### 数学代写|现代代数代写Modern Algebra代考|Math4120

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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|现代代数代写Modern Algebra代考|Ruler-and-Compass Constructibility of Regular Polygons

The ancient Greek mathematicians, who invented what we have come to call Euclidean geometry and the notion of a rigorous proof, bequeathed their successors a host of unsolved mathematical problems. Best-known amongst these are the questions of whether it is possible to trisect an angle, double a cube, or square a circle by means of a compass and an unmarked ruler alone. Here we treat a lesser-known, but equally natural, construction problem, namely, what regular polygons are constructible by ruler and compass alone? The other three problems are discussed informally at the end of the section.

The ruler-and-compass constructions of the equilateral triangle, the square, and the regular hexagon are standard fare in the high school curriculum. That the regular pentagon is also so constructible is true, but not so widely known. This is proved below in Proposition 2.14. A regular octagon is easily constructed by inscribing a square in a circle and then drawing the two diameters that are perpendicular to the sides of the square (Figure 2.5). In general, it is clear that given any regular $n$-gon it is possible to derive from it a regular $2 n$-gon by drawing radii perpendicular to its sides. Hence the regular $n$-gon is constructible for $n=2^{m+2}, 3 \cdot 2^{m}$, and $5 \cdot 2^{m}$ for $m=0,1,2, \ldots$. If a regular pentagon and an equilateral triangle are inscribed in a circle so that they share a vertex, as in Figure 2.6, then $\operatorname{arc} A B$ is $2 / 5-1 / 3=1 / 15$ of the total circumference of the circle. It follows that the regular i 5 -sided polygon is also constructible by ruler and compass. This information is summarized as the following proposition.

## 数学代写|现代代数代写Modern Algebra代考|Orders of Roots of Unity

We have seen that the 4 -th roots of unity are $1, i,-1$, and $-\mathrm{i}$ and that the 6-th roots of unity are $1,-\omega^{2}, \omega,-1, \omega^{2}$, and $-\omega$. However, $-1$ is already a square root of 1 , and $\omega$ and $\omega^{2}$ are also cube roots of 1 . If $\zeta$ is any root of unity, then the order of $\zeta$, denoted $o(\omega)=3, o(-\omega)=6, o(i)=4$, and $o\left(-\omega^{2}\right)=6$

The following proposition on the order of roots may seem obvious, but it does require formal proof. The integer $m$ is said to be a divisor of the integer $n$ (and $n$ is said to be a multiple of $m$ ) if there is an integer $k$ such that $n=k m$, denoted by $m \mid n$. An integer that is greater than 1 and whose only positive divisors are 1 and itself is said to be prime. An integer that is greater than 1 and is not a prime is said to be composite.

Proposition $2.16$ If $\zeta$ is any complex root of unity and $n$ is any integer, then $\zeta^{n}=1$ if and only if $n$ is a multiple of $o(\zeta)$.

Proof. If $n$ is a multiple of $o(\zeta)$, then there exists an integer $m$ such that $n=o(\zeta) m$ and hence $\zeta^{n}=\left(\zeta^{o(\zeta)}\right)^{m}=1^{m}=1$. Conversely, suppose that $n$ is an integer such that $\zeta^{n}=1$. If $n$ is positive then the process of long division yields integers $q$ and $r$ such that $q \geq 0$, $o(\zeta)>r \geq 0$, and $n=o(\zeta) q+r .$ But then
$$\zeta^{r}=\zeta^{n-o(\zeta) q}=\frac{\zeta^{n}}{\left(\zeta^{(}(\zeta)\right)^{q}}=\frac{1}{1^{q}}=1$$
Since $0 \leq r<o(\zeta)$ and $o(\zeta)$ is the least positive integer $m$ such that $\zeta^{m}=1$, it follows that $r=0$ and hence $n=o(\zeta) q$.

## 数学代写|现代代数代写Modern Algebra代考|Orders of Roots of Unity

$$\zeta^{r}=\zeta^{n-o(\zeta) q}=\frac{\zeta^{n}}{\left(\zeta^{(}(\zeta)\right)^{q}}=\frac{1}{1^{q}}=1$$

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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。