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}