Related papers: Hyperplane arrangements in polymake
Over the complex numbers, the complement of a collection of hyperplanes is a widely-studied object; the cohomology ring, in particular, is known to have a structure depending only on the combinatorial properties of the intersection of…
In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…
We present an automated methodology to optimize laminated composite structures. Our approach, inspired by bi-level optimization scheme, avoids its prime deficiencies and is based on developed computationally inexpensive stacking sequences…
We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in the complex projective plane. Such pair of arrangements has an additional property: they admit conjugated equations on the ring…
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where…
An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented…
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
Hyperplane hashing aims at rapidly searching nearest points to a hyperplane, and has shown practical impact in scaling up active learning with SVMs. Unfortunately, the existing randomized methods need long hash codes to achieve reasonable…
Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. $\mathcal{H}^2$-matrices refine this representation following the ideas of fast multipole…
We present a multi-robot task and motion planning method that, when applied to the rearrangement of objects by manipulators, results in solution times up to three orders of magnitude faster than existing methods and successfully plans for…
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the…
In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham…
Based on the notion of vectors and linear subspaces for a matroid, we develop a theory of flats and hyperplane arrangements for T-matroids, where T is a tract. This leads to several cryptomorphic descriptions of T-matroids: in terms of its…
Low-rank approximations are essential in modern data science. The interpolative decomposition provides one such approximation. Its distinguishing feature is that it reuses columns from the original matrix. This enables it to preserve matrix…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…
We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the…
This paper is a survey of our work based on the stratified Morse theory of Goresky and MacPherson. First we discuss the Morse theory of Euclidean space stratified by an arrangement. This is used to show that the complement of a complex…
This paper studies the underlying combinatorial structure of a class of object rearrangement problems, which appear frequently in applications. The problems involve multiple, similar-geometry objects placed on a flat, horizontal surface,…
We study the interaction between two structures on the group of polynomial automorphisms of the affine plane: its structure as an amalgamated free product and as an infinite-dimensional algebraic variety. We introduce a new conjecture, and…
The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed…