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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Solvability by Radicals

Whereas thousands of years elapsed between the solutions of the quadratic and the general cubic equation, only a few more years passed before Cardano assigned the problem of finding a formula for the general fourth-degree equation, or quartic equation, to his disciple Lodovico Ferrari and the latter succeeded in solving it. Subsequently, several other mathematicians presented their own solutions of the quartic. From our perspective, the most significant of these is Lagrange’s solution which is presented in detail in Section 6.5. Quite naturally, mathematicians next turned their attention to the general quintic equation, or fifth-degree equation. This equation, however, presented new difficulties, and no substantial progress was made for another 250 years. During the second half of the eighteenth century some mathematicians recognized the fact that even the innocent looking equation
$$x^{n}-1=0$$ presented them with challenges. They were aware that one of the solutions of the equation $x^{11}-1=0$ is the complex number
$$\cos \frac{2 \pi}{11}+i \sin \frac{2 \pi}{11}$$
and consequently this number can be expressed in terms of a radical of the $\mathrm{I}$ th order (namely $\sqrt[11]{1}$ ). However, in I77 I Vandermonde and Lagrange showed that the same number could also be expressed in terms of radicals of the 2 nd and 5 th orders, thus uncovering some surprising relationships between radicals of the $2 \mathrm{nd}, 5$ th, and I ith orders.

## 数学代写|现代代数代写Modern Algebra代考|Other Types of Solutions

While this book’s main theme is the issue of solvability of polynomial equations by means of algebraic operations, it might be pedagogically advantageous at this point to discuss some other senses in which an equation could be solvable. Consider the equation
$$x^{5}-6 x+3=0$$
If we set $f(x)=x^{5}-6 x+3$, then
$$\lim {x \rightarrow \infty} f(x)=\infty \text { and } \lim {x \rightarrow-\infty} f(x)=-\infty$$
Since $f(x)$ is a continuous function, it follows that its graph must cross the $x$-axis at some point and that point clearly yields a solution to Equation 3.8. This argument generalizes easily to the following:

Theorem 3.9 If $n$ is an odd positive integer, and $a_{1}, a_{2}, \ldots, a_{n}$ are real numbers, then the equation
$$x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n-1} x+a_{n}=0$$
has a real solution.
Thus, it is possible to argue the existence of a solution to an equation without having any information at all about the value of the solution. In fact, the mathematicians of the seventeenth and eighteenth centuries became convinced of the validity of the following sweeping statement:

Theorem 3.10 Every equation of the form $x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n-1} x+a_{n}=0$ has a solution.

# 现代代数代考

## 数学代写|现代代数代写Modern Algebra代考|Solvability by Radicals

$$x^{n}-1=0$$

$$\cos \frac{2 \pi}{11}+i \sin \frac{2 \pi}{11}$$

## 数学代写|现代代数代写Modern Algebra代考|Other Types of Solutions

$$x^{5}-6 x+3=0$$

$$\lim x \rightarrow \infty f(x)=\infty \text { and } \lim x \rightarrow-\infty f(x)=-\infty$$

$$x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n-1} x+a_{n}=0$$

## 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
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