### 数学代写|拓扑学代写Topology代考|MATH3531

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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|PRODUCTS OF SETS

We shall of ten have occasion to weld together the sets of a given class into a single new set called their product (or their Cartesian product). The ancestor of this concept is the coordinate plane of analytic geometry, that is, a plane equipped with the usual rectangular coordinate system. We give a brier description of this fundamental idea with a view to paving the way for our discussion of products of sets in general.

First, a few preliminary comments about the real line. We have already used this term several times without any explanation, and of course what we mean by it is an ordinary geometric straight line (see Fig. 9) whose points have been identified with-or coordinatized by-the set $R$ of all real numbers. We use the letter $R$ to denote the real line as well as the set of all real numbers, and we often speak of real numbers as if they were points on the real line, and of points on the real line as if they were real numbers. Let no one be deceived into thinking that the real line is a simple thing, for its structure is exceedingly intricate. Our present view of it, however, is as naive and uncomplicated as the picture of it given in Fig. 9. Generally speaking, we assume that the reader is familiar with the simpler properties of the real line-those relating to inequalities (see Problem 1-2) and the basic algebraic operations of addition, subtraction, multiplication, and division. One of the most significant facts about the real number system is perhaps less well known. This is the so-called least upper bound property, which asserts that every non-empty set of real numbers which has an upper bound has a least upper bound. It is an easy consequence of this that every nonempty set of real numbers which has a lower bound has a greatest lower bound. All these matters can be developed rigorously on the basis of a small number of axioms, and detailed treatments can of ten be found in books on elementary abstract algebra.

To construct the coordinate plane, we now proceed as follows. We take two identical replicas of the real line, which we call the $x$ axis and the $y$ axis, and paste them on a plane at right angles to one another in such a way that they cross at the zero point on each. The usual picture is given in Fig. 10. Now let $P$ be a point in the plane. We project $P$ perpendicularly onto points $P_{x}$ and $P_{y}$ on the axes.

## 数学代写|拓扑学代写Topology代考|PARTITIONS AND EQUIVALENCE RELATIONS

In the first part of this section we consider a non-empty set $X$, and we study decompositions of $X$ into non-empty subsets which fill it out and have no elements in common with one another. We give special attention to the tools (equivalence relations) which are normally used to generate such decompositions.

A partition of $X$ is a disjoint class $\left{X_{i}\right}$ of non-empty subsets of $X$ whose union is the full set $X$ itself. The $X_{i}$ ‘s are called the partition sets. Expressed somewhat differently, a partition of $X$ is the result of splitting it, or subdividing it, into non-empty subsets in such a way that each element of $X$ belongs to one and only one of the given subsets.

If $X$ is the set ${1,2,3,4,5}$, then ${1,3,5},{2,4}$ and ${1,2,3},{4,5}$ are two different partitions of $X$. If $X$ is the set $R$ of all real numbers, then we can partition $X$ into the set of all rationals and the set of all irrationals, or into the infinitely many closed-open intervals of the form $[n, n+1)$ where $n$ is an integer. If $X$ is the set of all points in the coordinate plane, then we can partition $X$ in such a way that each partition set consists of all points with the same $x$ coordinate (vertical lines), or so that each partition set consists of all points with the same $y$ coordinate (horizontal lines).

Other partitions of each of these sets will readily occur to the reader. In general, there are many different ways in which any given set can be partitioned. These manufactured examples are admittedly rather uninspiring and serve only to make our ideas more concrete. Later in this section we consider some others which are more germane to our present purposes.

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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。