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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|One-dimension: curves

Consider a one-dimensional curve in $n$-dimensional space, $\mathbb{R}^{n}$. This curve can be viewed as a map $f: \mathbb{R} \rightarrow \mathbb{R}^{n}$, defined by the coordinate functions, $\left{f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right}$. The velocity vector $v(t)=\boldsymbol{f}^{\prime}(t)$ is obviously always tangent to the curve, pointing in the direction of increasing $t$. The magnitude, $v(t)=|v(t)|$ is the speed of the curve in the given parameterization. The curve is said to have a critical point at $t$ if $v(t)=0$; otherwise it is regular there.
The arclength traversed during interval $(0, I)$ is
$$s(t)=\int_{t_{0}}^{t}\left|f^{\prime}\left(t^{\prime}\right)\right| d t^{\prime}=\int_{0}^{t}\left|\frac{\partial f}{d t^{\prime}}\right| d t^{\prime}$$
The derivative of arclength with respect to the parameter equals the speed:
$$\frac{d s}{d t}=\left|f^{\prime}\right|=v$$
Often, curves are parameterized by arclength, $t=s$, so that they are of unit speed, $v(t)=1$ for all $t$. Henceforth, we assume such arclength parameterization unless stated otherwise.
The curvature $k$ of the curve is the magnitude of the acceleration:
$$k(s)=\left|f^{\prime \prime}(s)\right|=\left|v^{\prime}(s)\right|,$$ or in other words, the rate at which the tangent vector’s direction changes as one moves along the curve. The radius of curvature at any point is given by
$$R(s)=\frac{1}{k(s)} .$$

## 数学代写|拓扑学代写Topology代考|Two-dimensions and beyond

A homeomorphism is a map that is one-to-one, onto (surjective), and continuous, with a continuous inverse. Homeomorphisms represent continuous deformations of manifolds, such as bending, stretching, and twisting; but ripping new holes or filling in old holes are not allowed. A regular surface is a two-dimensional manifold $S$ such that there is a mapping $f$ from an open neighborhood $V$ of each point to an open neighborhood $U$ in $\mathbb{R}^{2}$, such that

1. $f$ is differentiable,
2. $f$ is a homeomorphism, and
3. $f_{z}$ is one-to-one.
$f_{\text {s }}$ is called the differential or pushforward of $\boldsymbol{f}$, and is sometimes denoted $d \boldsymbol{f}$. It maps vectors tangent to $\mathcal{S}$ to vectors in $\mathbb{R}^{n}$. The differential map serves as a higher dimensional generalization of the parameter derivative $f^{\prime}$ in the one-dimensional case.

The set of all vectors tangent to two-dimensional surface $\mathcal{S}$ at a point $p \in \mathcal{S}$ is the tangent space $T V_{p}$ of $V$ at $p$. Similarly, there is a tangent space at the image of $p: T U_{f(p)}$. Let $\left{x^{1}, x^{2}\right}$ be coordinates on $U \subset \mathbb{R}^{2}$ and $\left{y^{1}, y^{2}\right}$ be coordinates on $V \subset \mathcal{S}$. Since the differential $f_{}$ takes vectors (which can be thought of as column matrices) to vectors, $f_{}$ can be thought of as a matrix. We can define coordinate basis vector fields $\partial / \partial y^{\alpha}$ and $\partial / \partial x^{\mu}$ on $V$ and $U$, respectively, where $\alpha, \mu \in{1,2}$. Vectors tangent to $\mathcal{S}$ and $\mathbb{R}^{2}$ can then be written in component form as $\boldsymbol{w}=w^{\alpha}\left(\partial / \partial y^{\alpha}\right)$ and $\boldsymbol{r}=r^{\mu}\left(\partial / \partial x^{\mu}\right)$. Then the differential $f_{s}$ takes $\boldsymbol{w}$ to the new vector $\boldsymbol{r}$ :
$$\boldsymbol{r}=f_{*}(\boldsymbol{w})$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|One-dimension: curves

$$s(t)=\int_{t_{0}}^{t}\left|f^{\prime}\left(t^{\prime}\right)\right| d t^{\prime}=\int_{0}^{t}\left|\frac{\partial f}{d t^{\prime}}\right| d t^{\prime}$$
arclength 对参数的导数等于速度：
$$\frac{d s}{d t}=\left|f^{\prime}\right|=v$$

$$k(s)=\left|f^{\prime \prime}(s)\right|=\left|v^{\prime}(s)\right|,$$

$$R(s)=\frac{1}{k(s)} .$$

## 数学代写|拓扑学代写Topology代考|Two-dimensions and beyond

1. F是可微分的，
2. F是同胚，并且
3. F和是一对一的。
Fs 被称为微分或前推F, 有时表示为dF. 它映射切线向量小号向量中Rn. 微分映射用作参数导数的更高维泛化F′在一维的情况下。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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