拆分多维项链

Combinatorial Algorithms for Multidimensional Necklaces

A necklace is an equivalence class of words of length $n$ over an alphabet under the cyclic shift (rotation) operation. As a classical object, there have been many algorithmic results for key operations on necklaces, including counting, generating, ranking, and unranking. This paper generalises the concept of necklaces to the multidimensional setting. We define multidimensional necklaces as an equivalence classes over multidimensional words under the multidimensional cyclic shift operation. Alongside this definition, we generalise several problems from the one dimensional setting to the multid…

Splitting multidimensional necklaces and measurable colorings of Euclidean spaces

A necklace splitting theorem of Goldberg and West asserts that any k-colored (continuous) necklace can be fairly split using at most k cuts. Motivated by the problem of Erdős on strongly nonrepetitive sequences, Alon et al. proved that there is a (t+3)-coloring of the real line in which no necklace has a fair splitting using at most t cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is k-coloring of R^{d} such that no cube has a fair splitting of size t (using at most t hyperplanes orthogonal to each of the axes), provided k>(t+4)^{d}-(t+3)^…

Multidimensional necklaces and measurable colorings of R^n

A well known generalization of Alon's "splitting nacklace theorem" by Longueville and Zivaljevic states that every k-colored n-dimensional cube can be fairly split using only k cuts in each dimension. Here we prove that for every t there exist a finite coloring (with at least (t+4)^d - (t+3)^d + (t+2)^d - 2^d + d(t+2) +3 different colors) of R^n such that no n-dimensional cube can be fairly split using at most t cuts in each dimension. In particular there is a finite coloring of R^n such that no two disjoint n-dimensional cubes have the same measure of each color.

Splitting necklaces, with constraints

We prove several versions of N. Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace (including "degenerate pieces" if they exist), provided the number of thieves $r=p^ν$ is a prime power. (2) The "binary splitting theorem" claims that if $r=2^d$ and the thieves are associated with…

基于精确射线分裂迹公式的二元项链的组合恒等式

Combinatorial identities for binary necklaces from exact ray-splitting trace formulae

基于一维射线分裂系统的精确迹公式，我们得到了循环二元序列（Pólya项链）的新的组合恒等式。

多面体复形：映射、链复形与几何项链

Polytopal complexes: maps, chain complexes and... necklaces

定义了两个多面体复形之间的多面体映射的概念。令人惊讶的是，这个定义非常简单，并且自然地扩展了单纯形和三次映射的定义。然后可以在相关的链复形之间定义诱导链映射。最后，我们利用这个新工具给出了Alon分裂项链定理的第一个组合证明。本文结束了开放的问题，如存在的Sperner引理的多面体复杂或存在的一个三次逼近定理。