Abstract:
The morphogenic Hedgehog (Hh) signaling regulates postnatal cerebellar development and its aberrant activation leads to medulloblastoma. The transcription factors Gli1 and Gli2 are the activators of Hh pathway and their function is finely controlled by different covalent modifications, such as phosphorylation and ubiquitination. We show here that Gli2 is endogenously acetylated and that this modification represents a key regulatory step for Hedgehog signaling. The histone acetyltransferase (HAT) coactivator p300, but not other HATs, acetylates Gli2 at the conserved lysine K757 thus inhibiting Hh target gene expression. By generating a specific anti acetyl-Gli2(Lys757) antisera we demonstrated that Gli2 acetylation is readily detectable at endogenous levels and is attenuated by Hh agonists. Moreover, Gli2 K757R mutant activity is higher than wild type Gli2 and is no longer enhanced by Hh agonists, indicating that acetylation represents an additional level of control for signal dependent activation. Consistently, in sections of developing mouse cerebella Gli2 acetylation correlates with the activation status of Hedgehog signaling. Mechanistically, acetylation at K757 prevents Gli2 entry into chromatin. Together, these data illustrate a novel mechanism of regulation of the Hh signaling whereby, in concert with Gli1, Gli2 acetylation functions as a key transcriptional checkpoint in the control of morphogen-dependent processes.

Abstract:
We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since \deg_q and \ord_q define two valuations on {\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition.

Abstract:
In this paper, we establish, under convenient diophantine assumptions, a complete analytic classification of $q$-difference modules over the field of germs of meromorphic functions at zero, proving some analytic analogs of the results by Soibelman and Vologodsky, cf. math.AG/0205117, and by Baranovsky and Ginzburg, cf. alg-geom/9607008.

Abstract:
In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, \S 13]. In the second part of the paper, we combine it with some formal q-analogous Fourier transformations, obtaining a statement on the irrationality of special values of the formal $q$-Borel transformation of a G_q-function.

Abstract:
In this survey we present the parameterized Galois theory of difference equations, as introduced by Hardouin-Singer. The purpose of this theory is to give a systematic approach to differential transcendence, also called hypertranscendence. As an example of the applications, we explain the Galoisian proof of H\"older's theorem on the differential transcendence of the Gamma function.

Abstract:
Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all finite places. It is equivalent to Katz's conjectural description of the generic Galois group. In this paper we prove an analogous statement for arithmetic q-difference equation.

Abstract:
Inspired by the theory of p-adic differential equations, this paper introduces an analogous theory for q-difference equations over a local field, when |q|=1. We define some basic concepts, for instance the generic radius of convergence, introduce technical tools, such as a twisted Taylor formula for analytic functions, and prove some fundamental statements, such as an effective bound theorem, the existence of a weak Frobenius structure and a transfer theorem in regular singular disks.

Abstract:
In this paper we investigate the interrelationships between fertility decisions and union dissolution in Italy and Spain. We argue that there might exist a spurious relationship between these two life trajectories. The analysis is based on the 1996 Fertility and Family Survey data for Italy and Spain. Results show that there is a spurious relationship between fertility and union dissolution in Italy but not in Spain. Nevertheless, in both countries, there is an evident direct effect of each process on the other: union dissolution decreases the risk of further childbearing, while childbirths decrease the risk of union dissolution.

Abstract:
We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cX_{\eta} of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the \emph{diameter} \delta_{\cX}(\xi,U) of U at \xi\in U, the \emph{radius} \rho_{\cX}(\xi) of the point \xi\in\cX_{\eta}, the \emph{radius of convergence} of solutions of (\cE,\na) at \xi, R(\xi) = R_{\cX}(\xi, U,(\cE, \na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that \delta_{\cX}(\xi,U), \rho_{\cX}(\xi) and R_{\xi}(U,\cE,\na) are upper semicontinuous functions of \xi; for Laurent domains in the affine space, \delta_{\cX}(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function.

Abstract:
This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of such a q-difference equation. In the second part, we work under the assumption q\in ]1,+\infty[. In this case, at least four different q-Borel sums of a divergent solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.