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Let L be the circular cone in R n which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f c (x) on R n by applying f to the spectral values of the spectral decomposition of x R n with respect to L . We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Frchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

Let G={h_t|t∈ℝ} be a continuous flow on a connected n-manifold M. The flow G is said to be strongly reversible by an involution τ if h_{−t}=τh_tτ for all t∈ℝ, and it is said to be periodic if h_s= identity for some s∈ℝ^∗. A closed subset K of M is called a global section for G if every orbit G(x) intersects K in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if G is periodic and strongly reversible by a reflection, then G has a global section.

We prove the nonlinear asymptotic stability of the gravitational hydrostatic equilibrium for the general equation of state of pressure-density relation in the framework of vacuum free boundary problem of spherically symmetric compressible Navier-Stokes-Poisson equations in three dimensions.The results apply to white dwarfs and polytropes with γ≥2, for which even the linearized asymptotic stability results are not available in literature, to the best of the authors' knowledge. Detailed decay rates of perturbations are given.

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.