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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|泛函分析作业代写Functional Analysis代考|Connected Sets

We have an intuitive notion of what it means for a shape to be in one piece. The following definition makes this idea precise:
Definition $5.1$
A subset $C$ of a metric space is disconnected when it can be divided into (at least) two disjoint non-empty subsets $C=A \cup B$ such that each subset is covered exclusively by an open set, i.e.,
$A \subseteq U, \quad B \cap U=\varnothing, \quad U$ open, $B \subseteq V, \quad A \cap V=\varnothing, \quad V$ open.
Otherwise a set is called connected.
Examples 5.2

1. Single points are always connected because they cannot be split into two nonempty sets. Similarly the empty set is connected.
2. Any subset of $\mathbb{Z}$ (or any discrete metric space) is disconnected except the single points and the empty set. Metric spaces with this property are called totally disconnected.

Proof Let $C$ contain more than one point, say $a$ and $b$. Take $A=U:={a}$ and $B=V:=C \backslash{a} \neq \varnothing$. Then $U$ and $V$ are open (any subset is open) and respectively contain $A$ and $B$ exclusively.

## 数学代写|泛函分析作业代写Functional Analysis代考|The connected subsets of R are precisely the intervals

Proof Every non-trivial subset of an interval $I \subseteq \mathbb{R}$ has a boundary point: Let $A$ be a non-trivial subset of $I$; that $A$ is non-trivial means that there exist $a_0 \in A$ and $b_0 \in I \backslash A$. We can assume $a_0<b_0$, otherwise switch the roles of $A$ and $I \backslash A$ in what follows. to get a nested sequence of intervals $\left[a_n, b_n\right]$ in $I$,
with $a_n \in A, b_n \in I \backslash A$.
By the bisection property (Example 4.3(3)), the sequences $\left(a_n\right)$ and $\left(b_n\right)$ are Cauchy and asymptotic, and since $\mathbb{R}$ is complete, they converge $a_n \rightarrow a$ and $b_n \rightarrow a$. The consequence is that, inside any open neighborhood $B_\epsilon(a)$, there are points $a_n \in A$ and $b_n \in I \backslash A$, making $a$ a boundary point of $A$. From the preceding proposition, this translates as “every interval is connected”.

Every connected subset $C$ of $\mathbb{R}$ has the interval property $a, b \in C \Rightarrow$ $[a, b] \subseteq C$ : Let $C$ be a connected set, and let $a, b \in C$ (say, $a<b)$. Any $x \in[a, b]$ which is not in $C$ would disconnect $C$ using the disjoint open sets $]-\infty, x$ [ and ]$x, \infty[$.

Every subset of $\mathbb{R}$ with the interval property is an interval: Let $A$ have the interval property. If $A \neq \varnothing$, say $x \in A$, and has an upper bound, then it has a least upper bound $b$. The interval $[x, b[$ is a subset of $A$ because there are points of $A$ arbitrarily close to $b$. Similarly if $a$ is the greatest lower bound then $] a, x] \subseteq A$. Going through all the possibilities of whether $A$ has upper bounds or lower bounds or none, and whether these belong to $A$ or not, results in all the possible cases of intervals. For example, if it contains its least upper bound $b$ but has no lower bound, then $[x, b] \subseteq A$ for any $x<b$, so that $A=]-\infty, b]$.

By contrast, the connected sets in other metric spaces may be very difficult to describe and imagine. Even in $\mathbb{R}^2$, there are infinite connected sets such that when a single point is removed, the remaining set is totally disconnected! (For further information search for “Cantor’s teepee”.) Connectedness is an important intrinsic property that a set may have: it is preserved by any continuous function. Even though the codomain space may be very different from the domain, a connected set remains in ‘one piece’.

# 泛函分析代写

## 数学代写|泛函分析作业代写Functional Analysis代考|Connected Sets

1. 单点总是连接的，因为它们不能分成两个非空集。类似地，空集是连通的。
2. 的任何子集 $\mathbb{Z}$ (或任何离散度量空间) 除了单点和空集之外是断开的。具有此属性的度量空间称为完全不连通的。
证明让 $C$ 包含多个点，比如说 $a$ 和 $b$. 拿 $A=U:=a$ 和 $B=V:=C \backslash a \neq \varnothing$. 然后 $U$ 和 $V$ 是开放的（任何子集都是开放的) 并分别包含 $A$ 和 $B$ 只。

## 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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