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• 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|抽象代数作业代写abstract algebra代考|Elementary Properties of Groups

Now that we have seen many diverse examples of groups, we wish to deduce some properties that they share. The definition itself raises some fundamental questions. Every group has an identity. Could a group have more than one? Every group element has an inverse. Could an element have more than one? The examples suggest not. But examples can only suggest. One cannot prove that every group has a unique identity by looking at examples, because each example inherently has properties that may not be shared by all groups. We are forced to restrict ourselves to the properties that all groups have; that is, we must view groups as abstract entities rather than argue by example. The next three theorems illustrate the abstract approach.

In a group $G$, there is only one identity element.
PROOF Suppose both $e$ and $e^{\prime}$ are identities of $\mathrm{G}$. Then,

1. $a e=a$ for all $a$ in $G$, and
2. $e^{\prime} a=a$ for all $a$ in $G$.
The choices of $a=e^{\prime}$ in (part 1) and $a=e$ in (part 2) yield $e^{\prime} e=e^{\prime}$ and $e^{\prime} e=e$. Thus, $e$ and $e^{\prime}$ are both equal to $e^{\prime} e$ and so are equal to each other.

Because of this theorem, we may unambiguously speak of “the identity” of a group and denote it by ‘ $e$ ‘ (because the German word for identity is Einheit).

## 数学代写|抽象代数作业代写abstract algebra代考|Uniqueness of Inverses

As was the case with the identity element, it is reasonable, in view of Theorem 2.3, to speak of “the inverse” of an element $g$ of a group; in fact, we may unambiguously denote it by $g^{-1}$. This notation is suggested by that used for ordinary real numbers under multiplication. Similarly, when $n$ is a positive integer, the associative law allows us to use $g^n$ to denote the unambiguous product:
$$\underbrace{g g \cdots g .}_{n \text { factors }}$$
We define $g^0=e$. When $n$ is negative, we define $g^n=\left(g^{-1}\right)^{|n|}$ [e.g., $g^{-3}=\left(g^{-1}\right)^3$. Unlike for real numbers, in an abstract group we do not permit noninteger exponents such as $g^{1 / 2}$. With this notation, the familiar laws of exponents hold for groups; that is, for all integers $m$ and $n$ and any group element $g$, we have $g^m g^n=g^{m+1}$ and $\left(g^m\right)^n=g^{m n}$. Although the way one manipulates the group expressions $g^m g^n$ and $\left(g^m\right)^n$ coincides with the laws of exponents for real numbers, the laws of exponents fail to hold for expressions involving two group elements. Thus, for groups in general, $(a b)^2=$ $a b a b$ rather than $a^2 b^2$. To remove parentheses in the expression $(a b)^{-2}$ we have $(a b)^{-2}=\left((a b)^{-1}\right)^2=\left(b^{-1} a^{-1}\right)^2=b^{-1} a^{-1} b^{-1} a^{-1}$ $(a b)^n \neq a^n b^n$. (See Exercises 25 and 36.)

The important thing about the existence of a unique inverse for each group element $a$ is that for every element $b$ in the group there is a unique solution in the group of the equations $a x=b$ and $x a=b$. Namely, $x=a^{-1} b$ in the first case and $x=b a^{-1}$ in the second case. In contrast, in the set ${0,1,2,3,4,5}$, the equation $2 x=4$ has the solutions $x=2$ and $x=5$ under the operation multiplication mod 6 . However, this set is not a group under multiplication $\bmod 6$.

Also, one must be careful with this notation when dealing with a specific group whose binary operation is addition and is denoted by “+.” In this case, the definitions and group properties expressed in multiplicative notation must be translated to additive notation. For example, the inverse of $g$ is written as $-g$. Likewise, for example, $g^3$ means $g+g+g$ and is usually written as $3 g$, whereas $g^{-3}$ means $(-g)+(-g)+(-g)$ and is written as $-3 g$. When additive notation is used, do not interpret “ng” as combining $n$ and $g$ under the group operation; $n$ may not even be an element of the group! Table $2.2$ shows the common notation and corresponding terminology for groups under multiplication and groups under addition.

# 抽象代数代写

## 数学代写|抽象代数作业代写abstract algebra代考|Elementary Properties of Groups

1. 一个和=一个对所有人一个在G， 和
2. 和′一个=一个对所有人一个在G.
的选择一个=和′在（第 1 部分）和一个=和在（第 2 部分）产量和′和=和′和和′和=和. 因此，和和和′都等于和′和所以彼此相等。

GG⋯G.⏟n 因素

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师