### 金融代写|金融数学代写Financial Mathematics代考|ACTL20001

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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

## 金融代写|金融数学代写Financial Mathematics代考|Euclidean space

If $n \in \mathbb{N}$, we use the symbol $\mathbb{R}^{n}$ to indicate the Cartesian ${ }^{1}$ product of $n$ copies of $\mathbb{R}$ with itself, i.e.:
$$\mathbb{R}^{n}:=\left{\left(x_{1}, x_{2}, \ldots, x_{n}\right) \mid x_{j} \in \mathbb{R} \text { for } j=1,2, \ldots, n\right} .$$
The concept of Euclidean ${ }^{2}$ space is not limited to the set $\mathbb{R}^{n}$, but it also includes the so-called Euclidean inner product, introduced in Definition 1.1. The integer $n$ is called dimension of $\mathbb{R}^{n}$, the elements $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of $\mathbb{R}^{n}$ are called points, or vectors or ordered $n$-tuples, while $x_{j}, j=1, \ldots, n$, are the coordinates, or components, of $\boldsymbol{x}$. Vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ are equal if $x_{j}=y_{j}$ for $j=1,2, \ldots, n$. The zero vector is the vector whose components are null, that is, $\mathbf{0}:=(0,0, \ldots, 0)$. In low dimension situations, i.e. for $n=2$ or $n=3$, we will write $\boldsymbol{x}=(x, y)$ and $\boldsymbol{x}=(x, y, z)$, respectively.

For our purposes, that is extending differential calculus to functions of several variables, we need to define an algebraic structure in $\mathbb{R}^{n}$. This is done by introducing operations in $\mathbb{R}^{n}$.

Definition 1.1. Let $\boldsymbol{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right), \boldsymbol{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right) \in \mathbb{R}^{n}$ and $\alpha \in \mathbb{R}$
(i) The sum of $\boldsymbol{x}$ and $\boldsymbol{y}$ is the vector:
$$\boldsymbol{x}+\boldsymbol{y}:=\left(x_{1}+y_{1}, x_{2}+y_{2}, \ldots, x_{n}+y_{n}\right) ;$$

(ii) The difference of $\boldsymbol{x}$ and $\boldsymbol{y}$ is the vector:
$$\boldsymbol{x}-\boldsymbol{y}:=\left(x_{1}-y_{1}, x_{2}-y_{2}, \ldots, x_{n}-y_{n}\right)$$
(iii) The $\alpha$-multiple of $\boldsymbol{x}$ is the vector:
$$\alpha \boldsymbol{x}=\left(\alpha x_{1}, \alpha x_{2}, \ldots, \alpha x_{n}\right) ;$$
(iv) The Euclidean inner product of $\boldsymbol{x}$ and $\boldsymbol{y}$ is the real number:
$$\boldsymbol{x} \cdot \boldsymbol{y}:=x_{1} y_{1}+x_{2} y_{2}+\ldots+x_{n} y_{n} .$$

## 金融代写|金融数学代写Financial Mathematics代考|Topology of R

Topology, that is the description of the relations among subsets of $\mathbb{R}^{n}$, is based on the concept of open and closed sets, that generalises the notion of open and closed intervals. After introducing these concepts, we state their most basic properties. The first step is the natural generalisation of intervals in $\mathbb{R}^{n}$

Definition 1.13. Open and closed balls are defined as follows:
(i) $\forall r>0$, the open ball, centered at $a$, of radius $r$, is the set of points:
$$B_{r}(\boldsymbol{a}):=\left{\boldsymbol{x} \in \mathbb{R}^{n} \mid|\boldsymbol{x}-\boldsymbol{a}|<r\right} ;$$
(ii) $\forall r \geq 0$, the closed ball, centered at $\boldsymbol{a}$, of radius $r$, is the set of points:
$$\bar{B}_{r}(\boldsymbol{a})\left{\boldsymbol{x} \in \mathbb{R}^{n}|| \mid \boldsymbol{x}-\boldsymbol{a} | \leq r\right}$$
Note that, when $n=1$, the open ball centered at $a$ of radius $r$ is the open interval $(a-r, a+r)$, and the corresponding closed ball is the closed interval $[a-r, a+r]$. Here we adopt the convention of representing open balls as dashed circumferences, while closed balls are drawn as solid circumferences, as shown in Figure $1.4 .$

## 金融代写|金融数学代写Financial Mathematics代考|Euclidean space

$\backslash$ Imathbb ${R} \wedge{n}:=\backslash$ left $\left{\backslash\right.$ left $\left(x_{-}{1}, x_{-}{2}, \backslash\right.$ dots, $x_{-}{n} \backslash$ right) $\backslash$ mid $\left.\left.x_{-}\right} j\right}$ in $\backslash$ mathbb ${R} \backslash$ text ${$ for $} j=1,2, \backslash$ dots, $n \backslash$ right $}$ 。

$(-)$ 总和 $\boldsymbol{x}$ 和 $\boldsymbol{y}$ 是向量:
$$\boldsymbol{x}+\boldsymbol{y}:=\left(x_{1}+y_{1}, x_{2}+y_{2}, \ldots, x_{n}+y_{n}\right)$$
(ii) 差异 $\boldsymbol{x}$ 和 $\boldsymbol{y}$ 是向量:
$$\boldsymbol{x}-\boldsymbol{y}:=\left(x_{1}-y_{1}, x_{2}-y_{2}, \ldots, x_{n}-y_{n}\right)$$
(iii) $\alpha$ – 倍数 $\boldsymbol{x}$ 是向量:
$$\alpha \boldsymbol{x}=\left(\alpha x_{1}, \alpha x_{2}, \ldots, \alpha x_{n}\right)$$
(iv) 的欧几里得内积 $\boldsymbol{x}$ 和 $\boldsymbol{y}$ 是实数:
$$\boldsymbol{x} \cdot \boldsymbol{y}:=x_{1} y_{1}+x_{2} y_{2}+\ldots+x_{n} y_{n}$$

## 金融代写|金融数学代写Financial Mathematics代考|Topology of R

(i) $\forall r>0$ ，开球, 以 $a$, 半径 $r$ ，是点的集合:
(二) $\forall r \geq 0$, 封闭球, 以 $\boldsymbol{a}$, 半径 $r$, 是点的集合:

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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。