Explicit differential characterization of the Newtonian free particle system in m > 1 dependent variables

In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0. This result, preliminary to the deep group-theoretic classification of second order analytic ordinary differential equations, was parachieved later in 1896 by Arthur Tresse, a French student of S. Lie. In the present paper, following closely the original strategy of proof of S. Lie, which we firstly expose and restitute in length, we generalize this explicit characterization to the case of several second order ordinary differential equations. Let K=R or C, or more generally any field of characteristic zero equipped with a valuation, so that K-analytic functions make sense. Let x in K, let m > 1, let y := (y^1, ..., y^m) in K^m and let y_xx^j = F^j(x,y,y_x^l), j = 1,...,m be a collection of m analytic second order ordinary differential equations, in general nonlinear. We provide an explicit necessary and sufficient condition in order that this system is equivalent, under a point transformation (x, y^1, ..., y^m) --> (X(x,y), Y^1(x,y),..., Y^m(x, y)), to the Newtonian free particle system Y_XX^1 = ... = Y_XX^m = 0. Strikingly, the (complicated) differential system that we obtain is of first order in the case m > 1, whereas it is of second order in S. Lie's original case m = 1.Comment: 76 pages, no figur

The estrogen-responsive B box protein: a novel enhancer of interleukin-1beta secretion

The estrogen-responsive B box protein (EBBP) and Pyrin belong to a family of structurally related proteins. While mutations in the pyrin gene cause an autoinflammatory disease, the biological function of EBBP is unknown. In this study, we identified the proinflammatory cytokine interleukin-1beta (IL-1beta) as an EBBP-binding partner. Furthermore, caspase-1 and NACHT, LRR and Pyrin domain containing protein (NALP) 1, two components of the recently identified inflammasome, a platform for the activation of caspase-1, also interact with EBBP. These proteins bind to the RFP domain of EBBP, suggesting that this domain of so far unknown function is an important protein-binding domain. EBBP was secreted in a caspase-1-dependent manner from cultured cells, and its secretion was enhanced by IL-1beta. Vice versa, endogenous and overerexpressed EBBP increased IL-1beta secretion. These results provide evidence for a role of EBBP in innate immunity by enhancing the alternative secretion pathway of IL-1beta