805091eade88 — Benedikt Fluhr <http://bfluhr.com> 5 years ago
Some Formulas on their own Line
3 files changed, 27 insertions(+), 14 deletions(-)

M 00_03_joinTrees.md
M 00_05_monoidalPosets.md
M 00_07_InterleavingsInDCats.md
M 00_03_joinTrees.md +12 -8
@@ 302,15 302,19 @@ is completely analogous.
 
 * **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

          
M 00_05_monoidalPosets.md +8 -4
@@ 100,9 100,13 @@ Next we provide explicit formulas for th
 
 * **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
-$(\epsilon \circ \delta)((a, b; c, d)) =
- \frac{1}{2} (\max \{-c, b\} + \max \{-a, d\})$.
+$$
+(\epsilon \circ \delta)((a, b; c, d)) =
+\frac{1}{2} (\max \{-c, b\} + \max \{-a, d\}) .
+$$
 

          
M 00_07_InterleavingsInDCats.md +7 -2
@@ 399,9 399,14 @@ The proof that the second triangle commu
 
 (@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}