## 数学代写|勒贝格积分代写Lebesgue Integration代考|MATH6210

2022年12月30日

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cantor’s 1872 Paper

In 1872, Cantor published Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Riehen (On the extension of a theorem from the theory of trigonometric series) where he proved results on the uniqueness of trigonometric series that converges to zero except possibly at an infinite set of points. The preface to this paper contains Cantor’s construction of the irrational numbers, a step he recognized as necessary before he could work with them.

Cantor’s discussion of infinite sets to which he could extend his results on uniqueness of the trigonometric series began with the set ${1,1 / 2,1 / 3,1 / 4, \ldots}$. This set has the very nice property that if we consider any open interval that contains 0 and remove the points that are in that interval, then we are left with a finite set. The point 0 is called an accumulation point of this set, and Cantor designated this set as an infinite set of type 1 .

Cantor actually defined type 1 sets to be infinite sets for which the derived set is finite, but he was only working with bounded sets. To extend his definition to unbounded sets, we count the number of times that we need to take the derived set in order to get to the empty set. A set with no accumulation points is considered to be type 0 . The set ${1,1 / 2,1 / 3,1 / 4, \ldots}$ is type 1 because its derived set is ${0}$ and the derived set of ${0}$ is the empty set.

If a derived set is infinite, then we can consider the derived set of its derived set. For example, starting with the set
$$T=\left{\frac{1}{m}+\frac{1}{n} \mid m, n \in \mathbb{N}\right},$$
its derived set contains ${1,1 / 2,1 / 3,1 / 4, \ldots}$, and with a little work (see Exercise 2.3.11) you can show that the derived set equals ${1,1 / 2,1 / 3,1 / 4, \ldots}$. The set $T$ is not type 1 , but $T^{\prime \prime \prime}=\emptyset$, and we say that $T$ is type 2 .

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Geometry of R

The real number line is, above all else, a line. While true lines may not exist in the world of our senses, we do see them at the intersection of flat or apparently flat surfaces such as the line of the horizon when looking across a sea or prairie. To imagine the line as infinite is easy, for that is simply imagining the absence of an end. For the line as a geometric construct, the natural operation is demarcation of distance. There are two critical properties of distances that become central to the nature of real numbers:

1. However small a distance we measure, it is always possible to imagine a smaller distance.
2. Any two distances are commensurate. However small one distance might be and large the other, one can always use the smaller to mark out the larger.
The second property is known as the Archimedean principle, ${ }^1$ that given any two distances, one can always find a finite multiple of the smaller that exceeds the larger.

Let us now take our line and mark a point on it, the origin. We conduct a mental experiment. We stretch the line, doubling distances from the origin. What does the line now look like? It cannot have gotten any thinner. It did not have any width to begin with. A point that was a certain distance from the origin is now twice as far, but the first property tells us that the line itself should look the same. No gaps or previously unseen structures are going to appear as we stretch it. No matter how many times we double the length, what we see does not change.

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cantor’s 1872 Paper

1872 年，康托尔发表了 Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Riehen（关于三角级数理论的定理 的扩展)，他在其中证明了三角级数唯一性的结果，该三角级数收玫 于零，除非可能在无限集合处点数。这篇论文的序言包含了康托尔对 无理数的构造，他认为这是在他可以使用它们之前的必要步㖩。

Cantor 对无限集的讨论是从集合开始的，他可以将他关于三角级数唯 一性的结果扩展到 $1,1 / 2,1 / 3,1 / 4, \ldots$. 这个集合有一个非常好的特 性，如果我们考虑任何包含 0 的开区间并删除该区间中的点，那么我 们将得到一个有限集。点 0 称为这个集合的一个細积点，康托尔指定 这个集合为类型 1 的无限集合。

$\mathrm{T}=\backslash$ left ${$ frac ${1}{\mathrm{m}}+\backslash$ frac ${1}{\mathrm{n}} \backslash m i d \mathrm{~m}, \mathrm{n} \backslash$ in $\backslash m a t h b b{N} \backslash$ right $}$,

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Geometry of R

1. 无论我们测量的距离有多小，总是可以想象更小的距离。
2. 任何两个距离都是相称的。无论一个距离有多小而另一个距离可能很大，我们总是可以用较小的距离来标记较大的距离。
第二个性质被称为阿基米德原理，1给定任意两个距离，总能找到大于较大距离的较小距离的有限倍数。

## 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。