相关论文: Shelling and triangulating the (extra)ordinary pol…
We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin.…
This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces…
Rectangulations are decompositions of a square into finitely many axis-aligned rectangles. We describe realizations of $(n-1)$-dimensional polytopes associated with two combinatorial families of rectangulations composed of $n$ rectangles.…
We report on enumerating the triangulations of cyclic polytopes with the new software mptopcom. This is relevant for its connection with higher Stasheff-Tamari orders, which occur in category theory and algebraic combinatorics.
We show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized Schur polynomials introduced by Sergeev and Veselov.
We classify all wormhole singularities, i.e. cyclic quotient surface singularities admitting at least two extremal P-resolutions, thereby solving an open problem posed by Urz\'ua. Our approach introduces a new combinatorial framework based…
A well-known conjecture of McMullen, proved by Billera, Lee and Stanley, describes the face numbers of simple polytopes. The necessary and sufficient condition is that the toric g-vector of the polytope is an M-vector, that is, the vector…
Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all…
Very recently, Galashin, Postnikov, and Williams introduced the notion of higher secondary polytopes, generalizing the secondary polytope of Gelfand, Kapranov, and Zelevinsky. Given an $n$-point configuration $\mathcal{A}$ in…
A subgroup H of a reductive group G is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on G/H as a semigroup of convex polytopes. From this we…
Polytope numbers for a given polytope are an integer sequence defined by the combinatorics of the polytope. Recent work by H. K. Kim and J. Y. Lee has focused on writing polytope number sequences as sums of simplex number sequences. We…
Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit…
The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix…
In this paper, we explore the geometry and the arithmetic of a family of polytopal sphere packings induced by regular polytopes in any dimension. We prove that every integral polytope is crystallographic, and we show that there are 11…
A polytope is called {\em regular-faced} if every one of its facets is a regular polytope. The 4-dimensional regular-faced polytopes were determined by G. Blind and R. Blind \cite{BlBl2,roswitha,roswitha2}. The last class of such polytopes…
We introduce a new algebraic construction, {\em monop}, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of…
Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings…
We prove a bijection between the triangulations of the 3-dimensional cyclic polytope C(n+2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to…
Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$-polynomial for the quotient ring $R/I_G$, where $I_G$ is the toric ideal of $G$, in terms of…
In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order…