ﻻ يوجد ملخص باللغة العربية
Let $n geq r geq s geq 0$ be integers and $mathcal{F}$ a family of $r$-subsets of $[n]$. Let $W_{r,s}^{mathcal{F}}$ be the higher inclusion matrix of the subsets in ${mathcal F}$ vs. the $s$-subsets of $[n]$. When $mathcal{F}$ consists of all $r$-subsets of $[n]$, we shall simply write $W_{r,s}$ in place of $W_{r,s}^{mathcal{F}}$. In this paper we prove that the rank of the higher inclusion matrix $W_{r,s}$ over an arbitrary field $K$ is resilient. That is, if the size of $mathcal{F}$ is close to ${n choose r}$ then $mbox{rank}_{K}(W_{r,s}^{mathcal{F}}) = mbox{rank}_{K}(W_{r,s})$, where $K$ is an arbitrary field. Furthermore, we prove that the rank (over a field $K$) of the higher inclusion matrix of $r$-subspaces vs. $s$-subspaces of an $n$-dimensional vector space over $mathbb{F}_q$ is also resilient if ${rm char}(K)$ is coprime to $q$.
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i eq j$) is nonzero whenever ${i,j}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult par
Based on the ranks of reduced density matrices, we derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement. In a similar way we obtain necess
A {it sign pattern matrix} is a matrix whose entries are from the set ${+,-, 0}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is
We find the $ell$-weights and the $ell$-weight vectors for the highest $ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(mathcal L(mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Having t