We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to $+\infty $ , we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.

We prove a rigid analytic analogue of the Artin–Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space $X$ vanishes in all degrees above the dimension of $X$. Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces.

Properties and comparison theorems for the maximal solution of the periodic discrete-time Riccati equation are supplemented by an extension of some earlier results and analysis, for the discrete-time Riccati equation to the periodic case.

We address the question of controlling the Brownian path in several dimensions (d≧2) by continually choosing its drift from among vectors of the unit ball in ℝd. The past and present of the path are supposed to be completely observable, while no anticipation of the future is allowed. Imposing a suitable cost on distance from the origin, as well as a cost of effort proportional to the length of the drift vector, ‘reasonable’ procedures turn out to be of the following type: to apply drift of maximal length along the ray towards the origin if the current position is outside a sphere centred at the origin, and to choose zero drift otherwise. It is shown just how to compute the radius of such a sphere in terms of the data of the problem, so that the resulting procedure is optimal.

Comparison theorems are developed for the pair of first order Riccati equations (1) and (2) . The comparisons are of an integral type and involve an auxiliary function μ. Applications are given to disconjugacy theory for self-adjoint equations of the second and fourth order.