### 数学代写|黎曼曲面代写Riemann surface代考|MTH3022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

In this section we will count the cells in the chains $\varphi, \tau$, and $\psi$ that were defined in the previous section. Note that
$$\begin{gathered} \varphi=\sum_r(A K)^r \eta \ \tau=\sum_r(-K A)^r H \varphi \ \psi=\sum_r(-K A)^r K \nu \end{gathered}$$
We will show that the number of nondegenerate cubical cells in one of these chains is bounded by $C^n$, by parametrizing the cells with trees.

Suppose $z$ is a point in $Z_I$, with $|I|=n$. Consider the chain $F(K A)^r z$. It is a sum of cells of the form
$$F K_{k_r} \alpha_r \ldots K_{k_1} \alpha_1 z$$
Each of these cells is an $r$-cube. Our main construction will be to describe the points in these cells using graphs (which are trees).

Fix a sequence $\alpha_1, \ldots, \alpha_r$. In particular there is a sequence of indices $I=$ $I_0, I_1, \ldots, I_r$ such that $\alpha_j: I_{j-1} \rightarrow I_j$. We will associate a graph to this choice as follows. The vertices are arrayed in $r+1$ rows, with the top row having $n+2$ vertices and bottom row having $n+r+2$ vertices. The $j$ th row from the top has $n+j+2$ vertices. The vertices are numbered from right to left in each row, beginning with 0 , and we denote the $k$ th vertex in the $j$ th row by $v_{j k}$. The vertices at the ends of the rows, $v_{j 0}$ and $v_{j(n+j+1)}$, are called side vertices. The edges of the graph go from vertices in one row to vertices in the next. There is an edge connecting $v_{j-1 i}$ to $v_{j k}$ if and only if $\alpha_j^{+}(k)=i$. Thus in each row except the bottom one, there is exactly one vertex with two edges emanating from below, and all of the other vertices have one edge below. The edges, when drawn as straight lines, do not intersect, because the maps $\alpha^{+}$ are order preserving. The edges drawn from $v_{j 0}$ to $v_{j+10}$ and from $v_{j(n+j+1)}$ to $v_{(j+1)(n+j+2)}$ are called side edges.

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

We can decompose the graph into strands, with the strands joining forks. The forks are the vertices which are connected to three edges (in other words the vertices $v_{j k}$ such that $\alpha_{j+1}^{+}(k)=\alpha_{j+1}^{+}(k+1)=k$ ), as well as, by convention, the top and bottom vertices. The strands are the unbroken sequences of edges joining forks, in other words the sequences of edges which meet at interior vertices with only two edges. Side strands are those consisting of side edges. The graph formed by the forks and strands considered as vertices and edges, is a union of binary trees. If a number is assigned to each non-side edge, then one obtains a number for each non-side strand as follows. Suppose $\sigma$ is a strand, composed of edges $e_1, \ldots, e_m$. Set
$$t(\sigma)=\min \left(1, t\left(e_1\right)+\ldots+t\left(e_m\right)\right) .$$
In the above construction, the point $u$ depends only on the numbers $t(\sigma)$ assigned to the strands. Here is another description of the construction of $u$. For each strand $\sigma$ there are indices $i(\sigma)$ and $j(\sigma)$, representing the indices corresponding to the left and right sides of the edges in the strand, respectively. If the strand $\sigma$ contains an edge ending in a vertex $v_{j k}$, then $i(\sigma)=i_{j, k-1}$ and $j(\sigma)=i_{j, k}$. (The notation $i(e)$ and $j(e)$ will also be used for an edge $e$.) Realize the tree geometrically, with a strand $\sigma$ represented by a line segment of length 1. Let $T$ denote the geometric realization of the tree. Then the function $t$ from the set of strands into $[0,1]$, and the initial point $z$, determine a map $\Psi_{z, t}: T \rightarrow Z$. Write $z=\left(z_1, \ldots, z_n\right)$. The top vertices of the tree go to the points $z_k \in Z$. The left and right side strands are mapped to $P$ and $Q$ respectively. If $\sigma$ is any strand, $\Psi_{z, t}$ maps the segment corresponding to $\sigma$ into $Z$ using the flow $f_{i(\sigma) j(\sigma)}$, beginning with the point corresponding to the fork $v$ at the top of $\sigma$, and moving at speed $t(\sigma)$. The beginning point $\Psi_{z, t}(v)$ has already been constructed inductively. If $p$ is a point on the segment $\sigma$, at distance $y$ below the fork $v, \Psi_{z, t}(p)=f_{i(\sigma) j(\sigma)}\left(\Psi_{z, t}(v), t(\sigma) y\right)$. Finally, the the values of $\Psi_{z, t}$ on the $n+r$ bottom vertices provide the points $u_1, \ldots, u_{n+r}$ to determine $u=u(z, t) \in Z_{I_r}$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

$$\varphi=\sum_r(A K)^r \eta \tau=\sum_r(-K A)^r H \varphi \psi=\sum_r(-K A)^r K \nu$$

$$F K_{k_r} \alpha_r \ldots K_{k_1} \alpha_1 z$$

## 数学代写|黎曼曲面代写Riemann surface代考|THE MAIN LEMMA

$$t(\sigma)=\min \left(1, t\left(e_1\right)+\ldots+t\left(e_m\right)\right) .$$

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