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We prove that intermediate Banach spaces $$\mathcal{A}$$ and $$\mathcal{B}$$ with respect to arbitrary Hilbert couples $$\bar {H}$$ and $$\bar {K}$$ are exact interpolation if and onlyif they are exact$K$-monotonic, i.e. the condition $$f^0 \in \mathcal{A}$$ and the inequality $$K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$$ , imply$g$^{0}∈$B$and ‖$g$^{0}‖$B$≤‖$f$^{0}‖_{$A$}($K$is Peetre’s$K$-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator $$T:\bar {H} \to \bar {K}$$ such that$Tf$^{0}=$g$^{0}, and $$K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$$ . Verifying the latter property, it suffices to consider the “diagonal case” where $$\bar {H} = \bar {K}$$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.

We present novel algorithms for compressible flows that\n\nare efficient for all Mach numbers. The approach is based on several\n\ningredients: semi-implicit schemes, the gauge decomposition\n\nof the velocity field and a second order formulation of the density\n\nequation (in the isentropic case) and of the energy equation (in\n\nthe full Navier-Stokes case). Additionally, we show that our approach\n\ncorresponds to a micro-macro decomposition of the model,\n\nwhere the macro field corresponds to the incompressible component\n\nsatisfying a perturbed low Mach number limit equation and\n\nthe micro field is the potential component of the velocity. Finally,\n\nwe also use the conservative variables in order to obtain a proper\n\nconservative formulation of the equations when the Mach number\n\nis order unity. We successively consider the isentropic case, the\n\nfull Navier-Stokes case, and the isentropic Navier-Stokes-Poisson\n\ncase. In this work, we only concentrate on the question of the\n\ntime discretization and show that the proposed method leads to\n\nAsymptotic Preserving schemes for compressible flows in the low\n\nMach number limit.

This paper studies the Voronoi diagrams on 2-manifold meshes based on geodesic metric (a.k.a. geodesic Voronoi diagrams or GVDs), which have polyline generators. We show that our general setting leads to situations more complicated than conventional 2D Euclidean Voronoi diagrams as well as point-source based GVDs, since a typical bisector contains line segments, hyperbolic segments and parabolic segments. To tackle this challenge, we introduce a new concept, called local Voronoi diagram (LVD), which is a combination of additively weighted Voronoi diagram and line-segment Voronoi diagram on a mesh triangle. We show that when restricting on a single mesh triangle, the GVD is a subset of the LVD and only two types of mesh triangles can contain GVD edges. Based on these results, we propose an efficient algorithm for constructing the GVD with polyline generators. Our algorithm runs in O(nNlogN) time and takes O(nN) space on an n-face mesh with m generators, where N = max{m,n}. Computational results on real-world models demonstrate the efficiency of our algorithm.

In this paper, we derive some local a priori estimates for the Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.