Polar decomposition and Brion's theorem

Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones

We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using irrational decompositions, and a generalization of the Lawrence-Varchenko formula.

Ehrhart-Macdonald reciprocity extended

For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation between these two counting functions. A similar counting function and reciprocity law exists for the sum of all solid angles at integer points in dilates of P. We derive a unifying generalization of these reciprocity theorems which follows in a natural way from Brion's Theorem on conic decompositions of polytopes.

From Modular Decomposition Trees to Level-1 Networks: Pseudo-Cographs, Polar-Cats and Prime Polar-Cats

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". This information, however, gets lost whenever $(T,t)$ contains vertices with label "prime". In this cont…

有限极空间中的战术分解与非扩散的典型群作用

Tactical decompositions in finite polar spaces and non-spreading classical group actions

本文研究有限典型群的扩散置换群性质，借助战术分解的AB-引理证明多数有限典型群在其极空间点集上的自然作用为非扩散，并考察了这类群的其他本原作用。