### 数学代写|拓扑学代写Topology代考|MATH6204

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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|Sets and Functions

It is sometimes said that mathematics $i s$ the study of sets and functions. Naturally, this oversimplifies matters; but it does come as close to the truth as an aphorism can.

The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up, onto the highlands of mathematics itself, where these concepts are indispensable in almost all of pure mathematics as it is today. Needless to say, we follow the latter course. We regard sets and functions as tools of thought, and our purpose in this chapter is to develop these tools to the point where they are sufficiently powerful to serve our needs through the rest of this book.

As the reader proceeds, he will come to understand that the words set and function are not as simple as they may seem. In a sense, they are simple; but they are potent words, and the quality of simplicity they possess is that which lies on the far side of complexity. They are like seeds, which are primitive in appearance but have the capacity for vast and intricate development.

## 数学代写|拓扑学代写Topology代考|SETS AND SET INCLUSION

We adopt a naive point of view in our discussion of sets and assume that the concepts of an element and of a set of elements are intuitively clear. By an element we mean an object or entity of some sort, as, for example, a positive integer, a point on the real line ( = a real number),or a point in the complex plane ( $=$ a complex number). A set is a collection or aggregate of such elements, considered together or as a whole. Some examples are furnished by the set of all even positive integers, the set of all rational points on the real line, and the set of all points in the complex plane whose distance from the origin is $1(=$ the unit circle in the plane). We reserve the word class to refer to a set of sets. We might speak, for instance, of the class of all circles in a plane (thinking of each circle as a set of points). It will be useful in the work we do if we carry this hierarchy one step further and use the term family for a set of classes. One more remark: the words element, set, class, and family are not intended to be rigidly fixed in their usage; we use them fluidly, to express varying attitudes toward the mathematical objects and systems we study. It is entirely reasonable, for instance, to think of a circle not as a set of points, but as a single entity in itself, in which case we might justifiably speak of the set of all circles in a plane.

There are two standard notations available for designating a particular set. Whenever it is feasible to do so, we can list its elements between braces. Thus ${1,2,3}$ signifies the set consisting of the first three positive integers, ${1, i,-1,-i}$ is the set of the four fourth roots of unity, and ${\pm 1, \pm 3, \pm 5, \ldots}$ is the set of all odd integers. This manner of specifying a set, by listing its elements, is unworkable in many circumstances. We are then obliged to fall back on the second method, which is to use a property or attribute that characterizes the elements of the set in question. If $P$ denotes a certain property of elements, then ${x: P}$ stands for the set of all elements $x$ for which the property $P$ is meaningful and true. For example, the expression
${x: x$ is real and irrational $}$,
which we read the set of all $x$ such that $x$ is real and irrational, denotes the set of all real numbers which cannot be written as the quotient of two integers. The set under discussion contains all those elements (and no others) which possess the stated property. The three sets of numbers described at the beginning of this paragraph can be written either way:
\begin{aligned} {1,2,3} &={n: n \text { is an integer and } 0<n<4} \ \text { and } \quad{1, i,-1,-i} &=\left{z: z \text { is a complex number and } z^{4}=1\right} \ {\pm 1, \pm 3, \pm 5, \ldots}={n: n \text { is an odd integer }} \end{aligned}

## 数学代写|拓扑学代写Topology代考| SETS AND SET INCLUSION

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## 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。