AENC/resampling_chain Changeset - b554f1fbc443 · Centrum Wiskunde & Informatica (CWI)

Changeset - b554f1fbc443

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Andras Gilyen - 8 years ago 2017-09-08 17:09:07
gilyen@clayoquot.swat.cwi.nl

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1 file changed with 5 insertions and 3 deletions:

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@@ @@ -62,6 +62,8 @@ @@
 \newcommand{\paths}[1]{\mathcal{P}\left(#1\to\mathbf{1}\right)}
 \newcommand{\start}[1]{\textsc{start}\left(#1\right)}
 \newcommand{\initone}[1]{\textsc{InitOne}\left(#1\right)}
 \newcommand{\patch}[1]{\textsc{Patch}\left(#1\right)}
 \newcommand{\patches}[1]{\textsc{Patches}\left(#1\right)}
 \newcommand{\maxgap}[1]{\mathrm{maxgap}\left(#1\right)}
 \newcommand{\gaps}[1]{#1_{\mathrm{gaps}}}
 \renewcommand{\P}{\mathbb{P}}
 \end{proof}
 	\begin{definition}[Connected patches]
-		Let $P\subseteq V$ be a connected component of $G$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $P\in \mathcal{P}$ denote the event that one of the patches is equal to $P$.
+		Let $P\subseteq V$ be a connected component of $G$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $\patch{P}$ denote the event that one of the patches is equal to $P$.
 		In other words
 		\begin{align*}
-		P\in\mathcal{P} := \NZ{\overline{\partial}P} \cap \Z{P}.
+		\patch{P} := \NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}
-		For $\mathcal{I}'\subseteq 2^{2^V}$ a set of patches we denote by $\mathcal{P}'\in \mathcal{P}$ the event that $\mathcal{P}'$ is a subset of the patches, i.e.,
+		For a set of patches $\mathcal{P}$ 	</i>
 		\begin{align*}
 			\mathcal{P}'\in \mathcal{P} := \bigcup_{P\in \mathcal{P}'}\NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}

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