assignmentutor-lab™ 为您的留学生涯保驾护航 在代写概率论Probability theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写概率论Probability theory代写方面经验极为丰富，各种代写概率论Probability theory相关的作业也就用不着说。

• 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|概率论代写Probability theory代考|Abundance of Compact Subsets

First we cite a theorem from [Bishop and Bridges 1985] that guarantees an abundance of compact subsets.

Theorem 3.1.1. Abundance of compact sets. Let $f: K \rightarrow R$ be a continuous function on a compact metric space $(K, d)$ with domain $(f)=K$. Then, for all but countably many real numbers $\alpha>\inf _K f$, the set
$$(f \leq \alpha) \equiv{x \in K: f(x) \leq \alpha}$$
is compact.
Proof. See theorem (4.9) in chapter 4 of [Bishop and Bridges 1985].
Classically, the set $(f \leq \alpha)$ is compact for each $\alpha \geq \inf _K f$, without exception. Such a general theorem would, however, imply the principle of infinite search and is therefore nonconstructive. Theorem 3.1.1 is sufficient for all our purposes.

Definition 3.1.2. Convention for compact sets $(f \leq \alpha)$. We hereby adopt the convention that if the compactness of the set $(f \leq \alpha)$ is required in a discussion, compactness has been explicitly or implicitly verified, usually by proper prior selection of the constant $\alpha$, enabled by an application of Theorem 3.1.1.

The following simple corollary of Theorem 3.1.1 guarantees an abundance of compact neighborhoods of a compact set.

Corollary 3.1.3. Abundance of compact neighborhoods. Let $(S, d)$ be a locally compact metric space, and let $K$ be a compact subset of $S$. Then the subset
$$K_r \equiv(d(\cdot, K) \leq r) \equiv{x \in S: d(x, K) \leq r}$$
is compact for all but countably many $r>0$.

## 数学代写|概率论代写Probability theory代考|Binary Approximation

Let $(S, d)$ be an arbitrary locally compact metric space. Then $S$ contains a countable dense subset. A binary approximation, defined presently, is a structured and well-quantified countable dense subset.

Recall that (i) $|A|$ denotes the number of elements in an arbitrary finite set $A$; (ii) a subset $A$ of $S$ is said to be metrically discrete if for each $y, z \in A$, either $y=z$ or $d(y, z)>0$; and (iii) a finite subset $A$ of a subset $K \subset S$ is called an $\varepsilon$-approximation of $K$ if for each $x \in K$, there exists $y \in A$ with that $d(x, y) \leq$ $\varepsilon$. Classically, each subset of $(S, d)$ is metrically discrete. Condition (iii) can be written more succinctly as
$$K \subset \bigcup_{x \in A}(d(\cdot, x) \leq \varepsilon) .$$
Definition 3.2.1. Binary approximation and modulus of local compactness. Let $(S, d)$ be a locally compact metric space, with an arbitrary but fixed reference point $x_0$. Let $A_0 \equiv\left{x_0\right} \subset A_1 \subset A_2 \subset \ldots$ be a sequence of metrically discrete and finite subsets of $S$. For each $n \geq 1$, write $\kappa_n \equiv\left|A_n\right|$. Suppose
$$\left(d\left(\cdot, x_{\circ}\right) \leq 2^n\right) \subset \bigcup_{x \in A(n)}\left(d(\cdot, x) \leq 2^{-n}\right)$$
and
$$\bigcup_{x \in A(n)}\left(d(\cdot, x) \leq 2^{-n+1}\right) \subset\left(d\left(\cdot, x_{\circ}\right) \leq 2^{n+1}\right)$$
for each $n \geq 1$. Then the sequence $\xi \equiv\left(A_n\right){n=1,2, \ldots}$ of subsets is called a binary approximation for $(S, d)$ relative to $x{\circ}$, and the sequence of integers
$$|\xi| \equiv\left(\kappa_n\right){n=1,2, \ldots} \equiv\left(\left|A_n\right|\right){n=1,2, \ldots}$$
is called the modulus of local compactness of $(S, d)$ corresponding to $\xi$.

# 概率论代考

## 数学代写|概率论代写Probability theory代考|Abundance of Compact Subsets

$$(f \leq \alpha) \equiv x \in K: f(x) \leq \alpha$$

$$K_r \equiv(d(\cdot, K) \leq r) \equiv x \in S: d(x, K) \leq r$$

## 数学代写|概率论代写Probability theory代考|Binary Approximation

$$K \subset \bigcup_{x \in A}(d(\cdot, x) \leq \varepsilon) .$$

$$\left(d\left(\cdot, x_{\circ}\right) \leq 2^n\right) \subset \bigcup_{x \in A(n)}\left(d(\cdot, x) \leq 2^{-n}\right)$$
$$\bigcup_{x \in A(n)}\left(d(\cdot, x) \leq 2^{-n+1}\right) \subset\left(d\left(\cdot, x_{\circ}\right) \leq 2^{n+1}\right)$$

$$|\xi| \equiv\left(\kappa_n\right) n=1,2, \ldots \equiv\left(\left|A_n\right|\right) n=1,2, \ldots$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师