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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|拓扑学代写Topology代考|Homeomorphisms

This section concerns a key concept in topology – the homeomorphism between topological spaces. We will see along with examples that many spaces are essentially the same up to homeomorphism.

Definition 3.16. Given topological spaces $X$ and $Y$. An invertible map
$$h: X \rightarrow Y$$
is a homeomorphism if both $h$ and $h^{-1}$ are continuous. The space $X$ is homeomorphic or topologically equivalent to $Y$, denoted $X \simeq Y,{ }^{7}$ if there exists a homeomorphism between them.

Proposition 3.17. For any continuous bijection $h: X \rightarrow Y$ between topological spaces, the following are equivalent:

• The map $h$ is a homeomorphism.
• The map $h$ is open.
• The map $h$ is closed.
Proof. Note that when a map $h$ is invertible, the inverse $h^{-1}$ is continuous if and only if $h$ is open. The results then follow from Definition 3.16.
Since the identity map of any topological space is a homeomorphism, and both composition and inverse preserve homeomorphisms, we conclude that being homeomorphic is an equivalent relation. Proposition $3.17$ reveals the importance of open maps and closed maps.

数学代写|拓扑学代写Topology代考|Topological properties

In order to prove that two spaces are homeomorphic, it suffices to present a homeomorphism between them. In contrast, to prove that two spaces are not homeomorphic, we usually look for a property or a characteristic shared by homeomorphic spaces such that one of the spaces has it and the other one does not.

Definition 3.31. A topological invariant is an invariant of a topological space which is invariant under homeomorphisms. A topological property is a topological invariant which is a property of that topological space.
Obvious examples include the cardinality or infiniteness of a space. Less obvious properties are the main object of the subsequent chapters. Proposition $3.32$ is a criterion for topological properties which will be frequently of use in the sequel.

Proposition 3.32. Let $X$ be a topological space satisfying a property $P$. Then $P$ is a topological property if it is satisfied by every topological space $Y$ such that a continuous surjection $f: X \rightarrow Y$ exists.
Proof. Let
$$f: X \rightarrow Y$$
be a homeomorphism. Then $Y$ is the continuous image of $X$. From premise, if $\bar{X}$ satisfies a property $P$, then so does $Y$. This proves that $\bar{P}$ is a topological property.

Proposition 3.33. Any topological property which is preserved by any continuous image is inherited by a retract.

拓扑学代考

数学代写|拓扑学代写Topology代考|Homeomorphisms

$$h: X \rightarrow Y$$

• 地图 $h$ 是同胚。
• 地图 $h$ 开了。
• 地图 $h$ 已经关了。
证明。请注意，当地图 $h$ 是可逆的，逆 $h^{-1}$ 是连续的当且仅当 $h$ 开了。结果遒循定义 3.16。
由于任何拓扑空间的恒等映射都是同胚，并且组合和逆都保留同胚，我们得出结论，同胚是等价关系。主张 $3.17$ 揭示了开放地图和封闭地图的重要性。

数学代写|拓扑学代写Topology代考|Topological properties

$$f: X \rightarrow Y$$

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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