Related papers: Generalized Pitman-Stanley polytope: vertices and …
The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…
In quantum mechanics textbooks, a single-particle scattering theory is introduced. In the present work, a generalized scattering theory is presented, which can be in principle applied to the scattering problems of arbitrary number of…
Given a finite quiver (directed graph) without loops and multiedges, the convex hull of the column vector of the incidence matrix is called the directed edge polytope and is an interesting example of lattice polytopes. In this paper, we…
Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as…
The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge's $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a…
In this paper we motivate some new directions of research regarding the lattice width of convex bodies. We show that convex bodies of sufficiently large width contain a unimodular copy of a standard simplex. This implies that every lattice…
A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbol{\sigma}(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbol{\sigma}: \mathbb{R}^{n-1} \rightarrow…
For two non-congruent regular polygons of the same type, the method of finding the points in the plane at the equal distances to the vertices, is established. The existence of two points with this property is proved for two polygons with a…
A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full…
Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…
The Chan-Robbins-Yuen polytope ($CRY_n$) of order $n$ is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \ldots , 0, -1)$. The volume…
Let $\rho$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq \rho(x,y)$. The question on classifying…
Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the…
We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann-Petty…
Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling…
In the vicinity of a massive object of various scales (ranging from young stars to galactic nuclei), mass flow creates a spectacular structure combining a thin disk and collimated jet. Despite a wide range of scaling parameters (such as…
Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data…
We give a new definition of lattice-face polytopes by removing an unnecessary restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and show that with the new definition, the Ehrhart polynomial of a lattice-face polytope…
We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call (generalized) oranges. Such partitions are composed of a finite number of maximal faces with exactly one shared medial face. We…
We initiate a systematic study of non-planar on-shell diagrams in N=4 SYM and develop powerful technology for doing so. We introduce canonical variables generalizing face variables, which make the dlog form of the on-shell form explicit. We…