## 数学代写|表示论代写Representation theory代考|MATH4314

2022年12月30日

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## 数学代写|表示论代写Representation theory代考|Computing Persistence of Simplicial Towers

In this section, we present an algorithm for computing the persistence of a simplicial tower. Consider a simplicial tower $\mathcal{K}: K_0 \stackrel{f_0}{\rightarrow} K_1 \stackrel{f_1}{\rightarrow} K_2 \stackrel{f_2}{\rightarrow} \cdots \stackrel{f_{n-1}}{\rightarrow} K_n$ and the map $f_{i j}: K_i \rightarrow K_j$ where $f_{i j}=f_{j-1} \circ \cdots \circ f_{i+1} \circ f_i$. To compute the persistent homology for a simplicial filtration, the persistence algorithm in the previous chapter essentially maintains a consistent basis by computing the image $f_{i j_z}\left(B_i\right)$ of a basis $B_i$ of $\mathrm{H}\left(K_i\right)$. As the algorithm moves through an inclusion in the filtration, the homology basis elements get created (birth) or are destroyed (death). Here, for towers, instead of a consistent homology basis, we maintain a consistent cohomology basis. We need to be aware that, for cohomology, the induced maps from $f{i j}: K_i \rightarrow K_j$ are reversed, that is, $f_{i j}^: \mathrm{H}^p\left(K_i\right) \leftarrow \mathrm{H}^p\left(K_j\right)$; refer to Section 2.5.4. So, if $B^i$ is a cohomology basis of $\mathrm{H}^p\left(K_i\right)$ maintained hy the algorithm, it computes implicitly the pre-image $f_{i j}^{*-1}\left(B^i\right)$. Dually, this implicitly maintains a consistent homology basis and thus captures all information about persistent homology as well.

We maintain a consistent cohomology basis using a notion called annotations [60] which are binary vectors assigned to simplices. These annotations are updated as we go forward through the sequence in the given tower. This implicitly maintains a cohomology basis in the reverse direction where the birth and death of cohomology classes coincide with the death and birth, respectively of homology classes.

Definition 4.6. (Annotation) Given a simplicial complex $K$, let $K(p)$ denote the set of $p$-simplices in $K$. An annotation for $K(p)$ is an assignment a : $K(p) \rightarrow \mathbb{Z}2^g$ of a binary vector $\mathrm{a}\sigma=\mathrm{a}(\sigma)$ of length $g$ to each $p$-simplex $\sigma \in K$. The binary vector $\mathrm{a}\sigma$ is called the annotation for $\sigma$. Each entry ” 0 ” or ” 1 ” of $\mathrm{a}\sigma$ is called its element. Annotations for simplices provide an annotation for every $p$-chain $c_p: \mathbf{a}{c_p}=\sum{\sigma \in c_p} \mathbf{a}_\sigma$.

An annotation a : $K(p) \rightarrow \mathbb{Z}_2^g$ is valid if the following two conditions are satisfied:

• $g=\operatorname{rank} \mathrm{H}_{\mathrm{p}}(\mathrm{K})$, and
• two $p$-cycles $z_1$ and $z_2$ have $\mathbf{a}{z_1}=\mathbf{a}{z_2}$ if and only if their homology classes are identical, that is, $\left[z_1\right]=\left[z_2\right]$.

## 数学代写|表示论代写Representation theory代考|Algorithm

Consider the persistence module $\mathrm{H}p \mathcal{K}$ induced by a simplicial tower $\mathcal{K}:\left{K_i \stackrel{f_i}{\rightarrow} K{i+1}\right}$ where every $f_i$ is a so-called elementary simplicial map which we will introduce shortly:
$$\mathrm{H}p \mathcal{K}: \mathrm{H}_p\left(K_0\right) \stackrel{f{0 *}}{\rightarrow} \mathrm{H}p\left(K_1\right) \stackrel{f{1 z}}{\rightarrow} \mathrm{H}p\left(K_2\right) \stackrel{f{2 *}}{\rightarrow} \ldots \stackrel{f_{n-1 }}{\rightarrow} \mathrm{H}p\left(K_n\right)$$ Instead of tracking a consistent homology basis for the module $\mathrm{H}_p \mathcal{K}$, we track a cohomology basis in the module $\mathrm{H}^p \mathcal{K}$ where the homomorphisms are in reverse direction: $$\mathrm{H}^{p f} \mathcal{K}: \mathrm{H}^p\left(K_0\right) \stackrel{f_0^}{\leftarrow} \mathrm{H}^p\left(K_1\right) \stackrel{f_1^}{\leftarrow} \mathrm{H}^p\left(K_2\right) \stackrel{f_2^}{\longleftarrow} \cdots \stackrel{f{n-1}^{\circ}}{\longleftarrow} \mathrm{H}^p\left(K_n\right) .$$
As we move from left to right in the above sequence, the annotations implicitly maintain a cohomology basis whose elements are also timestamped to signify when a basis element is born or dies. We keep in mind that the birth and death of a cohomology basis element coincides with the death and birth of a homology basis element because the two modules run in opposite directions.
To jump start the algorithm, we need annotations for simplices in $K_0$ at the beginning whose nonzero elements are timestamped with 0 . This can be achieved by considering an arbitrary filtration of $K_0$ and then applying the generic algorithm as we describe for inclusions in Section 4.2.3. The first vertex in this filtration gets the annotation of [1].

Before describing the algorithm, we observe a simple fact that simplicial maps can be decomposed into elementary maps which let us design simpler atomis steps for the algorithm.

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Computing Persistence of Simplicial Towers

$\mathcal{K}: K_0 \stackrel{f_0}{\rightarrow} K_1 \stackrel{f_1}{\rightarrow} K_2 \stackrel{f_2}{\rightarrow} \cdots \stackrel{f_{n-1}}{\rightarrow} K_n$ 和地图
$f_{i j}: K_i \rightarrow K_j$ 在哪里 $f_{i j}=f_{j-1} \circ \cdots \circ f_{i+1} \circ f_i$.

fij $: K_i \rightarrow K_j$ 是相反的，也就是说，
$f_{i j}^{:} \mathrm{H}^p\left(K_i\right) \leftarrow \mathrm{H}^p\left(K_j\right)$ ；请参阅第 2.5.4 节。因此，

$K(p) \rightarrow \mathbb{Z} 2^g$ 一个二元向量 $\mathrm{a} \sigma=\mathrm{a}(\sigma)$ 长度 $g$ 每一个 $p$ 单纯形 $\sigma \in K$. 二元向量 $\mathrm{a} \sigma$ 被称为注解 $\sigma$. 每个条目 “0” 或”1″的a $\sigma$ 称为它的元素。单纯形的注释为每个 $p$-链 $c_p: \mathbf{a} c_p=\sum \sigma \in c_p \mathbf{a}_\sigma$.

• $g=\operatorname{rank} \mathrm{H}_{\mathrm{p}}(\mathrm{K})$ ，和
• 二 $p$-周期 $z_1$ 和 $z_2$ 有 \$\mathbfa$}\left{z\right.$1}=Imathbf{a}$\left{z{-} 2\right}$ifandonlyiftheirhomologyclassesareide Veft[z_1\right]=Vleft[z_2\right]$\ 。

## 数学代写|表示论代写Representation theory代考|Algorithm

$\mathrm{H} p \mathcal{K}: \mathrm{H}p\left(K_0\right) \stackrel{f 0 *}{\rightarrow} \mathrm{H} p\left(K_1\right) \stackrel{f 1 z}{\rightarrow} \mathrm{H} p\left(K_2\right) \stackrel{f 2 *}{\rightarrow} \ldots \stackrel{f{n-1}}{\rightarrow}$

Imathrm ${\mathrm{H}} \wedge{\mathrm{p} f} \backslash m a t h c a \mid{\mathrm{K}}: \backslash m a t h r m{\mathrm{H}} \wedge p \backslash l$ eft(K_O\right) $\backslash S$

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## MATLAB代写

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