相关论文: 2-neighborly 0/1-polytopes of dimension 7

Inspired by the definition of balanced neighborly spheres, we define balanced neighborly polynomials and study the existence of these polynomials. The goal of this article is to construct balanced neighborly polynomials of type $(k,k,k,k)$…

We show that there are 105 isometric classes of fractal cubes of order 3 with 7 copies possessing one-point intersection property which are subdivided to 5 bi-Lipschitz classes of dendrites and 7 bi-Lipschitz classes of non-dendrites

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative…

Motivated by the problem of bounding the number of rays of plane tropical curves we study the following question: Given $n\in\mathbb{N}$ and a unimodular $2$-simplex $\Delta$ what is the maximal number of vertices a lattice polytope…

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

It is demonstrated that each nearly neighbourly family of standard boxes in $\mathbb{R}^3$ has at most 12 elements. A combinatorial classification of all such families that have exactly 12 elements is given. All families satisfying an extra…

For each $d\geq 3$, $n \geq 10$, and $k_1, k_2, \ldots, k_{d-1}\geq 2$ with $k_1+k_2+\ldots+k_{d-1}\leq n-1$, we construct a regular $d$-polytope whose automorphism group is of order $2^n$ and whose Schl\"afli type is $\{2^{k_1},2^{k_2},…

In the article, a series of neigbourly polyhedra is constructed. They have $N=2d+4$ vertices and are embedded in $\mathbb R^{2d}$. Their (affine) Gale diagrams in $\mathbb R^2$ have $d+3$ black points that form a convex polygon. These Gale…

A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…

We enumerate topologically-inequivalent compact Calabi-Yau threefold hypersurfaces. By computing arithmetic and algebraic invariants and the Gopakumar-Vafa invariants of curves, we prove that the number of distinct simply connected…

We provide explicit combinatorial formulas for Ottaviani's degree 15 invariant which detects cubics in 5 variables that are sums of 7 cubes. Our approach is based on the chromatic properties of certain graphs and relies on computer searches…

We define an analogue of the cube and an analogue of the 5-wedge in higher dimensions, each with $2d+2$ vertices and $d^2+2d-3$ edges. We show that these two are the only minimisers of the number of edges, amongst d-polytopes with $2d+2$…

We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in…

We study two simple modifications of self-avoiding polygons. Osculating polygons are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest neighbour…

In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that…

We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52,989,400,714,478 and up to isomorphism to be 105,978,177,936,292. We obtained these results by combining computer search with recently…

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the…