A general method for common intervals

Given an elementary chain of vertex set V, seen as a labelling of V by the set {1, ...,n=|V|}, and another discrete structure over $V$, say a graph G, the problem of common intervals is to compute the induced subgraphs G[I], such that $I$ is an interval of [1, n] and G[I] satisfies some property Pi (as for example Pi= "being connected"). This kind of problems comes from comparative genomic in bioinformatics, mainly when the graph $G$ is a chain or a tree (Heber and Stoye 2001, Heber and Savage 2005, Bergeron et al 2008). When the family of intervals is closed under intersection, we present here the combination of two approaches, namely the idea of potential beginning developed in Uno, Yagiura 2000 and Bui-Xuan et al 2005 and the notion of generator as defined in Bergeron et al 2008. This yields a very simple generic algorithm to compute all common intervals, which gives optimal algorithms in various applications. For example in the case where $G$ is a tree, our framework yields the first linear time algorithms for the two properties: "being connected" and "being a path". In the case where $G$ is a chain, the problem is known as: common intervals of two permutations (Uno and Yagiura 2000), our algorithm provides not only the set of all common intervals but also with some easy modifications a tree structure that represents this set

A General, Mass-Preserving Navier-Stokes Projection Method

The conservation of mass is common issue with multiphase fluid simulations. In this work a novel projection method is presented which conserves mass both locally and globally. The fluid pressure is augmented with a time-varying component which accounts for any global mass change. The resulting system of equations is solved using an efficient Schur-complement method. Using the proposed method four numerical examples are performed: the evolution of a static bubble, the rise of a bubble, the breakup of a thin fluid thread, and the extension of a droplet in shear flow. The method is capable of conserving the mass even in situations with morphological changes such as droplet breakup.Comment: Submitted to Computer Physics Communication