Martingale optimal transport duality

We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski's $S$-topology on the Skorokhod space.Comment: 29 page

The first order convergence law fails for random perfect graphs

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to\infty$.Comment: 11 pages. Minor corrections since last versio

Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree

"In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1,2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97^k n^4 log W) using O(n^3 log nW log k) space. This is the first single-exponential time, polynomial-space FPT algorithm for the weighted Steiner tree problem." PLEASE NOTE:This is an author-created version that the author has self-archived to the "Aaltodoc" (aaltodoc.aalto.fi) faculty-level repository at Aalto University. The final publication is available at link.springer.com via the link http://dx.doi.org/10.1007/978-3-662-47672-7_40Peer reviewe

The Orphan Adhesion-GPCR GPR126 Is Required for Embryonic Development in the Mouse

Adhesion-GPCRs provide essential cell-cell and cell-matrix interactions in development, and have been implicated in inherited human diseases like Usher Syndrome and bilateral frontoparietal polymicrogyria. They are the second largest subfamily of seven-transmembrane spanning proteins in vertebrates, but the function of most of these receptors is still not understood. The orphan Adhesion-GPCR GPR126 has recently been shown to play an essential role in the myelination of peripheral nerves in zebrafish. In parallel, whole-genome association studies have implicated variation at the GPR126 locus as a determinant of body height in the human population. The physiological function of GPR126 in mammals is still unknown. We describe a targeted mutation of GPR126 in the mouse, and show that GPR126 is required for embryonic viability and cardiovascular development

Almost All F-Free Graphs Have The Erdos-Hajnal Property

International audienceErdős and Hajnal conjectured that, for every graph H, there exists a constant ɛ(H) > 0 such that every H-free graph G (that is, not containing H as an induced subgraph) must contain a clique or an independent set of size at least |G|ɛ( H). We prove that there exists ɛ(H) such that almost every H-ïvee graph G has this property, meaning that, amongst the if-free graphs with n vertices, the proportion having the property tends to one as n → ∞