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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|抽象代数作业代写abstract algebra代考|Useful CAS Commands

In linear algebra, the theorem that every vector space has a basis gives us a standard method to describe elements in vector spaces, especially finite dimensional ones: (1) find a basis $\mathcal{B}$ of the vector space; (2) express a given vector by its coordinates with respect to $\mathcal{B}$. No corresponding theorem exists in group theory. Hence, one of the initial challenging questions of group theory is how to describe a group and its elements in a standard way. This is particularly important for implementing computational packages that study groups. There exist a few common methods and we will introduce them in parallel with the development of needed theory.

In Maple version 16 or below, the command with (group); accesses the appropriate package. In Maple version 17 or higher, the group package was deprecated in favor of with (GroupTheory) ;. The help files, whether provided by the program or those available online ${ }^2$ provide a list of commands and capabilities. Doing a search on “GroupTheory” locates the help file for the GroupTheory package. The student might find useful the LinearAlgebra package or, to support Example 1.2.11, the linear algebra package for modular arithmetic.

Consider the following lines of Maple code, in which the left justified text is the code and the centered text is the printed result of the code.

The first line makes active the linear algebra package for modular arithmetic. The next two code lines define matrices $A$ and $B$ respectively, both defined in $\mathbb{Z} / 11 \mathbb{Z}$. The next three lines calculate respective the determinant of $A$, the produce of $A B$, and the power $A^5$, always assuming we work in $\mathbb{Z} / 11 \mathbb{Z}$.

For SAGE, a browser search for “SageMath groups” will bring up references manuals and tutorials for group theory. Perhaps the gentlest introductory tutorial is entitled “Group theory and Sage.” ${ }^3$ We show here below the commands and approximate look for the same calculations in the console for SageMath for those we did above in Maple.

## 数学代写|抽象代数作业代写abstract algebra代考|Cyclic Groups

The process of simply considering the successive powers of an element gives rise to an important class of groups.
Definition 1.3.11
A group $G$ is called cyclic if there exists an element $x \in G$ such that every element of $g \in G$ we have $g=x^k$ for some $k \in \mathbb{Z}$. The element $x$ is called a generator of $G$.

For example, we notice that for all integers $n \geq 2$, the group $\mathbb{Z} / n \mathbb{Z}$ (with addition as the operation) is a cyclic group because all elements of $\mathbb{Z} / n \mathbb{Z}$ are multiples of $\overline{1}$. As we saw in Section A.6, one of the main differences with usual arithmetic is that $n \cdot \overline{1}=\overline{0}$. The intuitive sense that the powers (or multiples) of an element “cycle back” motivate the terminology of cyclic group.

Remark 1.3.12. We point out that a finite group $G$ is cyclic if and only if there exists an element $g \in G$ such that $|g|=|G|$.

Cyclic groups do not have to be finite though. The group $(\mathbb{Z},+)$ is also cyclic because every element in $\mathbb{Z}$ is obtained by $n \cdot 1$ with $n \in \mathbb{Z}$.
Example 1.3.13. Consider the group $U(14)$. The elements are
$$U(14)={\overline{1}, \overline{3}, \overline{5}, \overline{9}, \overline{11}, \overline{13}} .$$
This group is cyclic because, for example, the powers of $\overline{3}$ gives all the elements of $U(14)$ :
\begin{tabular}{c|cccccc}
$i$ & 1 & 2 & 3 & 4 & 5 & 6 \
\hline$\overline{3}^i$ & $\overline{3}$ & $\overline{9}$ & $\overline{13}$ & $\overline{11}$ & $\overline{5}$ & $\overline{1}$
\end{tabular}
We note that the powers of $\overline{3}$ will then cycle around, because $\overline{3}^7=\overline{3}^6 \cdot \overline{3}=\overline{3}$, then $\overline{3}^8=\overline{9}$, and so on.

Example 1.3.14. At first glance, someone might think that to prove that a group is not cyclic we would need to calculate the order of every element. If no element has the same order as the cardinality of the group, only then we could say that the group is not cyclic. However, by an application of the theorems in this section, we may be able to conclude the group is not cyclic with much less work.

As an example, suppose we wish to determine if $U(200)$ is cyclic. Note that $|U(200)|=\phi(200)=80$. The most obvious thing to try is to start calculating the powers of $\overline{3}$. Without showing all the powers here, we can check that $|\overline{3}|=$ 20. So we conclude immediately that $\overline{3}$ is not a generator of $U(200)$. In this list, we would find that $\overline{3}^{10}=\overline{49}$. This implies (and also using Proposition 1.3.8) that $|\overline{49}|=2$. It is easy to see that $\overline{199}^2=(\overline{-1})^2=\overline{1}$, so $|\overline{199}|=2$. Now if $U(200)$ were cyclic with generator $\bar{a}$, then $|\bar{a}|=80$. Also by Proposition 1.3.8, an element $\bar{a}^k$ has order 2 if and only if $\operatorname{gcd}(k, 80)=40$. However, the only integer $k$ with $1 \leq k \leq 80$ such that $\operatorname{gcd}(k, 80)=40$ is 40 itself. Hence, a cyclic of order 80 has at most one element of order 2. Since $U(200)$ has more than one element of order 2 , it is not a cyclic group.

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## 数学代写|抽象代数作业代写abstract algebra代考|Cyclic Groups

. 简单地考虑一个元素的连续幂的过程产生了一类重要的群。定义1.3.11

1.3.12.

$$U(14)={\overline{1}, \overline{3}, \overline{5}, \overline{9}, \overline{11}, \overline{13}} .$$

\begin{tabular}{c|cccccc}
$i$ & 1 & 2 & 3 & 4 & 5 & 6 \
\hline$\overline{3}^i$ & $\overline{3}$ & $\overline{9}$ & $\overline{13}$ & $\overline{11}$ & $\overline{5}$ & $\overline{1}$
\end{tabular}

## 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