### 数学代写|matlab代写|Finite Fields

MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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• Statistical Inference 统计推断
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• (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|matlab代写|Finite Fields

Finite fields play an important role in several of the applications that we will present in this book. In this section, we will describe the theoretical basis of constructing finite fields.

It can easily be shown that the ring $\mathbb{Z}{p}={0,1,2, \ldots, p-1}$ for prime $p$ is a field with the usual operations of addition and multiplication modulo $p$ (i.e., divide the result by $p$ and take the remainder). This shows that there are finite fields of order $p$ for every prime $p$. In the following discussion, we show how the fields $\mathbb{Z}{p}$ can be used to construct finite fields of order $p^{n}$ for every prime $p$ and positive integer $n$.

Suppose $m$ is an irreducible element in a Euclidean domain $D$, and let $B=(m)$. Then by Theorem $1.13$, we know that $D / B$ must be a field. If $D$ is the ring $\mathbb{Z}$ of integers and $m>0$, then $m$ must be a prime $p$. Note that if we perform the addition and multiplication operations in $D / B$ without including $B$ in the notation, these operations will be exactly the addition and multiplication operations in $\mathbb{Z}{p}$. Thus, we can view $D / B$ as $\mathbb{Z}{p}$.

Now, suppose $D$ is the integral domain $\mathbb{Z}{p}[x]$ of polynomials over $\mathbb{Z}{p}$ for prime $p$, and let $B=(f(x))$ for some irreducible polynomial $f(x)$ of degree

$n$ in $D$. Then again by Theorem $1.13$, we know that $D / B$ must be a field. Each element in $D / B$ is a coset of the form $g(x)+B$ for some $g(x) \in \mathbb{Z}{p}[x]$. Since $\mathbb{Z}{p}[x]$ is a Euclidean domain, there exists $r(x) \in \mathbb{Z}{p}[x]$ for which $g(x)+B=r(x)+B$ with either $r(x)=0$ or $\operatorname{deg}(r(x)){p}[x]$ with either $r(x)=0$ or $\operatorname{deg}(r(x))<n$. Since a polynomial $r(x) \in \mathbb{Z}{p}[x]$ with either $r(x)=0$ or $\operatorname{deg}(r(x)){p}$ ), there are $p^{n}$ polynomials $r(x) \in \mathbb{Z}_{p}[x]$ with either $r(x)=0$ or $\operatorname{deg}(r(x))<n$. Thus, the field $D / B$ will contain $p^{n}$ distinct elements. The operations on this field are the usual operations of addition and multiplication modulo $f(x)$ (i.e., divide the result by $f(x)$ and take the remainder). For convenience, when we write elements and perform the addition and multiplication operations in $D / B$, we will not include $B$ in the notation. That is, we will express the elements $r(x)+B$ in $D / B$ as just $r(x)$. Because it is possible to find an irreducible polynomial of degree $n$ over $\mathbb{Z}_{p}$ for every prime $p$ and positive integer $n$, the comments in the preceding paragraph indicate that there are finite fields of order $p^{n}$ for every prime $p$ and positive integer $n$. It is also true that all finite fields have order $p^{n}$ for some prime $p$ and positive integer $n$ (see Theorem 1.14).

## 数学代写|matlab代写|Finite Fields with Maple

In this section, we will show how Maple can be used to construct the nonzero elements as powers of $x$ in a finite field $\mathbb{Z}{p}[x] /(f(x))$ for prime $p$ and primitive polynomial $f(x) \in \mathbb{Z}{p}[x]$. We will consider the field used in Examples $1.9$ and $1.10$.

We will begin by defining the polynomial $f(x)=x^{2}+x+2 \in \mathbb{Z}{3}[x]$ used to construct the field elements. $>f:=x \rightarrow x^{\wedge} 2+x+2 ;$ $$f:=x \rightarrow x^{2}+x+2$$ We can use the Maple Irreduc function as follows to verify that $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$. The following command will return true if $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$, and false otherwise. $>\operatorname{Irreduc}(\mathrm{f}(\mathrm{x})) \bmod 3 ;$ true Thus, $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$, and $\mathbb{Z}{3}[x] /(f(x))$ is a field. However, in order for us to be able to construct all of the nonzero elements in this field as powers of $x$, it must be the case that $f(x)$ is also primitive. We can use the Maple Primitive function as follows to verify that $f(x)$ is primitive in $\mathbb{Z}{3}[x]$. The following command will return true if $f(x)$ is primitive in $\mathbb{Z}{3}[x]$, and false otherwise. $>\operatorname{Primitive}(\mathrm{f}(\mathrm{x})) \bmod 3 ;$ true Thus, $f(x)$ is primitive in $\mathbb{Z}{3}[x]$.
To construct the nonzero elements in $\mathbb{Z}{3}[x] /(f(x))$ as powers of $x$, we can use the Maple Powmod function. For example, the following command returns the field element that corresponds to $x^{6}$ in $\mathbb{Z}{3}[x] /(f(x))$.
$>\operatorname{Powmod}(\mathrm{x}, 6, \mathrm{f}(\mathrm{x}), \mathrm{x}) \bmod 3$;
$$x+2$$
The operation performed as a consequence of entering the preceding command is the polynomial $x$ given in the first parameter raised to the power 6 given in the second parameter, with the output displayed after the result is reduced modulo the third parameter $f(x)$ defined over the specified coefficient modulus 3 . The fourth parameter is the variable used in the first and third parameters.

We will now use a Maple for loop to construct and display the field elements that correspond to each of the first 8 powers of $x$ in $\mathbb{Z}{3}[x] /(f(x))$. Note that since $f(x)$ is primitive and $\mathbb{Z}{3}[x] /(f(x))$ only has a total of eight nonzero elements, this will cause each of the nonzero elements in $\mathbb{Z}_{3}[x] /(f(x))$ to be displayed exactly once. In the following commands, we store the results returned by Powmod for each of the first 8 powers of $x$ in the variable temp, and display these results using the Maple print function. Note where we use colons and semicolons in this loop, and note also that we use back quotes (“) in the print statement.

## 数学代写|matlab代写|Finite Fields with MATLAB

In this section, we will show how MATLAB can be used to construct the nonzero elements as powers of $x$ in a finite field $\mathbb{Z}{p}[x] /(f(x))$ for prime $p$ and primitive polynomial $f(x) \in \mathbb{Z}{p}[x]$. We will consider the field used in Examples $1.9$ and $1.10$.

We will begin by declaring the variable $x$ as symbolic, and defining the polynomial $f(x)=x^{2}+x+2 \in \mathbb{Z}_{3}[x]$ used to construct the field elements.
$>>$ syms $x$
$>y=Q(x) x^{\sim} 2+x+2$
$f=$
$Q(x) x^{\sim} 2+x+2$

To verify that $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$, we will use the user-written function Irreduc, which we have written separately from this MATLAB session and saved as the M-file Irreduc.m. The following command illustrates how the function Irreduc can be used. The function will return TRUE if $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$, and FALSE otherwise.
$>>\operatorname{Irreduc}(\mathrm{f}(\mathrm{x}), 3)$
ans $=$
TRUE
Thus, $f(x)$ is irreducible in $\mathbb{Z}{3}[x]$, and $\mathbb{Z}{3}[x] /(f(x))$ is a field. However, in order for us to be able to construct all of the nonzero elements in this field as powers of $x$, it must be the case that $f(x)$ is also primitive. To verify that $f(x)$ is primitive in $\mathbb{Z}{3}[x]$, we will use the user-written function Primitive, which we have written separately from this MATLAB session and saved as the M-file Primitive.m. The following command will return TRUE if $f(x)$ is primitive in $\widetilde{Z}{3}[x]$, and FALSE otherwise.
$>$ Primitive $(f(x), 3)$
ans =
TRUE
Thus, $f(x)$ is primitive in $\mathbb{Z}{3}[x]$. To construct the nonzero elements in $\mathbb{Z}{3}[x] /(f(x))$ as powers of $x$, we will use the user-written function Powmod, which we have written separately from this MATLAB session and saved as the M-file Powmod.m. For example, the following command returns the field element that corresponds to $x^{6}$ in $\mathbb{Z}_{3}[x] /(f(x))$.
$$\operatorname{Powmod}(\mathrm{x}, 6, \mathrm{f}(\mathrm{x}), \mathrm{x}, 3)$$
$$\text { ans }=$$
$$x+2$$

## 数学代写|matlab代写|Finite Fields

n在D. 然后再由定理1.13， 我们知道D/乙必须是一个字段。中的每个元素D/乙是形式的陪集G(X)+乙对于一些G(X)∈从p[X]. 自从从p[X]是欧几里得域，存在r(X)∈从p[X]为此G(X)+乙=r(X)+乙与r(X)=0或者你⁡(r(X))p[X]与r(X)=0或者你⁡(r(X))<n. 由于多项式r(X)∈从p[X]与r(X)=0或者你⁡(r(X))p）， 有pn多项式r(X)∈从p[X]与r(X)=0或者你⁡(r(X))<n. 因此，场D/乙将包含pn不同的元素。该字段的运算是加法和乘法模的常用运算F(X)（即，将结果除以F(X)并取余）。为方便起见，当我们写元素并执行加法和乘法运算时D/乙, 我们不会包括乙在符号中。也就是说，我们将表达元素r(X)+乙在D/乙就像r(X). 因为有可能找到一个不可约的次数多项式n超过从p对于每个素数p和正整数n，上一段中的注释表明存在有限的序域pn对于每个素数p和正整数n. 所有有限域都有顺序也是真的pn对于一些素数p和正整数n（见定理 1.14）。

## 数学代写|matlab代写|Finite Fields with Maple

F:=X→X2+X+2我们可以使用 Maple Irreduc 函数如下验证F(X)是不可约的从3[X]. 以下命令将返回 true 如果F(X)是不可约的从3[X]，否则为假。>艾瑞杜克⁡(F(X))反对3;因此，F(X)是不可约的从3[X]， 和从3[X]/(F(X))是一个字段。然而，为了让我们能够将这个领域中的所有非零元素构造为X, 一定是这样的F(X)也是原始的。我们可以使用 Maple Primitive 函数如下验证F(X)是原始的从3[X]. 以下命令将返回 true 如果F(X)是原始的从3[X]，否则为假。>原始⁡(F(X))反对3;因此，F(X)是原始的从3[X].

>战俘⁡(X,6,F(X),X)反对3;

X+2

## 数学代写|matlab代写|Finite Fields with MATLAB

>>符号X
>是=问(X)X∼2+X+2
F=

>>艾瑞杜克⁡(F(X),3)

TRUE

>原始(F(X),3)
ans =
TRUE

$$\operatorname Powmod} (\mathrm {x}, 6, \ mathrm {f} (\mathrm {x}), \ mathrm {x}, 3) \文本{答案}=x+2$$

## 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。