### 数学代写|拓扑学代写Topology代考|MATH4204

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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|THE ALGEBRA OF SETS

In this section we consider several useful ways in which sets can be combined with one another, and we develop the chief properties of these operations of combination.

As we emphasized above, all the sets we mention in this section are assumed to be subsets of our universal set $U$. $U$ is the frame of reference, or the universe, for our present discussions. In our later work the frame of reference in a particular context will naturally depend on what ideas we happen to be considering. If we find ourselves studying sets of real numbers, then $U$ is the set $R$ of all real numbers. If we wish to study sets of complex numbers, then we take $U$ to be the set $C$ of all complex numbers. We sometimes want to narrow the frame of reference and to consider (for instance) only subsets of the closed unit interval $[0,1]$, or of the closed unit disc ${z:|z| \leq 1}$, and in these cases we choose $U$ accordingly. Generally speaking, the universal set $U$ is at our disposal, and we are free to select it to fit the needs of the moment. For the present, however, $U$ is to be regarded as a fixed but arbitrary set. This generality allows us to apply the ideas we develop below to any situation which arises in our later work.

It is extremely helpful to the imagination to have a geometric picture available in terms of which we can visualize sets and operations on sets. A convenient way to accomplish this is to represent $U$ by a rectangular area in a plane, and the elements which make up $U$ by the points of this area. Sets can then be pictured by areas within this rectangle, and diagrams can be drawn which illustrate operations on sets and relations between them. For instance, if $A$ and $B$ are sets, then Fig. 1 represents the circumstance that $A$ is a subset of $B$ (we think of each set as consisting of all points within the corresponding closed curve). Diagrammatic thought of this kind is admittedly loose and imprecise; nevertheless, the reader will find it invaluable. No mathematics, however abstract it may appear, is ever carried on without the help of mental images of some kind, and these are often nebulous, personal, and difficult to describe.

## 数学代写|拓扑学代写Topology代考|FUNCTIONS

Many kinds of functions occur in topology, in a great variety of situations. In our work we shall need the full power of the general concept of a function, and since its modern meaning is much broader and deeper than its elementary meaning, we discuss this concept in considerable detail and develop its main abstract properties.

Let us begin with a brief inspection of some simple examples. Consider the elementary function
$$y=x^{2}$$
of the real variable $x$. What do we have in mind when we call this a function and say that $y$ is a function of $x$ ? In a nutshell, we are drawing attention to the fact that each real number $x$ has linked to it a specific real number $y$, which can be calculated according to the rule (or law of correspondence) given by the formula. We have here a process which, applied to any real number $x$, does something to it (squares it) to produce another real number $y$ (the square of $x$ ). Similarly,
$$y=x^{3}-3 x \quad \text { and } y=\left(x^{2}+1\right)^{-1}$$
are two other simple functions of the real variable $x$, and each is given by a rule in the form of an algebraic expression which specifies the exact manner in which the value of $y$ depends on the value of $x$.

## 数学代写|拓扑学代写Topology代考| THE ALGEBRA OF SETS

$z:|z| \leq 1$ ，在这些情况下，我们选择 $U$ 因此。一般来说，通用集 $U$ 随时可供我们使用，我们可以自由选择它 以满足当前的需求。然而，就目前而言， $U$ 被视为一个固定但任意的集合。这种普遍性使我们能够将下面发展 的想法应用于我们以后工作中出现的任何情况。

## 数学代写|拓扑学代写Topology代考| FUNCTIONS

$$y=x^{2}$$

$$y=x^{3}-3 x \quad \text { and } y=\left(x^{2}+1\right)^{-1}$$

$x$.

## 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。