Maximality of hyperspecial compact subgroups avoiding Bruhat-Tits theory

Let $k$ be a complete non-archimedean field (non trivially valued). Given a reductive $k$-group $G$, we prove that hyperspecial subgroups of $G(k)$ (i.e. those arising from reductive models of $G$) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of $\textrm{GL}_n$ and avoids all considerations on the Bruhat-Tits building of $G$.Comment: To appear at "Annales de l'Institut Fourier". This version avoids completely Berkovich geometr

Notions of Stein spaces in non-archimedean geometry

Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension $k'$ of $k$, every coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When $X$ has no boundary the characterization is simpler: in~2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in 3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space. v2: major revision to handle also the case of spaces with boundaryComment: 31 page

A Decomposition Algorithm for Nested Resource Allocation Problems

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is $O(n \log m \log (B/n))$, and the complexity of obtaining an $\epsilon$-approximate solution for the continuous case is $O(n \log m \log (B/\epsilon))$, $n$ being the number of variables, $m$ the number of ascending constraints (such that $m < n$), $\epsilon$ a desired precision, and $B$ the total resource. This algorithm attains the best-known complexity when $m = n$, and improves it when $\log m = o(\log n)$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Rigid-analytic functions on the universal vector extension

Let $K$ be a non-trivially valued complete non-Archimedean field. Given an algebraic group over $K$ such that every regular function is constant, every rigid-analytic function on it is shown to be constant. In particular, an algebraic group whose analytification is Stein (in Kiehl's sense) is necessarily affine--a remarkable difference between the complex and the non-Archimedean worlds.Comment: 49 pages, comments welcome