Algorithms to test open set condition for self-similar set related to P.V. numbers

Fix a P.V. number $\lambda ^{-1}>1.$ Given $\mathbf{p}=(p_{1},\cdots,p_{m})\in \mathbb{N}^{m}$, \mathbf{b}=(b_{1},\cdots,b_{m})\in \mathbb{Q^{m}, for the self-similar set $E_{\mathbf{p},\mathbf{b}}=\cup_{i=1}^{m}(\lambda ^{p_{i}}E_{\mathbf{p},\mathbf{b}}+b_{i})$ we find an efficient algorithm to test whether $E_{\mathbf{p},\mathbf{b}}$ satisfies the open set condition (strong separation condition) or not

Order continuous extensions of positive compact operators on Banach lattices

Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and $T: G\rightarrow F$ be an order continuous positive compact (resp. weakly compact) operators. We show that if $G$ is an ideal or an order dense sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly compact) positive extension to $E$ which is likewise order continuous on $E$. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on $E$.Comment: 7 page