On quantum limits on flat tori.
On quasistars in -cubes
On rainbowness of semiregular polyhedra
We introduce the rainbowness of a polyhedron as the minimum number such that any colouring of vertices of the polyhedron using at least colours involves a face all vertices of which have different colours. We determine the rainbowness of Platonic solids, prisms, antiprisms and ten Archimedean solids. For the remaining three Archimedean solids this parameter is estimated.
On rhombic dodecahedra.
On Schläfli's reduction formula.
On simple polytopes.
On solution sets of information inequalities
We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce structural properties of Bayesian networks, which is important within causal inference.
On splitting infinite-fold covers
Let X be a set, κ be a cardinal number and let ℋ be a family of subsets of X which covers each x ∈ X at least κ-fold. What assumptions can ensure that ℋ can be decomposed into κ many disjoint subcovers? We examine this problem under various assumptions on the set X and on the cover ℋ: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of ℝⁿ by polyhedra and by arbitrary convex sets. We focus on...
On Surface-Minimizing Polyhedral Decompositions.
On the chiral Archimedean solids.
On the Classification of Toric Fano Varieties.
On the computation of Clebsch-Gordan coefficients and the dilation effect.
On the convex hull of projective planes
We study the finite projective planes with linear programming models. We give a complete description of the convex hull of the finite projective planes of order 2. We give some integer linear programming models whose solution are, either a finite projective (or affine) plane of order n, or a (n+2)-arc.
On the diameter of matroid ports.
On the Difficulty of Triangulating Three-Dimensional Nonconvex Polyhedra.
On the Expected Number of k-Sets.
On the Face-Vector of a 5-Valent Convex 3-Polytope
On the face-vectors of trivalent convex polyhedra
On the graph labellings arising from phylogenetics
We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.
On the Graver complexity of codimension 2 matrices.