数学代写|matlab代写|Preliminary Mathematics

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
• 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|matlab代写|Permutation Groups

There are two purposes to this chapter. We very quickly and concisely review some of the basic algebraic concepts that are probably familiar to many readers, and also introduce some topics for specific use in later chapters. We will generally not pursue topics any further than necessary to obtain the material needed for the applications that follow. Topics reviewed in this chapter include permutation groups, the ring of integers, polynomial rings, finite fields, and examples that incorporate these topics using the philosophies of concepts covered in later chapters.

Suppose a set $G$ is closed under an operation *. That is, suppose $a * b \in G$ for all $a, b \in G$. Then $*$ is called a binary operation on $G$. We will use the notation $(G, *)$ to represent the set $G$ with this operation. Suppose $(G, *)$ also satisfies the following three properties.

1. $(a * b) * c=a *(b * c)$ for all $a, b, c \in G$.
2. There exists an identity element $e \in G$ for which $e * a=a * e=a$ for all $a \in G$.
3. For each $a \in G$, there exists an inverse element $b \in G$ for which $a * b=b * a=e$. The inverse of $a$ is usually denoted by $a^{-1}$ if $$is a general operation or multiplication, and -a if$$ is addition.

Then $(G, *)$ is called a group. For example, it can easily be verified that for the set $\mathbb{Z}$ of integers, $(\mathbb{Z},+)$ is a group with identity element 0 , but $(\mathbb{Z}, \cdot)$ with normal integer multiplication is not a group.Let $S$ be a set, and let $B(S)$ be the collection of all bijections (i.e., one-to-one and onto mappings) on $S$. Then any $\alpha \in B(S)$ can be uniquely expressed by its action $\alpha(s)$ on the elements $s \in S$.

数学代写|matlab代写|Cosets and Quotient Groups

Let $H$ be a subgroup of a group $G$. For an element $g \in G$, we define $g H={g h \mid h \in H}$, called a left coset of $H$ in $G$. Since $g h_{1}=g h_{2}$ implies $h_{1}=h_{2}$ for all $h_{1}, h_{2} \in H$, there is a one-to-one correspondence between the elements in $g H$ and $H$. Thus, if $H$ is finite, $|g H|=|H|$. Now, suppose $g_{1}, g_{2} \in G$. If $x \in g_{1} H \cap g_{2} H$ for some $x \in G$, then $x=g_{1} h_{1}=g_{2} h_{2}$ for some $h_{1}, h_{2} \in H$, and $g_{1}=g_{2} h_{2} h_{1}^{-1} \in g_{2} H$. Then for any $y \in g_{1} H$, it follows that $y=g_{1} h_{3}$ for some $h_{3} \in H$, and so $y=g_{1} h_{3}=g_{2} h_{2} h_{1}^{-1} h_{3} \in g_{2} H$. Thus, $g_{1} H \subseteq g_{2} H$. Similarly, $g_{2} H \subseteq g_{1} H$, and so $g_{1} H=g_{2} H$. The preceding discussion implies that if $g_{1}, g_{2} \in G$, then either $g_{1} H=g_{2} H$, or $g_{1} H$ and $g_{2} H$ are disjoint. As a result, $G$ is the union of pairwise disjoint left cosets of $H$ in $G^{1}$

Example 1.5 Consider the subgroup $A_{n}$ of $S_{n}$. If $\alpha$ is an odd permutation in $S_{n}$, then $\alpha A_{n}$ and $A_{n}$ will be disjoint. If $\beta$ is also an odd permutation in $S_{n}$, then $\beta^{-1} \alpha$ will be even. Thus, $\beta^{-1} \alpha \in A_{n}$, and $\alpha A_{n}=\beta A_{n}$. From this we can conclude that there are exactly two distinct left cosets of $A_{n}$ in $S_{n}$, one consisting of the even permutations in $S_{n}$, and the other consisting of the odd permutations in $S_{n}$.

For a finite group $G$ with subgroup $H$, the following theorem is a fundamental algebraic result regarding the number of left cosets of $H$ in $G$.

数学代写|matlab代写|Rings and Euclidean Domains

Let $R$ be a set with two binary operations, an “addition” $+$ and a “multiplication”*. Suppose $R$ also satisfies the following three properties.

1. $(R,+)$ is an abelian group, with identity we will denote by 0 .
2. $(a * b) * c=a *(b * c)$ for all $a, b, c \in R$.
3. $a *(b+c)=(a * b)+(a * c)$ and $(a+b) * c=(a * c)+(b * c)$ for all $a, b, c \in R$.

Then $R$ is called a ring. If it is also true that $a * b=b * a$ for all $a, b \in R$, then $R$ is said to be a commutative ring. Also, if $R$ contains a multiplicative identity element (i.e., an element, usually denoted by 1 , that satisfies $1 * a=a * 1=a$ for all $a \in R$ ), then $R$ is said to be a ring with identity. As is customary (and as we have already done frequently when dealing with multiplication in groups), we will usually suppress the $*$ from the notation when performing the multiplication operation in rings.

All of the rings that we will use in this book are commutative rings with identity. A commutative ring $R$ with identity is called an integral domain if $a, b \in R$ with $a b=0$ implies either $a=0$ or $b=0$. For example, $\mathbb{Z}$ with ordinary addition and multiplication is an integral domain. A commutative ring $R$ with identity is called a field if every nonzero element in $R$ has a multiplicative inverse in $R$. For example, $\mathbb{R}$ with ordinary addition and multiplication is a field. Also, since all fields are integral domains, then $\mathbb{R}$ is an integral domain.

In addition to $\mathbb{Z}$, we will make extensive use of the ring $F[x]$ of polynomials in $x$ with coefficients in a field $F$, with the operations of ordinary addition and multiplication. Like $\mathbb{Z}$, the ring $F[x]$ is an integral domain but not a field.

Suppose now that $B$ is a nonempty subset of a commutative ring $R$. If $(B,+)$ is a subgroup of $(R,+)$, and if $r b \in B$ for all $r \in R$ and $b \in B$, then $B$ is called an ideal of $R$. For an ideal $B$ of $R$, if there exists an element $b \in B$ for which $B={r b \mid r \in R}$, then $B$ is called a principal ideal of $R$. In this case, $B$ is denoted by $(b)$, and called the ideal generated by $b$.

If $f(x) \in F[x]$ for some field $F$, then $(f(x))$ consists of all multiples of $f(x)$ over $F$. That is, $(f(x))$ consists of all polynomials in $F[x]$ that have $f(x)$ as a factor. A similar result holds for integers $n \in \mathbb{Z}$. We will show in Theorem $1.9$ that all ideals of $F[x]$ and $\mathbb{Z}$ are principal ideals. Since $F[x]$ and $\mathbb{Z}$ are integral domains in which every ideal is principal, they are called principal ideal domains.

Ideals play a role in ring theory similar to the role played by normal subgroups in group theory. For example, we can use an ideal to construct a new ring from an existing one. Suppose $B$ is an ideal of a commutative ring $R$. Then since $(B,+)$ is a subgroup of the abelian group $(R,+)$, it follows that $R / B={r+B \mid r \in R}$ is an abelian group with addition operation $(r+B)+(s+B)=(r+s)+B$. As it turns out, $R / B$ is also a commutative ring with multiplication operation $(r+B)(s+B)=(r s)+B$.

数学代写|matlab代写|Permutation Groups

1. (一个∗b)∗C=一个∗(b∗C)对所有人一个,b,C∈G.
2. 存在一个标识元素和∈G为此和∗一个=一个∗和=一个对所有人一个∈G.
3. 对于每个一个∈G, 存在逆元b∈G为此一个∗b=b∗一个=和. 的倒数一个通常表示为一个−1如果 $一世s一个G和n和r一个l这p和r一个吨一世这n这r米在l吨一世pl一世C一个吨一世这n,一个nd-一种一世F$是加法。

数学代写|matlab代写|Rings and Euclidean Domains

1. (R,+)是一个阿贝尔群，我们用 0 表示。
2. (一个∗b)∗C=一个∗(b∗C)对所有人一个,b,C∈R.
3. 一个∗(b+C)=(一个∗b)+(一个∗C)和(一个+b)∗C=(一个∗C)+(b∗C)对所有人一个,b,C∈R.

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。