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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|Vectors and forms

The space of linear functionals on $T_{\mathrm{p}} M$ (the set of linear mappings from $T_{\mathrm{p}} M$ to the real numbers, $\mathbb{R}$ ), is called the cotangent space or space of dual vectors $T_{p}^{} M$ at $p$. Given a local set of coordinates $\left{x^{\mu}\right}$ on $M$, the partial derivative operators $\left{\partial / \partial x_{\mu}\right}$ and the differentials $\left{d x^{\mu}\right}$, respectively, define bases of $T_{\mathrm{p}} M$ and $T_{p}^{} M$, often called coordinate bases. Any basis not obtained in this way from a set of coordinates on $M$ is called a noncoordinate basis.

On flat spaces like $\mathbb{R}^{n}$, the spaces $T_{\mathrm{p}} M$ and $T_{p} M^{}$ are often treated as interchangeable and no distinction is made between them. More generally, they are isomorphic to each and the metric can be used to define mappings between them: \begin{aligned} &\text { b: } T_{p} M \rightarrow T_{p}^{} M \ &#: T_{p}^{*} M \rightarrow T_{p} M . \end{aligned}
Under these mappings, a vector with components $V^{a}$ defines a one-form $V^{b}=V_{\mu}^{b} d x^{\mu}$ with components
$$V_{\mu}^{b}=g_{\mu \nu} V^{\nu} .$$
Similarly, a one-form $\omega$ with components $\omega_{\alpha}$ defines a vector $\omega_{\sharp}=\omega_{\sharp}^{\mu} \partial_{\mu}$ with components
$$\omega_{\sharp}^{\mu}=g^{\mu \nu} \omega_{\nu} .$$
Most often in physics, the $\sharp$ and b symbols are omitted, and the type of object is simply denoted by the positions of the indices, up or down. In this case, we would then simply write
$$V_{\mu}=g_{\mu \nu} V^{\nu}$$
and
$$\omega^{\mu}=g^{\mu \nu} \omega_{\nu},$$
with the metric being used to raise and lower indices.

## 数学代写|拓扑学代写Topology代考|Curvature

We wish to have a means of describing how a manifold curves. To do this requires defining a few preliminary notions. Given an $n$-dimensional manifold $M$, any point can be specified by a set of $n$ numbers, the coordinates relative to the appropriate local coordinate system. Any curve on $M$ can then be described by a set of functions $\left{f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right}$, where $t$ is a parameter along the curve and the functions give the coordinates of the point on the curve at parameter value $t$. (More precisely, what we are calling $f_{i}(t)$ should be written as $f_{i}(\alpha(t))$, where $f_{i}$ is the coordinate mapping defined in the section $4.1$ and $\alpha(t)$ defines the curve.) These $n$ coordinate functions can of course be combined into a single vector-valued function $f(t)$. The ‘velocity’ vector or tangent vector to the curve at $t$ is the vector $v(t)=\boldsymbol{f}^{\prime}(t)$ whose components are $\left{d f_{1} / d t, d f_{2} / d t, \ldots, d f_{n} / d t\right}$. At a given point $p$, the tangent space $T_{\mathrm{p}} M$ is the $n$-dimensional plane spanned by the tangent vectors of all possible curves in $M$ passing through $p$. The tangent plane provides a linear (or flat) approximation to $M$ in a neighborhood of $p$; it can be thought of as a piece of $\mathrm{R}^{n}$ sewn onto $M$ at $p$ (figure 4.3). The curvature at $p$ can then be viewed as a measure of how fast $M$ pulls away from the flat tangent space as one moves away from $p$. Before generalizing, first consider the simplest case, in which the manifold itself is a one-dimensional curve.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Vectors and forms

《left 的分隔符缺失或无法识别 上 $M$, 偏导算子 $\backslash$ left 的分隔符缺失或无法识别 差异
《1eft 的分隔符缺失或无法识别，分别定义 $T_{\mathrm{p}} M$ 和 $T_{p} M$ ，通常称为坐标基。任何不是以这种方式从一组坐标中获得的基础 $M$ 称为非坐标基。

$$V_{\mu}^{b}=g_{\mu \nu} V^{\nu} .$$

$$\omega_{\sharp}^{\mu}=g^{\mu \nu} \omega_{\nu} .$$

$$V_{\mu}=g_{\mu \nu} V^{\nu}$$

$$\omega^{\mu}=g^{\mu \nu} \omega_{\nu}$$

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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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