### 数学代写|拓扑学代写Topology代考|MAST90023

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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|拓扑学代写Topology代考|Manifolds

A manifold is a topological space that is locally connected in a particular way. A 1-manifold has this local connectivity looking like a segment. A 2manifold (with boundary) has the local connectivity looking like a complete or partial disk. In layman’s terms. a 2-manifold has the structure of a piece of paper or rubber sheet, possibly with the boundaries glued together to form a closed surface – a category that includes disks, spheres, tori, and Möhius bands.

Definition 1.22. (Manifold) A topological space $M$ is an m-manifold, or simply a manifold, if every point $x \in M$ has a neighborhood homeomorphic to $\mathbb{B}_{o}^{m}$ or $\mathrm{H}^{m}$. The dimension of $M$ is $m$.

Every manifold can be partitioned into boundary and interior points. Observe that these words mean very different things for a manifold than they do for a metric space or topological space.

Definition 1.23. (Boundary; Interior) The interior Int $M$ of an $m$-manifold $M$ is the set of points in $M$ that have a neighborhood homeomorphic to $\mathbb{B}_{o}^{m}$. The boundary $\mathrm{Bd} M$ of $M$ is the set of points $M \backslash$ Int $M$. The boundary $\mathrm{Bd} M$, if not empty, consists of the points that have a neighborhood homeomorphic to $\mathrm{H}^{m}$. If $\mathrm{Bd} M$ is the empty set, we say that $M$ is without boundary.

## 数学代写|拓扑学代写Topology代考|Smooth Manifolds

A purely topological manifold has no geometry. But if we embed it in a Euclidean space, it could appear smooth or wrinkled. We now introduce a “geometric” manifold by imposing a differential structure on it. For the rest of this chapter, we focus on only manifolds without boundary.

Consider a map $\phi: U \rightarrow W$ where $U$ and $W$ are open sets in $\mathbb{R}^{k}$ and $\mathbb{R}^{d}$, respectively. The map $\phi$ has $d$ components, namely $\phi(x)=$ $\left(\phi_{1}(x), \phi_{2}(x), \ldots, \phi_{d}(x)\right)$, where $x=\left(x_{1}, x_{2}, \ldots, x_{k}\right)$ denotes a point in $\mathbb{R}^{k}$. The Jacobian of $\phi$ at $x$ is the $d \times k$ matrix of the first-order partial derivatives
$$\left[\begin{array}{ccc} \frac{\partial \phi_{1}(x)}{\partial x_{1}} & \cdots & \frac{\partial \phi_{1}(x)}{\partial x_{k}} \ \vdots & \ddots & \vdots \ \frac{\partial \phi_{d}(x)}{\partial x_{1}} & \cdots & \frac{\partial \phi_{d}(x)}{\partial x_{k}} \end{array}\right]$$
The map $\phi$ is regular if its Jacobian has rank $k$ at every point in $U$. The map $\phi$ is $C^{i}$-continuous if the $i$-th-order partial derivatives of $\phi$ are continuous.

The reader may be familiar with parametric surfaces, for which $U$ is a twodimensional parameter space and its image $\phi(U)$ in $d$-dimensional space is a parametric surface. Unfortunately, a single parametric surface cannot easily represent a manifold with a complicated topology. However, for a manifold to be smooth, it suffices that each point on the manifold has a neighborhood that looks like a smooth parametric surface.

## 数学代写|拓扑学代写Topology代考|Functions on Smooth Manifolds

In previous sections, we introduced topological spaces, including the special case of (smooth) manifolds. Very often, a space can be equipped with continuous functions defined on it. In this section, we focus on real-valued functions of the form $f: X \rightarrow \mathbb{R}$ defined on a topological space $X$, also called scalar functions; see Figure 1.8(a) for the graph of a function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$. Scalar functions appear commonly in practice that describe space/data of interest (e.g., the elevation function defined on the surface of the Earth). We are interested in the topological structures behind scalar functions. In this section, we limit our discussion to nicely behaved scalar functions (called Morse functions) defined on smooth manifolds. Their topological structures are characterized by the so-called critical points which we will introduce below. Later in the book we will also discuss scalar functions on simplicial complex domains, as well as more complex maps defined on a space $X$, for example, a multivariate function $f: X \rightarrow \mathbb{R}^{d}$

## 数学代写|拓扑学代写Topology代考|Smooth Manifolds

[∂φ1(X)∂X1⋯∂φ1(X)∂Xķ ⋮⋱⋮ ∂φd(X)∂X1⋯∂φd(X)∂Xķ]

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## MATLAB代写

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