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

The Vector Valued Quartile Operator

Certain vector-valued inequalities are shown to hold for a Walsh analog of the bilinear Hilbert transform. These extensions are phrased in terms of a recent notion of quartile type of a UMD (Unconditional Martingale Differences) Banach space X. Every known UMD Banach space has finite quartile type, and it was recently shown that the Walsh analog of Carleson's Theorem holds under a closely related assumption of finite tile type. For a Walsh model of the bilinear Hilbert transform however, the quartile type should be sufficiently close to that of a Hilbert space for our results to hold. A full set of inequalities is quantified in terms of quartile type.Comment: 32 pages, 5 figures, incorporates referee's report, to appear in Collect. Mat

Core congestion is inherent in hyperbolic networks

We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset $X$ of vertices of a $\delta$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk $D(m,4 \delta)$ of radius $4 \delta$ centered at $m$ intercepts at least one half of the total flow between all pairs of vertices of $X$, where the flow between two vertices $x,y\in X$ is carried by geodesic (or quasi-geodesic) $(x,y)$-paths. A set $S$ intercepts the flow between two nodes $x$ and $y$ if $S$ intersect every shortest path between $x$ and $y$. Differently from what was conjectured by Jonckheere et al., we show that $m$ is not (and cannot be) the center of mass of $X$ but is a node close to the median of $X$ in the so-called injective hull of $X$. In case of non-uniform traffic between nodes of $X$ (in this case, the unit flow exists only between certain pairs of nodes of $X$ defined by a commodity graph $R$), we prove a primal-dual result showing that for any $\rho>5\delta$ the size of a $\rho$-multi-core (i.e., the number of disks of radius $\rho$) intercepting all pairs of $R$ is upper bounded by the maximum number of pairwise $(\rho-3\delta)$-apart pairs of $R$