### 数学代写|matlab代写|The Euclidean Algorithm

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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|matlab代写|The Euclidean Algorithm

Suppose $a$ and $b$ are nonzero elements in a Euclidean domain $D$, and consider an element $d \in D$ for which $d \mid a$ and $d \mid b$. Suppose that $d$ also has the property that for all $x \in D$, if $x \mid a$ and $x \mid b$, then $x \mid d$. Then $d$ is called a greatest common divisor of $a$ and $b$, denoted $\operatorname{gcd}(a, b)$.

Greatest common divisors do not always exist for pairs of nonzero elements in rings, although as we will show in Theorem $1.16$, greatest common divisors do always exist for pairs of nonzero elements in Euclidean domains. Also, greatest common divisors, when they exist, need not be unique. For example, in the Euclidean domain $\mathbb{Z}$, both 1 and $-1$ are greatest common divisors of any pair of distinct primes. However, it can easily be verified that if both $d_{1}$ and $d_{2}$ are greatest common divisors of a pair of nonzero elements in a Euclidean domain $D$, then $d_{1}$ and $d_{2}$ will be associates of each other in $D$.

Theorem 1.16 Suppose $a$ and b are nonzero elements in a Euclidean domain D. Then there exists a greatest common divisor $d$ of $a$ and $b$ that can be expressed as $d=a u+b v$ for some $u, v \in D$.

Proof. Let $B$ be an ideal of $D$ of smallest order that contains both $a$ and $b$. It can easily be shown that $B={a r+b s \mid r, s \in D}$. Since $D$ is a Euclidean domain, we know by Theorem $1.9$ that $D$ must also be a principal ideal domain, and $B=(d)$ for some $d \in D$. Since $d$ generates $B$, and $a, b \in B$, then $d \mid a$ and $d \mid b$. Also, since $d \in B={a r+b s \mid r, s \in D}$, then $d=a u+b v$ for some $u, v \in D$. Now, if $x \in D$ with $x \mid a$ and $x \mid b$, then $a=x r$ and $b=x s$ for some $r, s \in D$. Thus, $d=a u+b v=x r u+x s v=x(r u+s v)$, and so $x \mid d$.

When considering specific rings, it is often convenient to place restrictions on greatest common divisors to make them unique. For example, for nonzero elements $a$ and $b$ in $\mathbb{Z}$, there is a unique positive greatest common divisor of $a$ and $b$. Also, for nonzero polynomials $a$ and $b$ in the ring $F[x]$ of polynomials over a field $F$, there is a unique monic (i.e., with a leading coefficient of 1 ) greatest common divisor of $a$ and $b$. Since these are the only two rings that we will use extensively in this book, for the remainder of this book we will assume that greatest common divisors are defined uniquely with these restrictions. We should also note that even though the greatest common divisor of two integers or polynomials $a$ and $b$ is uniquely defined with these restrictions, the $u$ and $v$ that yield $\operatorname{gcd}(a, b)=a u+b v$ need not be unique.

## 数学代写|matlab代写|Computer Exercises

1. For each of the following polynomials $f(x)$, all of which are primitive in $\mathbb{Z}{2}[x]$, construct the field elements that correspond to powers of $x$ in $\mathbb{Z}{2}[x] /(f(x))$. (Note: We will use the fields resulting from these $f(x)$ later in this book.)
(a) $f(x)=x^{5}+x^{3}+1$
(b) $f(x)=x^{6}+x^{5}+1$
(c) $f(x)=x^{7}+x+1$
(d) $f(x)=x^{8}+x^{4}+x^{3}+x^{2}+1$
2. For each of the following polynomials $f(x)$, both of which are primitive in $\mathbb{Z}{5}[x]$, construct the field elements that correspond to powers of $x$ in $\mathbb{Z}{5}[x] /(f(x))$. (Note: We will use the fields resulting from these $f(x)$ later in this book.)
(a) $f(x)=x^{5}+4 x+2$
(b) $f(x)=3 x^{7}+4 x+1$
3. Find a primitive polynomial of degree 4 in $\mathbb{Z}_{3}[x]$, and use this polynomial to construct the nonzero elements in a finite field. (Note: You will need a field of this size in the Chapter 2 Exercises.)
4. Find a primitive polynomial of degree 2 in $\mathbb{Z}_{11}[x]$, and use this polynomial to construct the nonzero elements in a finite field. (Note: You will need a field of this size in the Chapter 2 Exercises.)
5. Use a primitive polynomial to construct the nonzero elements in a finite field of order 127 .

## 数学代写|matlab代写|General Properties

Let $B_{1}, B_{2}, \ldots, B_{b}$ be subsets of a set $S=\left{a_{1}, a_{2}, \ldots, a_{v}\right}$. We will refer to the elements $a_{i}$ as objects and to the subsets $B_{j}$ as blocks. Suppose this collection of objects and blocks also satisfies the following three conditions.

1. Each object appears in the same number of blocks.
2. Each block contains the same number of objects.
3. Every possible pair of objects appears together in the same number of blocks.

Then this collection of objects and blocks is called a balanced incomplete block design. For convenience, we will refer to balanced incomplete block

designs as simply block designs. We will describe a block design using the parameters $(v, b, r, k, \lambda)$ if the design has $v$ objects and $b$ blocks, each object appears in $r$ blocks, each block contains $k$ objects, and every possible pair of objects appears together in $\lambda$ blocks.

In all of the $(v, b, r, k, \lambda)$ block designs that we will consider, we will assume that $k0$. It is reasonable to make these assumptions. Clearly $k \leq v$, and $k=v$ corresponds to the case in which each block contains all of the objects. In the example in the introduction to this chapter, this would represent the potentially unreasonable case in which each of the test-drivers (represented by the blocks) evaluates each of the vehicles (represented by the objects). Also, clearly $\lambda \geq 0$, and $\lambda=0$ corresponds to the case in which each block contains just one of the objects. In the example in the introduction to this chapter, this would represent the potentially unfair case in which each of the test-drivers evaluates just one of the vehicles.

## 数学代写|matlab代写|Computer Exercises

1. 对于以下每个多项式F(X), 所有这些都是原始的从2[X]，构造对应于幂的字段元素X在从2[X]/(F(X)). （注意：我们将使用这些产生的字段F(X)本书后面的内容。）
（a）F(X)=X5+X3+1
(二)F(X)=X6+X5+1
（C）F(X)=X7+X+1
(d)F(X)=X8+X4+X3+X2+1
2. 对于以下每个多项式F(X), 两者都是原始的从5[X]，构造对应于幂的字段元素X在从5[X]/(F(X)). （注意：我们将使用这些产生的字段F(X)本书后面的内容。）
（a）F(X)=X5+4X+2
(二)F(X)=3X7+4X+1
3. 在从3[X], 并使用此多项式构造有限域中的非零元素。（注意：在第 2 章练习中，您将需要此大小的字段。）
4. 在从11[X], 并使用此多项式构造有限域中的非零元素。（注意：在第 2 章练习中，您将需要此大小的字段。）
5. 使用本原多项式在 127 阶有限域中构造非零元素。

## 数学代写|matlab代写|General Properties

1. 每个对象出现在相同数量的块中。
2. 每个块包含相同数量的对象。
3. 每对可能的对象一起出现在相同数量的块中。

## 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。