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# 数学代写|现代代数代考Modern Algebra代写|MATH6451 Examples of Applications

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## 数学代写|现代代数代考Modern Algebra代写|Examples of Applications

Throughout history, the theory of numbers was considered as a purely theoretical science with no practical applications and connections to the real world. Mathematicians considered number theory as a purely intellectual aspect. C.F. Gauss said that “if mathematics is the queen of the sciences, then the theory of numbers is, because of its supreme uselessness, the queen of mathematics.”

According to another outstanding English mathematician, G.H. Hardy, “The ‘real’ mathematics of the ‘real mathematicians’, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless’ (and this is as true of ‘applied’ as of ‘pure’ mathematics)”. However, in the same book G.H. Hardy wrote: “If the theory of numbers could be employed for any practical and obviously honotable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications.”

## 数学代写|现代代数代考Modern Algebra代写|Euler’s function and its Properties

One of the most important functions in number theory and its applications is Euler’s totient function. It is an example of arithmetic and multiplicative functions.

Definition 7.1.
An arithmetic function is a function that maps the set of natural numbers to the set of complex numbers: $f: \mathbb{Z}^{+} \rightarrow \mathbb{C}$.

An arithmetic function $f: \mathbb{Z}^{+} \rightarrow \mathbb{R}$, which is not identically zero and satisfies the condition
$$f(m n)=f(m) f(n)$$
for all $m, n \in \mathbb{Z}^{+}$such that $\operatorname{gcd}(m, n)=1$, is called multiplicative.
An arithmetic function $f(x)$, which is not identically zero and satisfies the condition (7.1) for all $m, n \in \mathbb{Z}^{+}, f(x)$, is called totally multiplicative.

Recall that Euler’s totient function $\varphi(m)$ is the number of positive integers less than $m$ that are coprime to $m$. We assume that the number 1 is coprime to each positive integer $m$ and $\varphi(1)=1$. It is obvious that $\varphi(x)$ is an arithmetic function. We show that this function is also multiplicative.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写| Euler’s function and its Properties

$$f(m n)=f(m) f(n)$$

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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。