An algorithm for constructing certain differential operators in positive characteristic

Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some specific families of polynomials, we also study the level of such a differential operator $\delta$, i.e., the least integer $e$ such that $\delta$ is $R^{p^e}$-linear. In particular, we obtain a characterization of supersingular elliptic curves in terms of the level of the associated differential operator.Comment: 23 pages. Comments are welcom

Wettability of pristine and alkyl-functionalized graphane

Graphane is a hydrogenated form of graphene with high bandgap and planar structure insensitive to a broad range of chemical substitutions. We describe an atomistic simulation approach to predict wetting properties of this new material. We determine the contact angle to be 73°. The lower hydrophobicity compared to graphene is explained by the increased planar density of carbon atoms while we demonstrate that the presence of partial charges on carbonand hydrogen atoms plays only a minor role. We further examine the effects of graphane functionalization by alkyl groups of increasing chain lengths. The gradual increase in contact angle with chain length offers a precise control of surface wettability. A saturated contact angle of 114° is reached in butylated form. We find the saturation of contact angle with respect to the length of the functional groups to coincide with the loss of water\u27s ability to penetrate the n-alkyl molecular brush and interact with carbon atoms of the underlying lattice. Since no experimental data have yet become available, our modeling results provide the first estimate of the wettability of graphane. The results also show how its alkyl functionalization provides the basis for a variety of chemical modifications to tune hydrophilicity while preserving the planar geometry of the substrate