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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|Connections and covariant derivatives

Suppose that you are driving on a long, straight stretch of highway in your expensive new sports car. Intellectually, you know that the highway is not really straight, but that it in fact follows the curving of the Earth. However, your senses tell you the path is straight. Not only does it look straight, but you also feel no feeling of following a curved path: there is no sensation of centripetal acceleration perpendicular to the ground. This is partly because the radius of the Earth is so large, but it is also in part due to the fact that we tend to take the Earth’s surface as the reference by which we measure motion. Only when the curve of the road bends horizontally, tangent to the surface on which you are traveling, does the curving become completely apparent. Essentially, being confined to a two-dimensional surface, motions within the surface are clearly visible, but motions of the confining surface itself perpendicular to its tangent plane are largely invisible from our viewpoint. The curved motion along the ‘straight’ road is, however, clearly visible to an alien observer preparing his plan for world domination while orbiting on a satellite. Such an external observer, floating well outside the confines of the planet’s surface, clearly sees the road bending to follow the Earth’s surface and finds it obvious that our motion is fully three-dimensional.
Ordinary space derivatives like $d / d x$ or $\nabla$ will describe how this field appears to the outside observer on the satellite; this three-dimensional view of the system is the extrinsic view, which can see how the $2 \mathrm{D}$ earth and the 1D road curve within a larger three-dimensional space. But we would like to be able to also give an intrinsic description of curvature and motion, relying only on quantities that can be measured by an observer within the lower-dimensional space, to whom the extra dimensions outside the curve or surface are invisible. This leads us the ideas of connection and covariant derivative.

Consider a curve $\gamma(s)$ on a surface $\mathcal{S}$, and some vector field $v(s)$ defined along the curve and tangent to $\mathcal{S}$; our car’s velocity, for example. Although $v(s)$ is tangent to the surface, its derivative $v^{\prime}=d v / d s=\left(\partial v_{j} / \partial x_{i}\right)\left(d x_{i} / d s\right) \hat{e}_{j}$ may not be, since the curving of the surface itself may introduce a component of $v^{\prime}$ perpendicular to $\mathcal{S}$. The perceived change in $v$, as viewed by the observer moving along the curve, will

then be the projection of this non-tangential vector $v^{\prime}$ back into the tangent plane. This is the covariant derivative along the curve:
$$\begin{gathered} \nabla_{\gamma} v(s)=v^{\prime}-\left(\text { normal component of } v^{\prime}\right) \ =v^{\prime}-\left\langle v^{\prime}, N\right\rangle N, \end{gathered}$$

## 数学代写|拓扑学代写Topology代考|Fiber bundles

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|Fiber bundles

1. $f$ 是可微分的，
2. $f$ 是同胚，并且
3. $f_{z}$ 是一对一的。
与二维曲面相切的所有向量的集合 $\mathcal{S}$ 在某一点 $p \in \mathcal{S}$ 是切线空间 $T V_{p}$ 的 $V$ 在 $p$. 同样，在图像上也有一个切线空间 $p: T U_{f(p)}$. 让
\eft 的分隔符缺失或无法识别 是坐标 $U \subset \mathbb{R}^{2}$ 和\1eft 的分隔符缺失或无法识别 是坐标 $V \subset \mathcal{S}$. 由于差分 $f$ 将向量 (可以认为
列矩阵) 转换为向量， $f$ 可以认为是一个矩阵。我们可以定义坐标基向量场 $\partial / \partial y^{\alpha}$ 和 $\partial / \partial x^{\mu}$ 上 $V$ 和 $U$ ，分别在哪里 $\alpha, \mu \in 1,2$. 相切的向量 $\mathcal{S}$ 和 $\mathbb{R}^{2}$ 然后可以写成组 形式 $\boldsymbol{w}=w^{\alpha}\left(\partial / \partial y^{\alpha}\right)$ 和 $\boldsymbol{r}=r^{\mu}\left(\partial / \partial x^{\mu}\right)$. 然后是微分 $f_{s}$ 需要 $\boldsymbol{w}$ 到新向量 $\boldsymbol{r}$ :

## 有限元方法代写

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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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