b7414fe14a85 — Benedikt Fluhr <http://bfluhr.com> 7 years ago
Reformulations
4 files changed, 70 insertions(+), 22 deletions(-)

M 00_01_intro.md
M 00_02_ReebGraphs.md
M 00_03_joinTrees.md

M 00_00_00_links.md +1 -0
@@ 8,6 8,7 @@
[topology]: https://en.wikipedia.org/wiki/Topological_space#Definition_via_open_sets
[proper]: https://en.wikipedia.org/wiki/Proper_map
[continuous function]: https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces
+[properties quotient space]: https://en.wikipedia.org/wiki/Quotient_space_%28topology%29#Properties
[Reeb graph]: https://en.wikipedia.org/wiki/Reeb_graph
[Morse]: https://en.wikipedia.org/wiki/Morse_theory#Fundamental_theorems
[extended pseudometric]: https://en.wikipedia.org/wiki/Metric_(mathematics)#Extended_metrics


M 00_01_intro.md +9 -9
@@ 154,7 154,7 @@ such that for all $p \in X$ the estimate
hold?

Next we may ask, how large $\varepsilon$ needs to be
-in order for the answer to *yes*.
+in order for the answer to be *yes*.
This is the motivation behind the following

* **Definition.**

@@ 175,7 175,7 @@ Next we observe that $M$ satisfies the t
Let $h \colon Z \rightarrow \R$ be another continuous function,
then $M(f, h) \leq M(f, g) + M(g, h)$.

-Now we recall that we started with two continuous $f$ and $g$
+Now we recall that we started with two continuous functions $f$ and $g$
and each might describe the luminance of an image
or the distance to a certain shape
and this is all good,

@@ 282,7 282,7 @@ Completely analogously to the above we h
Let $h \colon Z \rightarrow \R$ be another continuous function,
then $\mu(f, h) \leq \mu(f, g) + \mu(g, h)$.

-Moreover we have the following relationship to the absolute distance.
+Moreover we have the following relation to the absolute distance.

* **Lemma.**
We have $\mu(f, g) \leq M(f, g)$.

@@ 374,9 374,9 @@ the *category of $\overline{\R}$-spaces*

* **Definition.**
For two continuous functions $f \colon X \rightarrow \overline{\R}$ and
-$g \colon Y \rightarrow \overline{\R}$
-a *homomorphism $\varphi$ from $f$ to $g$* also denoted by
-$\varphi \colon f \rightarrow g$ is a continuous map
+$g \colon Y \rightarrow \overline{\R}$,
+a *homomorphism $\varphi$ from $f$ to $g$*, also denoted by
+$\varphi \colon f \rightarrow g$, is a continuous map
$\varphi \colon X \rightarrow Y$ such that the diagram
$$\xymatrix{ @@ 396,12 396,12 @@ commutes. We define the composition of homomorphisms in the *category of \overline{\R}-spaces* by the composition of maps. - The *category of \R-spaces* is the full subcategory of + The *category of \R-spaces* is the [full subcategory][] of all real-valued continuous functions. * *Remark.* -With these definitions in place a reformulation of question @qAbs -is whether f and g are isomorphic as objects of the +With these definitions in place a reformulation of question @qAbs is, +whether f and g are isomorphic as objects of the category of \overline{\R}-spaces. ### Further Prior Art  M 00_02_ReebGraphs.md +56 -9 @@ 4,30 4,77 @@ In this section we introduce Reeb graphs * **Definition** (Reeb Graph)**.** Given a continuous function f \colon X \rightarrow \overline{\R} -and x \in X +and x \in X, let \pi_f (x) be the connected component of x in f^{-1}(f(x)). In this way we obtain a function \pi_f \colon X \rightarrow \pi_f (X) \subset 2^X and we endow -\pi_f (X) with the quotient topology. -By the universal property -of the quotient space there is a unique continuous function +\pi_f (X) with the quotient topology[^refQuoTop]. +By the +[universal property of the quotient space][properties quotient space] +there is a unique continuous function \mathcal{R} f \colon \pi_f (X) \rightarrow \overline{\R} such that +$$
+\xymatrix{
+  X
+  \ar[r]^{\pi_f}
+  \ar[dr]_f
+  &
+  \pi_f (X)
+  \ar[d]^{\mathcal{R} f}
+  \\
+  &
+  \overline{\R}
+}
+$$+commutes, i.e. \mathcal{R} f \circ \pi_f = f. For another continuous function g \colon Y \rightarrow \overline{\R} and a homomorphism \varphi \colon f \rightarrow g -we may use the universal property of \pi_f again +we may use the universal property of \pi_f again, to obtain a unique continuous map \mathcal{R} \varphi \colon \pi_f (X) \rightarrow \pi_g (Y) -with +such that +$$
+\xymatrix{
+  X
+  \ar[d]_{\pi_f}
+  \ar[r]^{\varphi}
+  &
+  Y
+  \ar[d]^{\pi_g}
+  \\
+  \pi_f (X)
+  \ar[r]_{\mathcal{R} \varphi}
+  &
+  \pi_g (Y)
+}
+$$+commutes, i.e. \mathcal{R} \varphi \circ \pi_f = \pi_g \circ \varphi. +[^refQuoTop]: see for example [@bredon1993, definition I.13.1] + * **Lemma.** Let f \colon X \rightarrow \overline{\R} and g \colon Y \rightarrow \overline{\R} be continuous functions and let \varphi \colon f \rightarrow g be a homomorphism, -then \mathcal{R} g \circ \mathcal{R} \varphi = \mathcal{R} f. +then the diagram +$$
+\xymatrix{
+  \pi_f (X)
+  \ar@/^/[rr]^{\mathcal{R} \varphi}
+  \ar[dr]_{\mathcal{R} f}
+  & &
+  \pi_g (Y)
+  \ar[dl]^{\mathcal{R} g}
+  \\
+  &
+  \overline{\R}
+}
+
+commutes.
Or in other words $\mathcal{R} \varphi$ is a homomorphism
from $\mathcal{R} f$ to $\mathcal{R} g$
in the category of $\overline{\R}$-spaces.

@@ 101,6 148,6 @@ universal property of $\pi_f$ that
The previous two lemmata imply that $\mathcal{R}$ is an endofunctor
on the category of $\overline{\R}$-spaces.
Later we will define an interleaving distance on Reeb graphs,
-but first we will introduce join trees and their interleavings
-as they are easier to understand and may provide us with some intuition
+but first we will introduce join trees and their interleavings.
+Join trees are easier to understand and may provide us with some intuition
for understanding the more sophisticated interleavings of Reeb graphs.


M 00_03_joinTrees.md +4 -4
@@ 46,11 46,11 @@ is shown in the image below.
Before we get to interleavings we make some auxiliary definitions.

* **Definition.**
-Let
+We set
$D^{\perp} := - \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$
-then $D^{\perp}$ is a monoid by component-wise addition.
+ \set{(a, b)}{(-\infty, \infty] \times (-\infty, \infty]}{a + b \geq 0}$. +Component-wise addition yields the structure of a monoid on$D^{\perp}$. Next we define two weightings on$D^{\perp}\$.

* **Definition.**

@@ 150,7 150,7 @@ the *relative interleaving distance of

We note that these definitions of an interleaving distance
are different from the one by @morozov2013.
-The subsequent corollary shows however that
+However the subsequent corollary shows that
our absolute interleaving distance is equal to the distance by @morozov2013.
The reason we gave a different definition is that
now we can phrase the computation