### 数学代写|实分析作业代写Real analysis代考|MAST20026

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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|The Real Numbers

The key to understanding many of the fundamental concepts of calculus, such as limits, continuity, and the integral, is the least upper bound property of the real number system $\mathbb{R}$. As we all know, the rational number system contains gaps. For example, there does not exist a rational number $r$ such that $r^{2}=2$, i.e., $\sqrt{2}$ is irrational. The fact that the rational numbers do contain gaps makes them inadequate for any meaningful discussion of the above concepts.

The standard argument used in proving that the equation $r^{2}=2$ does not have a solution in the rational numbers goes as follows: Suppose that there exists a rational number $r$ such that $r^{2}=2$. Write $r=\frac{m}{n}$ where $m, n$ are integers which are not both even. Thus $m^{2}=2 n^{2}$. Therefore $m^{2}$ is even, and hence $m$ itself must be even. But then $m^{2}$, and hence also $2 n^{2}$ are both divisible by 4 . Therefore $n^{2}$ is even, and as a consequence $n$ is also even. This however contradicts our assumption that not both $m$ and $n$ are even. The method of proof used in this example is proof by contradiction; namely, we assume the negation of the conclusion and arrive at a logical contradiction.
The above argument shows that there does not exist a rational number $r$ such that $r^{2}=2$. This argument was known to Pythagoras (around 500 B.C.), and even the Greek mathematicians of this era noted that the straight line contains many more points than the rational numbers. It was not until the nineteenth century, however, when mathematicians became concerned with putting calculus on a firm mathematical footing, that the development of the real number system was accomplished. The construction of the real number system is attributed to Richard Dedekind (1831-1916) and Georg Cantor (1845-1917), both of whom published their results independently in 1872. Dedekind’s aim was the construction of a number system, with the same completeness as the real line, using only the basic postulates of the integers and the principles of set theory. Instead of constructing the real numbers, we will assume their existence and examine the least upper bound property. As we will see, this property is the key to many basic facts about the real numbers which are usually taken for granted in the study of calculus.

## 数学代写|实分析作业代写Real analysis代考|Sets and Operations on Sets

Sets are constantly encountered in mathematics. One speaks of sets of points, collections of real numbers, and families of functions. A set is conceived simply as a collection of definable objects. The words set, collection, and family are all synonymous. The notation $x \in A$ means that $x$ is an element of the set $A$; the notation $x \notin A$ means that $x$ is not an element of the set $A$. The set containing no elements is called the empty set and will be denoted by $\emptyset$.
A set can be described by listing its elements, usually within braces {} . For example,
$$A={-1,2,5,4}$$
describes the set consisting of the numbers $-1,2,4$, and 5 . More generally, a set $A$ may be defined as the collection of all elements $x$ in some larger collection satisfying a given property. Thus the notation
$$A={x: P(x)}$$
defines $A$ to be the set of all objects $x$ having the property $P(x)$. This is usually read as “A equals the set of all elements $x$ such that $P(x)$.” For example, if $x$ ranges over all real numbers, the set $A$ defined by
$$A={x: 1<x<5}$$
is the set of all real numbers which lie between 1 and 5 . For this example, $3.75 \in A$ whereas $5 \notin A$. We will also use the notation $A={x \in X: P(x)}$ to indicate that only those $x$ which are elements of $X$ are being considered.
Some basic sets that we will encounter throughout the text are the following:
$\mathbb{N}=$ the set of natural numbers or positive integers $={1,2,3, \ldots}$ $\mathbb{Z}=$ the set of all integers $={\ldots,-2,-1,0,1,2, \ldots}$ $\mathbb{Q}=$ the set of rational numbers $={p / q: p, q \in \mathbb{Z}, q \neq 0}$, and $\mathbb{R}=$ the set of real numbers.

## 数学代写|实分析作业代写Real analysis代考|Sets and Operations on Sets

$$A=-1,2,5,4$$

$$A=x: P(x)$$

$$A=x: 1<x<5$$

$\mathbb{N}=$ 自然数或正整数的集合 $=1,2,3, \ldots \mathbb{Z}=$ 所有整数的集合 $=\ldots,-2,-1,0,1,2, \ldots \mathbb{Q}=$ 有理数集 $=p / q: p, q \in \mathbb{Z}, q \neq 0$ ，和 $\mathbb{R}=$ 实数集。

## 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。