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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|Components

It seems intuitively clear that every space is the disjoint union of connected subsets. To make this rigorous, let us present some more propositions that go some way in helping us show whether a set is connected, especially the principle that whenever connected sets intersect, their union is connected; this allows us to build connected sets from smaller ones.
Proposition $5.8$
If $C$ is connected then so is $C$ with some boundary points (in particular $\bar{C}$ ).
Proof Let $D$ be $C$ with the addition of some boundary points. Suppose it separates as $D=A \cup B$ each covered exclusively by open sets $U$ and $V$. Then $C$ would also split up in the same way, unless $C \subseteq U$ say. This cannot be the case, for let $x \in B$ be a boundary point covered by $V$. Then there is a ball $B_r(x) \subseteq V$ containing points of $C$, a contradiction. Thus $D$ disconnected implies $C$ is disconnected.
Theorem $5.9$
If $A_i, B$ are connected sets and $\forall i A_i \cap B \neq \varnothing$ then $B \cup \bigcup_i A_i$ is connected. If $A_n$ are connected for $n=1,2, \ldots$, and $A_n \cap A_{n+1} \neq \varnothing$ then $\bigcup_n A_n$ is connected.

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

A set $B$ is bounded when the distance between any two points in the set has an upper bound,
$$\exists r>0, \quad \forall x, y \in B, \quad d(x, y) \leqslant r .$$
The least such upper bound is called the diameter of the set:
$$\operatorname{diam} B:=\sup {x, y \in B} d(x, y) .$$ In everyday, but not very helpful, terms one can say that a bounded set does not “reach to infinity”, or even that it is “finite” in a geometric sense (the unit circle has an infinite number of points but is bounded in $\mathbb{R}^2$ ). The characteristic properties of bounded sets are Proposition 6.2 Any subset of a bounded set is bounded. The union of a finite number of bounded sets is bounded. Proof (i) Let $B$ be a bounded set with $d(x, y) \leqslant r$ for any $x, y \in B$. In particular this holds for $x, y$ in any subset $A \subseteq B$, so $A$ is bounded. (ii) Given a finite number of bounded sets $B_1, \ldots, B_N$, with diameters $r_1, \ldots, r_N$, respectively, let $r:=\max _n r_n$. Pick a representative point from each set, $a_n \in B_n$, and take the maximum distance between any two, $\widetilde{r}:=\max {m, n} d\left(a_m, a_n\right)$; it certainly exists as there are only a finite number of such pairs. Now, for any two points $x, y \in \bigcup_n B_n$, that is, $x \in B_i, y \in B_j$, and using the triangle inequality twice,

# 泛函分析代写

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

$$\exists r>0, \quad \forall x, y \in B, \quad d(x, y) \leqslant r .$$

$$\operatorname{diam} B:=\sup x, y \in B d(x, y) .$$

## 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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