Related papers: Skeletal Rigidity of Phylogenetic Trees
We introduce the concept of orthogonal structure on complex Grassmannians. Based on this structure, we define the notion of orthogonal mappings. This class of maps generalizes holomorphic maps between the Shilov boundaries of type-I bounded…
Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of…
In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction…
One regards spaces of trees as stratified spaces, to study distributions of phylogenetic trees. Stratified spaces with may have cycles, however spaces of trees with a fixed number of leafs are contractible. Spaces of trees with three leafs,…
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
There are several tools available to infer phylogenetic trees, which depict the evolutionary relationships among biological entities such as viral and bacterial strains in infectious outbreaks, or cancerous cells in tumor progression trees.…
Graphs are interesting structures: extremely useful to depict real-life problems, extremely easy to understand given a sketch, extremely complicated to represent formally, extremely complicated to compare. Phylogeny is the study of the…
This is the first in a series of papers on geometric mapping theory in Carnot groups -- and more generally equiregular manifolds -- in which we prove a number of new structural results for Sobolev (in particular quasisymmetric) mappings,…
Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have…
A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose internal edges have a positive weight. Given a set of n leaves, the collection of all phylogenetic trees with this leaf set can be assembled into a metric cube…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
Skeletal polyhedra and polygonal complexes are finite or infinite periodic structures in 3-space with interesting geometric, combinatorial, and algebraic properties. These structures can be viewed as finite or infinite periodic graphs…
Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a…
Many mechanical structures, both engineered and biological, combine heavy rigid elements such as bones and beams with lightweight flexible ones such as cables and membranes. These are referred to as tensegrities, reflecting that cables can…
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss…
Phylogenetic trees provide a fundamental representation of evolutionary relationships, yet the combinatorial explosion of possible tree topologies renders inference computationally challenging. Classical approaches to characterizing tree…
The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the…
We prove several hardness results on folding origami crease patterns. Flat-folding finite crease patterns is fixed-parameter tractable in the ply of the folded pattern (how many layers overlap at any point) and the treewidth of an…