The Simplex Algorithm in Dimension Three

Computing the Ehrhart quasi-polynomial of a rational simplex

We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the kth coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to k-dimensional faces of the polytope. We discuss possible extensions and open questions.

基于4维二元单纯码几何的对称$(15,8,4)$-设计

Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$

本文研究了$n$元集合的$2m$元子集点线几何中，对应于二元单纯码及对称$(n,2m,m)$-设计的极大团，并在$k=4$时为五个著名的对称$(15,8,4)$-设计提供了几何解释。

高维中的莫利单纯形：正则性、反射与反例

Morley Simplices in Higher Dimensions: Regularity, Reflections, and Counterexamples

本文研究高维单纯形中莫利单纯形为正则的条件，证明正则单纯形为孤立解，并构造反例表明在六维及以上存在无反射对称性的正则莫利单纯形，四维给出非正则精确示例。

Algorithmic Symplectic Packing

In this article we explore a symplectic packing problem where the targets and domains are $2n$-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to $\mathbb{Z}^n$, and we require the embeddings to induce isomorphisms between first homology groups. In this case, Maley, Mastrangeli and Traynor showed that the problem can be reduced to a combinatorial optimization problem, namely packing certain allowable simplices into a given standard simplex. They designed a computer program and presented computational results. In particular, they dete…

The worst way to collapse a simplex

In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose the collapses in such a way that one arrives at a nontrivial complex which admits no collapsing moves. Here we examine this phenomenon in the case of a simplex. In particular we characterize all values of $n$ and $d$ so that the n-simplex may collapse to a d-complex from which no further collapses are possible. Equivalently and in the language of high-dimensional generalizations of trees, we const…

对 M. S. Baladram《论单纯形 t-设计的显式构造》一文的评论与更正

Comment and correction for "On Explicit Construction of Simplex $t$-designs" by M. S. Baladram

本文指出 Baladram 关于 d=3 时存在 3 点单纯形 3-设计的论断有误，提出修正证明方法，并据此得到规模为 d^2 的单纯形 3-设计，进而提出对称受限单纯形 t-设计概念。