In this paper we discuss the derivation and use of local pressure boundary conditions for finite difference schemes for the unsteady incompressible Navier-Stokes equations in the velocity-pressure formulation. Their use is especially well suited for the computation of moderate to large Reynolds number flows. We explore the similarities between the implementation and use of local pressure boundary conditions and local vorticity boundary conditions in the design of numerical schemes for incompressible flow in 2D. In their respective formulations, when these local numerical boundary conditions are coupled with a fully explicit convectively stable time stepping procedure, the resulting methods are simple to implement and highly efficient. Unlike the vorticity formulation, the use of the local pressure boundary condition approach is readily applicable to 3D flows. The simplicity of the local pressure boundary condition approach and its easy adaptation to more general flow settings make the resulting scheme an attractive alternative to the more popular methods for solving the Navier-Stokes equations in the velocity-pressure formulation. We present numerical results of a second-order finite difference scheme on a nonstaggered grid using local pressure boundary conditions. Stability and accuracy of the scheme applied to Stokes flow is demonstrated using normal mode analysis. Also described is the extension of the method to variable density flows.

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We determine the smallest Schatten class containing all integral operators with kernels in$L$_{p}(L_{p', q})^{symm}, where 2 <$p$∞ and 1≦$q$≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].

For a positive integer$k$and a non-negative integer$t$, a class of simplicial complexes, to be denoted by$k$-CM_{$t$}, is introduced. This class generalizes two notions for simplicial complexes: being$k$-Cohen–Macaulay and$k$-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM_{$t$}(=1−CM_{$t$}) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM_{$t$}property under the operation of join of two simplicial complexes. We show that a complex is$k$-CM_{$t$}if and only if the links of its non-empty faces are$k$-CM_{$t$−1}. We prove that for an integer$s$≤$d$, the ($d$−$s$−1)-skeleton of a ($d$−1)-dimensional$k$-CM_{$t$}complex is ($k$+$s$)-CM_{$t$}. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.