On Lattice-Free Orbit Polytopes

Given a permutation group acting on coordinates of $\mathbb{R}^n$, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called \emph{core points} and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the $2$-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all $2$-homogeneous groups up to degree twelve. For transitive groups that are not $2$-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee comments; to appear in Discrete and Computational Geometr

On the number of conjugacy classes of a permutation group

We prove that any permutation group of degree $n \geq 4$ has at most $5^{(n-1)/3}$ conjugacy classes.Comment: 9 page

Galois groups of multivariate Tutte polynomials

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let \bv = \{v_e\}_{e\in E}, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over K(\bv), and give three self-contained proofs that the Galois group of $\hat Z_M$ over K(\bv) is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul

The development of sentence-interpretation strategies in monolingual German-learning children with and without specific language impairment

Previous research on sentence comprehension conducted with German-learning children has concentrated on the role of case marking and word order in typically developing children. This paper compares, the performance of German-learning children with language impairment (age 4-6 years) and without language impairment (aged 2-6, 8-9 years) in two experiments that systematically vary the cues animacy, case marking; word-order, and subject-verb agreement. The two experiments differ with regard to the choice of case marking: in the first it is distinct but in the second it is neutralized. The theoretical framework is the competition model developed by Bates and Mac Whinney and their collaborators, a variant of the parallel distributed processing models. It is hypothesized that children of either population first appreciate the cue animacy that can be processed locally, that is, "on the spot," before they turn to more distributed cues leading ultimately up to subject-verb agreement, which presupposes the comparison of various constituents before an interpretation can be established. Thus agreement is more "costly" in processing than animacy or the (more) local cue initial NP. In experiment I with unambiguous case markers it is shown that the typically developing children proceed from animacy to the nominative (predominantly in coalition with the initial NP) to agreement, while in the second experiment with ambiguous case markers these children turn from animacy to the initial NP and then to agreement. The impaired children also progress from local to distributed cues. Yet, in contrast to the control group, they do not acknowledge the nominative in coalition with the initial NP in the first experiment but only in support of agreement. However, although they do not seem to appreciate distinct case markers to any large extent in the first experiment, they are irritated if such distinctions are lacking: in experiment II all impaired children turn to. animacy (some in coalition with the initial NP and/or particular word orders). In the discussion, the relationship between short-term memory and processing as well as the relationship between production and comprehension of case markers and agreement are addressed. Further research is needed to explore in more detail "cue costs" in sentence comprehension