Abstract
The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion are examined by mathematical modelling. We find a transcritical bifurcation that might explain two alternative scenarios: in one case the appearance of autoimmune responses and in another the suppression of the immune response.

Abstract
We present the construction of new pseudo-smooth structures, near the singularities, such that the pseudo-Anosov maps are diffeomorphisms, in this pseudo-smooth structures, and have the property that the stable and unstable foliations are uniformly contracted and expanded by the pseudo-Anosov dynamics. We construct the C(1+) orthogonal charts of such pseudo-Anosov diffeomoprhisms.

Abstract
The theory of planned behaviour studies the decision-making mechanisms of individuals. We propose the Bayesian-Nash equilibria as one, of many, possible mechanisms of transforming human intentions in behaviour. This process corresponds to the best strategic individual decision taking in account the collective response. We show that saturation, boredom and frustration can lead to splitted strategies, in opposition to no saturation that leads to a constant strategy.

Abstract
We study the problem of determination of asset prices in an incomplete market proposing three different but related scenarios. One scenario uses a market game approach whereas the other two are based on risk sharing or regret minimizing considerations. Dynamical schemes modelling the convergence of the buyer's and of the seller's prices to a unique price are proposed.

Abstract
We present a new R&D investment function in a Cournot competition model inspired in the logistic equation. We do a full characterization of the associated game and study the short- and long-term economical effects derived from using this new R&D investment function. In particular, we find the existence of regions with multiple Nash investment equilibria. For low production costs, that can correspond to the production of old technologies, the long-term economical effects are not very sensitive to small changes in the efficiency of the R&D programmes neither to small changes in the market structure. However, for high production costs, that can correspond to the production of new technologies, the long-term economical effects are very sensitive to small changes in the efficiency of the R&D programmes and also to small changes in the market structure.

Abstract
We prove a one-to-one correspondence between C1+ conjugacy classes of diffeomorphisms with hyperbolic sets contained in surfaces and stable and unstable pairs of one-dimensional C1+ self-renormalizable structures.