# HG changeset patch # User Benedikt Fluhr # Date 1498096566 -7200 # Thu Jun 22 03:56:06 2017 +0200 # Node ID 805091eade88efeb11016f1166219abcf1daa92a # Parent 0c0495aee65cdf906cd927eda689433ae59c2fe1 Some Formulas on their own Line diff --git a/00_03_joinTrees.md b/00_03_joinTrees.md --- a/00_03_joinTrees.md +++ b/00_03_joinTrees.md @@ -302,15 +302,19 @@ * **Corollary** (Triangle Inequality)**.** We have -$M_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} h) - \leq - M_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + - M_{J} (\mathcal{R} \mathcal{E} g, \mathcal{R} \mathcal{E} h)$ +$$+M_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} h) +\leq +M_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + +M_{J} (\mathcal{R} \mathcal{E} g, \mathcal{R} \mathcal{E} h) +$$ and -$\mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} h) - \leq - \mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + - \mu_{J} (\mathcal{R} \mathcal{E} g, \mathcal{R} \mathcal{E} h)$. +$$+\mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} h) +\leq +\mu_{J} (\mathcal{R} \mathcal{E} f, \mathcal{R} \mathcal{E} g) + +\mu_{J} (\mathcal{R} \mathcal{E} g, \mathcal{R} \mathcal{E} h) . +$$ Next we show that the interleaving distances provide lower bounds for the corresponding distances of functions diff --git a/00_05_monoidalPosets.md b/00_05_monoidalPosets.md --- a/00_05_monoidalPosets.md +++ b/00_05_monoidalPosets.md @@ -100,9 +100,13 @@ * **Lemma.** For all $(a, b; c, d) \in \mathcal{D}$ we have -$(\epsilon \circ \gamma \circ \delta)((a, b; c, d)) = - \max \{-a, b, -c, d\}$ +$$+(\epsilon \circ \gamma \circ \delta)((a, b; c, d)) = +\max \{-a, b, -c, d\} +$$ and -$(\epsilon \circ \delta)((a, b; c, d)) = - \frac{1}{2} (\max \{-c, b\} + \max \{-a, d\})$. +$$+(\epsilon \circ \delta)((a, b; c, d)) = +\frac{1}{2} (\max \{-c, b\} + \max \{-a, d\}) . +$$ diff --git a/00_07_InterleavingsInDCats.md b/00_07_InterleavingsInDCats.md --- a/00_07_InterleavingsInDCats.md +++ b/00_07_InterleavingsInDCats.md @@ -399,9 +399,14 @@ (@triaIneq) **Corollary** (Triangle Inequality)**.** We have -$M_{\mathcal{S}} (A, C) \leq M_{\mathcal{S}} (A, B) + M_{\mathcal{S}} (B, C)$ +$$+M_{\mathcal{S}} (A, C) \leq M_{\mathcal{S}} (A, B) + M_{\mathcal{S}} (B, C) +$$ and -$\mu_{\mathcal{S}} (A, C) \leq \mu_{\mathcal{S}} (A, B) + \mu_{\mathcal{S}} (B, C)$. +$$+\mu_{\mathcal{S}} (A, C) \leq +\mu_{\mathcal{S}} (A, B) + \mu_{\mathcal{S}} (B, C) . +$$ ## Homomorphisms of *D*-Categories {#homomorphisms-D-cat}