Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $𝒟$-modules on $X$ induces a fully faithful functor on a subcategory of germs of formal holonomic $𝒟$-modules. Further, given a germ $ℳ$ of holonomic $𝒟$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $ℳ$ and the classical formal and analytic invariants of $ℳ$.

This paper is devoted to considering the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation $$f^{{}^{\text{'}\text{'}}}+A_1\left(z\right)f^{{}^{\text{'}}}+A_0\left(z\right)f=F,$$ where $A_1\left(z\right)$, $A_0\left(z\right)$$(\neg \equiv 0)...$

Suppose that $f\left(z\right)$ is a meromorphic or entire function satisfying $P\left(z,f,f^{\prime },\mathrm{\dots },f^{\left(n\right)}\right)=0$ where $P$ is a polynomial in all its arguments. Is there a limitation on the growth of $f$, as measured by its characteristic $T\left(r,f\right)$? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.

This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and $-1/4$. This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections....

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $xgt;0$, we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c\ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity. If $c=c_0$...

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...

On décrit des preuves galoisiennes des versions logarithmique et exponentielle de la conjecture de Schanuel, pour les variétés abéliennes sur un corps de fonctions.

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.

Dans cet article, nous étudions le problème de l’existence de polynômes de Darboux dans $\mathbf{C}\left\{x\right\}\left[y,y^{\prime }\right]$ pour la dérivation $\delta =R\left(y\right){\partial }_x+R\left(y\right)y^{\prime }{\partial }_y-V(y,y^{\prime }){\partial }_{y^{\prime }}$.