Related papers: Monotone Paths on Cross-Polytopes
The topologies permitted in joint ocular dominance (OD), orientation preference (OP), and direction preference (DP) maps in the primary visual cortex (V1) are considered, with the aim of finding a maximally symmetric periodic case that can…
The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(\Phi^3)$ theory. Its faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon…
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies…
The correlation of gaps in dimer systems was introduced in 1963 by Fisher and Stephenson, who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we…
The multiplihedra {M_n} form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of…
Surface plasmon polaritons (SSP), moving along a smooth curved interface between two isotropic media with slowly varying dielectric permittivities and magnetic permeabilities and supporting SSP, are studied theoretically. Solutions of…
Directed paths have been used extensively in the scientific literature as a model of a linear polymer. Such paths models in particular the conformational entropy of a linear polymer and the effects it has on the free energy. These directed…
According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the…
Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics,…
A concept of generalized regular polytope is introduced in this work. The number of its (1...n-1)-dimensional elements is not necessarily integer, though all the combinatorial and metric properties meet those of regular polytopes in a…
Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections…
Let $P$ be a polytope. The hitting number of $P$ is the smallest size of a hitting set of the facets of $P$, i.e., a subset of vertices of $P$ such that every facet of $P$ has a vertex in the subset. An extended formulation of $P$ is the…
Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets. These polytopes were first studied by Lov\'{a}sz as a means of establishing the chromatic number of distance graphs on spheres,…
We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with…
This paper introduces an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations. This notation allows us to view inclusion of faces as the process of contracting tree edges. Our notation…
A (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with…
Let $f$ be a Morse function on a smooth compact manifold $M$ with boundary. The path component $\mathrm{PH}_f^{-1}(D)$ containing $f$ of the space of Morse functions giving rise to the same Persistent Homology $D=\mathrm{PH}(f))$ is shown…
Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such…
Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a…
Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.