We construct a spanoid $\mathcal{S}$ on $n$ elements with $\textsf{rank}(\mathcal{S}) \ge n^c \textsf{f-rank}(\mathcal{S})$ where $c = \log_5 3 - \log_5 2.5 \approx 0.113283$. This answers a question of Dvir-Gopi-Wigderson [DGW18].
The maximal rank of an abstract regular polytope for M24, the Mathieu group of degree 24, is 5. There are four such polytopes of rank 5 and in this note we describe them using Curtis's MOG. This description is then used to give an upper bound for the diameter of the chamber graphs of these polytopes.
A multiplicative 3-matching in a group $G$ is a triple of sets $\{a_i\}, \{b_i\}, \{c_i\} \subset G$ such that $a_ib_jc_k = 1$ if and only if $i=j=k$. Here we record the fact that $\text{PSL}(2,p)$ has no multiplicative 3-matching of size greater than $O(p^{8/3})$, yet the slice rank of its group algebra's multiplication tensor is at least $Ω(p^3)$ over any field. This gives a negative answer to a conjecture of Petrov.
本文证明了一个连通图的线性秩宽为1当且仅当它是一个距离遗传图且它的分裂分解树是一条路.一个直接的结果是，我们可以在线性时间内判定一个图的线性秩宽是否至多为1，如果不是，则给出一个障碍.其他直接的结果是线性秩宽为1的图的几个特征。特别地，一个连通图具有线性秩宽1当且仅当它局部等价于一个毛毛虫当且仅当它是一条路的顶点子图[O-juanKwon and Sang-il Oum，Graphs of small rank-width are approximates-minor of graphs of small tree-width，arxiv：1203.3606]当且仅当它不包含co-K_2图、Net图和5-圈图作为顶点子图[Isolde Adler，Arthur M. Farley和Andrzej Proskurowski，线性秩宽度至多为1的障碍，arxiv：1106.2533]。
The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $\F$ is at most $k$ over the algebraic closure of $\F$, where $k$ is a given positive integer. We estimate the arithmetic complexity of our algorithm.
