

In this paper we develop a new technique to prove existence of solutions of Fokker–Planck equations on Hilbert spaces for Kolmogorov operators with non-trace-class second order coefficients or equivalently with an associated stochastic partial differential equation (SPDE) with non-trace-class noise. Applications include stochastic 2D and 3D-Navier–Stokes equations with non-trace-class additive noise.

For a harmonic map between two hyperkäher manifolds, we prove a Weitzenböck type formula for the defining quantity of quaternionic maps, and apply it to harmonic morphisms. We also provide a sufficient and necessary condition for a smooth map being quaternionic.

Given an arithmetic lattice of the unitary group U(3,1) arising from a hermitian form over a CM-field, we show that all unitary representations of U(3,1) with nonzero cohomology contribute to the cohomology of the attached arithmetic complex 3-manifold, at least when we pass to a finite-index subgroup of the given arithmetic lattice.

The Laplace–Beltrami operator (LBO) is the fundamental geometric object associated with manifold surfaces and has been widely used in various tasks in geometric processing.

By understanding that the LBO can be computed by differential quantities, we propose an approach for discretizing the LBO on manifolds by estimating differential quantities. For a point on the manifold, we first fit a quadratic surface to this point and its neighborhood by minimizing the least-square energy function. Then we compute the first- and second-order differential quantities by the approximated quadratic surface. Finally the discrete LBO at this point is computed from the estimated differential quantities and thus the Laplacian matrix over the discrete manifold is constructed.

Our approach has several advantages: it is simple and efficient and insensitive to noise and boundaries. Experimental results have shown that our approach performs better than most of the current approaches.

We also propose a feature-aware scheme for modifying the Laplacian matrix. The modified Laplacian matrix can be used in other feature preserving geometric processing applications.

Weighted U-statistics and generalized L-statistics are commonly used in statistical inference and their asymptotic properties have been well developed. In this paper sharp non-uniform Berry–Esseen bounds for weighted U-statistics and generalized L-statistic are established.