## 数学代写|拓扑学代写Topology代考|MATH4204

2022年12月30日

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• (Generalized) Linear Models 广义线性模型
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## 数学代写|拓扑学代写Topology代考|Constructing a GIC

One may wonder how to efficiently construct the graph induced complexes in practice. Experiments show that the following procedure runs quite efficiently in practice. It takes advantage of computing nearest neighbors within a range and, more importantly, computing cliques only in a sparsified graph.

Let the ball $B(q, \delta)$ in metric d be called the $\delta$-cover for the point $q$. A graph induced complex $\mathcal{G}^\alpha(P, Q$, d) where $Q$ is a $\delta$-sparse $\delta$-sample can be built easily by identifying $\delta$-covers with a rather standard greedy (farthest point) iterative algorithm. Let $Q_i=\left{q_1, \ldots, q_i\right}$ be the point set sampled so far from $P$. We maintain the invariants (i) $Q_i$ is $\delta$-sparse and (ii) every point $p \in P$ that is in the union of $\delta$-covers $\bigcup_{q \in Q_i} B(q, \delta)$ has its closest point $v(p) \in \operatorname{argmin}{q \in Q_i} \mathrm{~d}(p, q)$ in $Q_i$ identified. To augment $Q_i$ to $Q{i+1}=Q_i \cup\left{q_{i+1}\right}$, we choose a point $q_{i+1} \in P$ that is outside the $\delta$ covers $\bigcup_{q \in Q_i} B(q, \delta)$. Certainly, $q_{i+1}$ is at least $\delta$ units away from all points in $Q_i$ thus satisfying the first invariant. For the second invariant, we check every point $p$ in the $\delta$-cover of $q_{i+1}$ and update $v(p)$ to include $q_{i+1}$ if its distance to $q_{i+1}$ is smaller than the distance $\mathrm{d}(p, v(p))$. At the end, we obtain a sample $Q \subseteq P$ whose $\delta$-covers cover the entire point set $P$ and thus is a $\delta$ sample of $(P, \mathrm{~d})$ which is also $\delta$-sparse due to the invariants maintained. Next, we construct the simplices of $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. This needs identifying cliques in $G^\alpha(P)$ that have vertices with different closest points in $Q$. We delete every edge $p p^{\prime}$ from $G^\alpha(P)$ where $v(p)=v\left(p^{\prime}\right)$. Then, we determine every clique $\left{p_1, \ldots p_k\right}$ in the remaining sparsified graph and include the simplex $\left{v\left(p_1\right), \ldots, v\left(p_k\right)\right}$ in $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. The main saving here is that many cliques of the original graph are removed before it is processed for clique computation.
Next, we focus on the second topic of this chapter, namely homology groups. They are algebraic structures to quantify topological features in a space. They do not capture all topological aspects of a space in the sense that two spaces with the same homology groups may not be topologically equivalent. However, two spaces that are topologically equivalent must have isomorphic homology groups. It turns out that the homology groups are computationally tractable in many cases, thus making them more attractive in topological data analysis. Before we introduce their definition and variants in Section 2.5, we need the important notions of chains, cycles, and boundaries given in the following section.

## 数学代写|拓扑学代写Topology代考|Algebraic Structures

First, we recall briefly the definitions of some standard algebraic structures that are used in the book. For details we refer the reader to any standard book on algebra, for example, [14].

Definition 2.18. (Group; Homomorphism; Isomorphism) A set $G$ together with a binary operation “+” is a group if it satisfies the following properties: (i) for every $a, b \in G, a+b \in G$; (ii) for every $a, b, c \in G$, $(a+b)+c=a+(b+c)$; (iii) there is an identity element denoted 0 in $G$ so that $a+0=0+a=a$ for every $a \in G$; and (iv) there is an inverse $-a \in G$ for every $a \in G$ so that $a+(-a)=0$. If the operation “+” commutes, that is, $a+b=b+a$ for every $a, b \in G$, then $G$ is called abelian. A subsèt $H \subseteq G$ is a subgroup of $(G,+)$ if $(H,+)$ is also a group.

Definition 2.19. (Free abelian group; Basis; Rank; Generator) An abelian group $G$ is called free if there is a subset $B \subseteq G$ so that every element of $G$ can be written uniquely as a finite sum of elements in $B$ and their inverses disregarding trivial cancellations $a+b=a+c-c+b$. Such a set $B$ is called a basis of $G$ and its cardinality is called its rank. If the condition of uniqueness is dropped, then $B$ is called a generator of $G$ and we also say that $B$ generates $G$.

Definition 2.20. (Coset; Quotient) For a subgroup $H \subseteq G$ and an element $a \in G$, the left coset is $a H={a+b \mid b \in H}$ and the right coset is $H a={b+a \mid b \in H}$. For abelian groups, the left and right cosets are identical and hence are simply called cosets. If $G$ is abelian, the quotient group of $G$ with a subgroup $H \subseteq G$ is given by $G / H={a H \mid a \in G}$ where the group operation is inherited from $G$ as $a H+b H=(a+b) H$ for every $a, b \in G$.

Definition 2.21. (Homomorphism; Isomorphism; Kernel; Image; Cokernel) A map $h: G \rightarrow H$ between two groups $(G,+)$ and $(H, *)$ is called a homomorphism if $h(a+b)=h(a) * h(b)$ for every $a, b \in G$. If, in addition, $h$ is bijective, it is called an isomorphism. Two groups $G$ and $H$ with an isomorphism are called isomorphic and denoted as $G \cong H$. The kernel, image, and cokernel of a homomorphism $h: G \rightarrow H$ are defined as subgroups ker $h={a \in G \mid h(a)=0}, \operatorname{Im} h={b \in H \mid \exists a \in G$ with $h(a)=b}$, and the quotient group coker $h=H /$ im $h$, respectively.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Constructing a GIC

Ileft{p_1, Idots p_klright} 在剩余的稀疏图中并包括单纯 $\mathcal{G}^\alpha(P, Q, \mathrm{~d})$. 这里的主要节省是原始图的许多团在处 理团计算之前被删除。

## 数学代写|拓扑学代写Topology代考|Algebraic Structures

Image; Cokernel) 地图 $h: G \rightarrow H$ 两组之间 $(G,+)$ 和 $(H, *)$ 称为同态如果 $h(a+b)=h(a) * h(b)$ 每一个 $a, b \in G$. 如果，此外， $h$ 是双射的，称为同构。两组 $G$ 和 $H$ 具有同构的称为同构并表示为 $G \cong H$. 同态的内 核、图像和上核 $h: G \rightarrow H$ 被定义为子群ker $h=a \in G|h(a)=0, \operatorname{Im} h=b \in H| \exists a \in G \$$with$\$h$ ，和商群 cokerh $h=H /$ 在里面 $h$ ，分别。

## 有限元方法代写

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## MATLAB代写

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