94507d9882e3 — Benedikt Fluhr <http://bfluhr.com> 6 years ago
Now using YAML Structure File
18 files changed, 0 insertions(+), 30 deletions(-)

M 00_12_EqualityOfInterlDist.md => EqualityOfInterlDist.md
M 00_07_InterleavingsInDCats.md => InterleavingsInDCats.md
M 99_03_RGraphs.md => RGraphs.md
M 00_02_ReebGraphs.md => ReebGraphs.md
M 99_04_skeletenEpigraph.md => SkeletonEpigraph.md
M 00_09_complPersistenceEnhancements.md => complPersistenceEnhancements.md
M 99_02_constructibleRSpaces.md => constructibleRSpaces.md
M 00_06_interleavingReebGraphs.md => interleavingReebGraphs.md
M 00_01_intro.md => intro.md
M 00_03_joinTrees.md => joinTrees.md
M 00_00_00_macros.tex => macros.tex
M meta.md => meta.yaml
M 00_05_monoidalPosets.md => monoidalPosets.md
M 00_11_negEnhJoinTrees.md => negEnhJoinTrees.md
M 00_08_posPersistenceEnhancements.md => posPersistenceEnhancements.md
M 00_04_precosheaves.md => precosheaves.md
M 00_10_someEquivalences.md => someEquivalences.md

M 00_12_EqualityOfInterlDist.md => EqualityOfInterlDist.md +0 -2
@@ 1,5 1,3 @@
-# Equality of Interleaving Distances
-
Finally we get to proving theorem @interEq.
We reuse the notation and the definitions from the previous two subsections.
We aim to show that the Reeb precosheaf $\mathcal{C}$


M 00_07_InterleavingsInDCats.md => InterleavingsInDCats.md +0 -2
@@ 1,5 1,3 @@
-# Interleavings in *D*-Categories
-
Up to this point we have seen two notions of an interleaving,
the first for join trees and the second for precosheaves.
In order to show theorem @interEq we will use several more


M 99_03_RGraphs.md => RGraphs.md +0 -2
@@ 1,5 1,3 @@
-## Graphs over the Reals
-
* **Definition.**
Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
non-negative integer $n$.


M 00_02_ReebGraphs.md => ReebGraphs.md +0 -2
@@ 1,5 1,3 @@
-# Reeb Graphs
-
In this section we introduce Reeb graphs.

* **Definition** (Reeb Graph)**.**


M 99_04_skeletenEpigraph.md => SkeletonEpigraph.md +0 -2
@@ 1,5 1,3 @@
-## A Skeleton for the Epigraph {#skeleton-epigraph}
-
Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
non-negative integer $n$
and let $X$ be an $S$-skeleton for a bounded $\R$-space.


M 00_09_complPersistenceEnhancements.md => complPersistenceEnhancements.md +0 -2
@@ 1,5 1,3 @@
-# Complete Persistence-Enhancements
-
In the previous section we defined positive persistence enhancements
of functors on $\R$-spaces and provided one for $\mathcal{C}$,
thereby finally establishing that the interleaving distances of


M 99_02_constructibleRSpaces.md => constructibleRSpaces.md +0 -2
@@ 1,5 1,3 @@
-## Constructible Spaces over the Reals {#constructible-spaces}
-
* **Definition.**
Let $S = \{a_1 < a_2 < \dots < a_n\} \subset \R$ for some
non-negative integer $n$


M 00_06_interleavingReebGraphs.md => interleavingReebGraphs.md +0 -2
@@ 1,5 1,3 @@
-# Interleaving Reeb Graphs
-
In this section we define the interleaving distance of Reeb graphs
due to @deSilva2016.
Strictly speaking it is somewhat misleading to name this the interleaving


M 00_01_intro.md => intro.md +0 -2
@@ 1,5 1,3 @@
-# Introduction
-
The topic of the present text is the interleaving distance
of join trees by @morozov2013.
Just like this paper we focus on topological data analysis (TDA),


M 00_03_joinTrees.md => joinTrees.md +0 -2
@@ 1,5 1,3 @@
-# Join Trees
-
In this section we define join trees and their interleaving distance
due to @morozov2013.


M 00_00_00_links.md => links.md +0 -0

M 00_00_00_macros.tex => macros.tex +0 -0

M meta.md => meta.yaml +0 -0

M 00_05_monoidalPosets.md => monoidalPosets.md +0 -2
@@ 1,5 1,3 @@
-# Monoidal Posets for 1D-Interleavings
-
Before we defined [interleavings of join trees](#join-trees) we introduced the
poset $D^{\perp}$ and the two weightings
$\epsilon'$ and $\epsilon''$ on $D^{\perp}$.


M 00_11_negEnhJoinTrees.md => negEnhJoinTrees.md +0 -2
@@ 1,5 1,3 @@
-# A negative Enhancement of Join Trees
-
To provide a negative persistence-enhancement
for $\mathcal{R} \circ \mathcal{E}$
we first provide one for $\mathcal{E}$.


M 00_08_posPersistenceEnhancements.md => posPersistenceEnhancements.md +0 -2
@@ 1,5 1,3 @@
-# Positive Persistence-Enhancements
-
In the previous section we defined strict $D$-categories,
interleavings of objects in $D$-categories,
and showed that interleavings of precosheaves


M 00_04_precosheaves.md => precosheaves.md +0 -2
@@ 1,5 1,3 @@
-# Interlude on Precosheaves {#precosheaves}
-
In this section we develop the theory of precosheaves to the extend needed
for the interleaving distance of Reeb graphs by @deSilva2016
and subsequent sections.


M 00_10_someEquivalences.md => someEquivalences.md +0 -2
@@ 1,5 1,3 @@
-# Some Equivalences
-
In this section we move one step closer to proving theorem @interEq.
We consider interleavings of precosheaves in the image
of the functor $\mathcal{C} \mathcal{E}$ and transform those into