Related papers: Simplicial arrangements on convex cones
Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always…
We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in \cite{monod} to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we…
We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of…
We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex.
In this paper we study plus-one generated arrangements of conics and lines in the complex projective plane with simple singularities. We provide several degree-wise classification results that allow us to construct explicit examples of such…
In applications that use knowledge representation (KR) techniques, in particular those that combine data-driven and logic methods, the domain of objects is not an abstract unstructured domain, but it exhibits a dedicated, deep structure of…
In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introduce a new tool that enables us to characterize the property…
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…
We introduce moment maps for continuous unitary representations of general topological groups. For solvable separable locally compact groups, we prove that the closure of the image of the moment map of any representation is convex.
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain class of conic-line…
This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…
This paper contains a classication of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration…
The state space of an operator system of $n$-by-$n$ matrices has, in a sense, many normal cones. Merely this convex geometrical property implies smoothness qualities and a clustering property of exposed faces. The latter holds since each…
Let $S$ be a finite set with $n$ elements in a real linear space. Let $\cJ_S$ be a set of $n$ intervals in $\nR$. We introduce a convex operator $\co(S,\cJ_S)$ which generalizes the familiar concepts of the convex hull $\conv S$ and the…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
Small Coxeter groups are exactly those for which the Tits representation takes integral values, which makes the study of their congruence subgroups significant. In \cite{MR0938643}, Squier introduced a matrix representation of an Artin…
Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group…
We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main…
The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of $n + 1$ general hyperplanes in the $n-$dimensional projective…
Transversal structures (also known as regular edge labelings) are combinatorial structures defined over 4-connected plane triangulations with quadrangular outer-face. They have been intensively studied and used for many applications…