Distances between pairs of vertices and vertical profile in conditioned Galton–Watson trees

We consider a conditioned Galton–Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet–Mélou and Janson (Bousquet‐Mélou and Janson, Ann Appl Probab 16 (2006) 1597–1632), saying that the vertical profile of a randomly labelled conditioned Galton–Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 381–395, 2011     

Poset limits and exchangeable random posets

We develop a theory of limits of finite posets in close analogy to the recent theory of graph limits. In particular, we study representations of the limits by functions of two variables on a probability space, and connections to exchangeable random infinite posets.     

Phase separation and crystallinity in proton conducting membranes of styrene grafted and sulfonated poly(vinylidene fluoride)

A series of proton exchange membranes have been prepared by the preirradiation grafting method. Styrene was grafted onto a matrix of poly(vinylidene fluoride) (PVDF) after electron beam irradiation. Part of the samples was crosslinked with divinylbenzene (DVB) or bis(vinylphenyl)ethane (BVPE). Subsequent sulfonation gave membranes grafted with poly(styrene sulfonic acid) and marked PVDF‐g‐PSSA. It was found that the intrinsic crystallinity of the matrix decreased in both the grafting and the sulfonation reaction in all the membranes. The graft penetration and the ion conductivity are influenced strongly by the crosslinker. The ion conductivity is considerably lower in crosslinked membranes than in noncrosslinked ones. Generally, the mechanical strength decreases with crosslinking. The membranes show a regular phase separated structure in which the sulfonated grafts are incorporated in the amorphous parts of the matrix polymer. The phase separated domains are small, of the order of magnitude of 100–250 nm. These were resolved on transmission electron micrographs and on atomic force images but could not be resolved with microprobe Raman spectroscopy. © 1999 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 37: 1741–1753, 1999