An uncountable Mittag-Leffler condition with an application to ultrametric locally convex vector spaces

Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set $(I,\leq)$ admitting a $countable$ cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an ultrametric analogous of a result of V.P.Palamodov in relation with the acyclicity of Frechet spaces with respect to the completion functor.Comment: 19 page

\bs{p}-Adic Confluence of \bs{q}-Difference Equations

We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equation. This fact implies an equivalence, called ``Confluence'', between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves'' on the disk $\mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $\xi$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $\sigma_\xi$.Comment: 43 page