### 数学代写|实分析作业代写Real analysis代考|MATH315

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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|实分析作业代写Real analysis代考|Baire’s Category Theorem

The next two simple theorems are the most important general results of the theory of metric spaces.
1.5.1. Theorem. (THF. NFSTFD RAI.I. THFORFM) I.et $X$ he a complete metric. space and let $\left{B_n\right}$ be a sequence of closed balls with radii tending to zero such that $B_{n+1} \subset B_n$ for all $n$. Then $\bigcap_{n=1}^{\infty} B_n$ is not empty.

PROOF. Let us take $x_n \in B_n$. Since the balls decrease and their radii tend to zero, the sequence $\left{x_n\right}$ is Cauchy. By the completeness of $X$ it converges to some point, which belongs to all balls $B_n$ by their closedness.

It is clear that in place of balls one can take any decreasing closed sets of diameter $d_n \rightarrow 0$. Simple examples show that the completeness of $X$ and the closedness of balls are important (see also Exercise 1.9.33). One cannot omit the condition that the radii tend to zero (Exercise 1.9.34).

1.5.2. Theorem. (BAIRE’S CATEGORY THEOREM) Let $X$ be a complete metric space such that $X=\bigcup_{n=1}^{\infty} X_n$, where the sets $X_n$ are closed. Then at least one of them contains an open ball of a positive radius.

If $X=\bigcup_{n=1}^{\infty} A_n$, where $A_n$ are arbitrary sets, then at least one of $A_n$ is everywhere dense in some ball of a nonzero radius, i.e., a complete metric space cannot be the countable union of nowhere dense sets.

PRoOF. Suppose the contrary. Then for every $n$ in every open ball $U$ there is an open ball disjoint with $X_n$, since otherwise $U$ belongs to $\overline{X_n}=X_n$. Hence there exists a closed ball $B_1$ of radius $r_1>0$ disjoint with $X_1$. The ball $B_1$ contains a closed ball $B_2$ of a positive radius $r_2<r_1 / 2$ disjoint with $X_2$. By induction we obtain decreasing closed balls $B_n$ with positive radii tending to zero such that $B_n \cap X_n=\varnothing$. The previous theorem gives a common point for all $B_n$ not belonging to the union of $X_n$, which is a contradiction. The last assertion of the theorem is obvious from the first one applied to the closures of $A_n$.

## 数学代写|实分析作业代写Real analysis代考|Topological Spaces

A natural and very important generalization of the concept of metric space is a topological space.
1.6.1. Definition. A set $X$ with a distinguished family $\tau$ of its subsets is called a topological space if 1) $\varnothing, X \in \tau, 2)$ the intersection of every two sets from $\tau$ belongs to $\tau, 3)$ the union of every collection of sets from $\tau$ belongs to $\tau$. The sets from $\tau$ are called open and the family $\tau$ is called a topology.

A topology base is a collection of open set such that their unions give all open sets.
A neighborhood of a point is any open set containing it.
The complements of open sets are called closed sets. It is clear that any finite unions and arbitrary intersections of closed sets are closed. The empty set and the whole space are simultaneously open and closed.
1.6.2. Example. (i) The family $(\varnothing, X)$ is the minimal topology on a set $X$. (ii) The family $2^X$ of all subsets of $X$ is the maximal topology on $X$. (iii) The collection of open sets in a metric space $(X, d)$ (according to the terminology introduced for metric spaces!) is a topology. This topology is called the topology generated by the metric $d$. A topological space is called metrizable if its topology is generated by some metric.

Note that although the metric generates the indicated topology, this topology does not enable us to reconstruct the original metric. For example, the standard metric of the real line generates the same topology as the bounded metric defined by the formula $|\operatorname{arctg} x-\operatorname{arctg} y|$.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Baire’s Category Theorem

1.5.1. 定理。(THF.NFSTFD RAI.I.THFORFM) I.et $X$ 他是一个完整的指标。空间并让 $\left\langle L_工{\right.$ B_n右 是一系列半径趋于零的封闭球，使得 $B_{n+1} \subset B_n$ 对全部 $n$. 然后 $\bigcap_{n=1}^{\infty} B_n$ 不是空的。

1.5.2. 定理。(BAIRE 的范畴定理) 让 $X$ 是一个完备的度量空间使得 $X=\bigcup_{n=1}^{\infty} X_n$ ，其中集 合 $X_n$ 关闭。那么其中至少有一个包含一个正半径的空心球。

$r_2<r_1 / 2$ 与 $X_2$. 通过归纳我们得到递减的封闭球 $B_n$ 正半径趋于零，使得 $B_n \cap X_n=\varnothing$.

## 数学代写|实分析作业代写Real analysis代考|Topological Spaces

1.6.1. 定义。一套 $X$ 名门望族 $\tau$ 它的子集称为拓扑空间，如果 1)ø, $X \in \tau, 2$ 每两组的交集来 自 $\tau$ 属于 $\tau, 3)$ 每个集合集合的并集 $\tau$ 属于 $\tau$. 套从 $\tau$ 被称为开放和家庭 $\tau$ 称为拓扑。

1.6.2. 例子。(一) 家庭 $(\varnothing, X)$ 是集合上的最小拓扑 $X$. (二) 家庭 $2^X$ 的所有子集 $X$ 是上的最大拓 扑 $X$. (iii) 度量空间中开集的集合 $(X, d)$ （根据为度量空间引入的术语!）是一种拓扑。这种拓 扑称为度量生成的拓扑 $d$. 如果拓扑空间的拓扑是由某种度量生成的，则该拓扑空间称为可度量 的。

## 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。