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# 数学代写|泛函分析代写Functional Analysis代考|Examples of Abstract Systems

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## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Examples of Abstract Systems

We now list a number of important special abstract mathematical systems:
Groupoid. A groupoid is any abstract system consisting of a set on which a closed operation has been defined.
Semi-group. A semi-group is an associative groupoid.
Monoid. A monoid is a semi-group with an identity element.
Finally, we arrive at an important type of abstract system that occurs frequently in mathematical analysis:
Group. An abstract system $\mathcal{G}$ consisting of a set $G$ and one binary operation $$on G is called a group iff the following conditions are satisfied. (i) Operation$$ is associative.
(ii) There exists an identity element $e$ in $G$.
(iii) Every element $a$ in $G$ has its inverse $a^{-1}$ in $G$.
Additionally, if the operation $$is commutative, the group is called an Abelian or commutative group. It is understood that$$ is closed on $G$; i.e., $a * b \in G$ for all $a, b \in G$.

Example 1.14.1
(Dynamics of Mechanical Systems)

A reader familiar with the dynamics of mechanical systems will appreciate this important example of semi-groups. Many dynamical systems are governed by differential equations of the form
\begin{aligned} \frac{d \boldsymbol{q}(t)}{d t} & =\boldsymbol{A} \boldsymbol{q}(t), \quad t>0 \ \boldsymbol{q}(0) & =\boldsymbol{q}_0 \end{aligned}
wherein $\boldsymbol{q}(t)=\left{q_1(t), q_2(t), \ldots, q_n(t)\right}^T$ is an $n$-vector whose components are functions of time $t$ and $\boldsymbol{A}$ is an $n \times n$ invertible matrix (those unfamiliar with these terms may wish to return to this example after reading Chapter 2).

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The Real Number System

The study of properties of the real number system lies at the heart of mathematical analysis, and much of higher analysis is a direct extension or generalization of intrinsic properties of the real line. This section briefly surveys its most important features, which are essential to understanding many of the ideas in subsequent chapters.

Our objective in this section is to give a concise review of topological properties of real line $\mathbb{R}$ or, more generally, the Cartesian product $\mathbb{R}^n$. By topological properties we mean notions like open and closed sets, limits, and continuity. We shall elaborate on all these subjects in a much more general context in Chapter 4 where we study the general theory of topological spaces and then the theory of metric spaces, of which the space $\mathbb{R}^n$ is an example.

It may seem somehow inefficient and redundant that we shall go over some of the notions studied in this section again in Chapter 4 in a much broader setting. This “didactic conflict” is a result of the fact that we do need the results presented in this section to develop the notion of the Lebesgue integral in Chapter 3 , which in turn serves as a primary tool to construct the most important examples of structures covered in Chapter 4.
We begin with a short review of the algebraic properties of the set of real numbers $\mathbb{R}$.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Examples of Abstract Systems

(ii) $G$中存在一个身份元素$e$。
(iii) $G$中的每个元素$a$在$G$中都有它的逆$a^{-1}$。

(机械系统动力学)

\begin{aligned} \frac{d \boldsymbol{q}(t)}{d t} & =\boldsymbol{A} \boldsymbol{q}(t), \quad t>0 \ \boldsymbol{q}(0) & =\boldsymbol{q}_0 \end{aligned}

## 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。