Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by the property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a basis of the subgroup that it generates.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1011.5369, Monatsh. Math. (2015

We consider a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the possible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words.

The properties of high-pH suspensions of mixtures of silica with low-molecular-weight samples of the water-soluble polymer polyethylenimine (PEI) have been studied. At pH > 10 and low ionic strength, silica nanoparticles are stabilized by a negative surface charge, and PEI has only a very low positive charge. The adsorption of PEI induces a localized positive charge on the segments of polymer closest to the silica surface. The parts of the molecule furthest away from the surface have little charge because of the high pH of the medium. The polymer-covered particle remains negatively charged, imparting some electrostatic stabilization. Suspensions of silica and low-molecular-weight PEI are low-viscosity fluids immediately after mixing, but aggregation occurs leading to the eventual gelation (or sedimentation at lower concentrations) of these mixtures, indicating colloidal instability. The gelation time passes through a minimum with increasing surface coverage. The rate of gelation increases exponentially with molecular weight: for molecular weight> or = 10,000 Da PEI, the instability is so severe that uniform suspensions cannot be produced using simple mixing techniques. The gelation rates increase rapidly with temperature, ionic strength, and reduction in pH. The rate of gelation increases with increasing particle concentration at low surface coverage but decreases at high coverage as a consequence of a small increase in pH. Gels are broken by application of high shear into aggregates that re-gel more rapidly than the original discrete coated particles.

Abstract. We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p.

International audienceWe investigate the relation between bifix codes and interval exchange 5 transformations. We prove that the class of natural codings of regular 6 interval echange transformations is closed under maximal bifix decoding