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We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that, for n ≥ 3, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established by I. Babenko and F. Balacheff [Sur la forme de la boule unit\'e de la norme stable unidimensionnelle, Manuscripta Math. 119 (2006) 347--358] and M. Jotz [Hedlund metrics and the stable norm, Diff. Geometry Appl. 27 (2009) 543--550] in the form of stable norms as an extension of G. A. Hedlund's classical result [Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932) 719--739]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

The nonlinear and non-stationary nature of Navier-Stokes equations produces fluid flows that can be noticeably different in appearance with subtle changes. In this paper we introduce a method that can analyze the intrinsic multiscale features of flow fields from a decomposition point of view, by using the Hilbert-Huang transform method on 3D fluid simulation. We show how this method can provide insights to flow styles and help modulate the fluid simulation with its internal physical information. We provide easy-toimplement algorithms that can be integrated with standard grid-based fluid simulation methods, and demonstrate how this approach can modulate the flow field and guide the simulation with different flow styles. The modulation is straightforward and relates directly to the flow¡¯s visual effect, with moderate computational overhead.

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We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs.