## 数学代写|信息论作业代写information theory代考|GAMBLING AND SIDE INFORMATION

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论作业代写information theory代考|GAMBLING AND SIDE INFORMATION

Suppose the gambler has some information that is relevant to the outcome of the gamble. For example, the gambler may have some information about the performance of the horses in previous races. What is the value of this side information?

One definition of the financial value of such information is the increase in wealth that results from that information. In the setting described in Section 6.1 the measure of the value of information is the increase in the doubling rate due to that information. We will now derive a connection between mutual information and the increase in the doubling rate.

To formalize the notion, let horse $X \in{1,2, \ldots, m}$ win the race with probability $p(x)$ and pay odds of $o(x)$ for 1 . Let $(X, Y)$ have joint probability mass function $p(x, y)$. Let $b(x \mid y) \geq 0, \sum_x b(x \mid y)=1$ be an arbitrary conditional betting strategy depending on the side information $Y$, where $b(x \mid y)$ is the proportion of wealth bet on horse $x$ when $y$ is observed. As before, let $b(x) \geq 0, \sum b(x)=1$ denote the unconditional betting scheme.
Let the unconditional and the conditional doubling rates be
$$W^(X)=\max {\mathbf{b}(x)} \sum_x p(x) \log b(x) o(x),$$ $$W^(X \mid Y)=\max {\mathbf{b}(x \mid y)} \sum_{x, y} p(x, y) \log b(x \mid y) o(x)$$
and let
$$\Delta W=W^(X \mid Y)-W^(X)$$
We observe that for $\left(X_i, Y_i\right)$ i.i.d. horse races, wealth grows like $2^{n W^(X \mid Y)}$ with side information and like $2^{n W^(X)}$ without side information.

## 数学代写|信息论作业代写information theory代考|DEPENDENT HORSE RACES AND ENTROPY RATE

The most common example of side information for a horse race is the past performance of the horses. If the horse races are independent, this information will be useless. If we assume that there is dependence among the races, we can calculate the effective doubling rate if we are allowed to use the results of previous races to determine the strategy for the next race.

Suppose that the sequence $\left{X_k\right}$ of horse race outcomes forms a stochastic process. Let the strategy for each race depend on the results of previous races. In this case, the optimal doubling rate for uniform fair odds is
\begin{aligned} W^( & \left.X_k \mid X_{k-1}, X_{k-2}, \ldots, X_1\right) \ & =E\left[\max {\mathbf{b}\left(\cdot \mid X{k-1}, X_{\left.k-2, \ldots, X_1\right)}\right.} E\left[\log S\left(X_k\right) \mid X_{k-1}, X_{k-2}, \ldots, X_1\right]\right] \ & =\log m-H\left(X_k \mid X_{k-1}, X_{k-2}, \ldots, X_1\right), \end{aligned}
which is achieved by $b^\left(x_k \mid x_{k-1}, \ldots, x_1\right)=p\left(x_k \mid x_{k-1}, \ldots, x_1\right)$. At the end of $n$ races, the gambler’s wealth is
$$S_n=\prod_{i=1}^n S\left(X_i\right),$$
and the exponent in the growth rate (assuming $m$ for 1 odds) is
\begin{aligned} \frac{1}{n} E \log S_n & =\frac{1}{n} \sum E \log S\left(X_i\right) \ & =\frac{1}{n} \sum\left(\log m-H\left(X_i \mid X_{i-1}, X_{i-2}, \ldots, X_1\right)\right) \ & =\log m-\frac{H\left(X_1, X_2, \ldots, X_n\right)}{n} . \end{aligned}

# 信息论代写

## 数学代写|信息论作业代写information theory代考|GAMBLING AND SIDE INFORMATION

$$W^(X)=\max {\mathbf{b}(x)} \sum_x p(x) \log b(x) o(x),$$$$W^(X \mid Y)=\max {\mathbf{b}(x \mid y)} \sum_{x, y} p(x, y) \log b(x \mid y) o(x)$$

$$\Delta W=W^(X \mid Y)-W^(X)$$

## 数学代写|信息论作业代写information theory代考|DEPENDENT HORSE RACES AND ENTROPY RATE

\begin{aligned} W^( & \left.X_k \mid X_{k-1}, X_{k-2}, \ldots, X_1\right) \ & =E\left[\max {\mathbf{b}\left(\cdot \mid X{k-1}, X_{\left.k-2, \ldots, X_1\right)}\right.} E\left[\log S\left(X_k\right) \mid X_{k-1}, X_{k-2}, \ldots, X_1\right]\right] \ & =\log m-H\left(X_k \mid X_{k-1}, X_{k-2}, \ldots, X_1\right), \end{aligned}

$$S_n=\prod_{i=1}^n S\left(X_i\right),$$

\begin{aligned} \frac{1}{n} E \log S_n & =\frac{1}{n} \sum E \log S\left(X_i\right) \ & =\frac{1}{n} \sum\left(\log m-H\left(X_i \mid X_{i-1}, X_{i-2}, \ldots, X_1\right)\right) \ & =\log m-\frac{H\left(X_1, X_2, \ldots, X_n\right)}{n} . \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|BOUNDS ON THE OPTIMAL CODE LENGTH

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论作业代写information theory代考|BOUNDS ON THE OPTIMAL CODE LENGTH

We now demonstrate a code that achieves an expected description length $L$ within 1 bit of the lower bound; that is,
$$H(X) \leq L<H(X)+1$$
Recall the setup of Section 5.3: We wish to minimize $L=\sum p_i l_i$ subject to the constraint that $l_1, l_2, \ldots, l_m$ are integers and $\sum D^{-l_i} \leq 1$. We proved that the optimal codeword lengths can be found by finding the $D$-adic probability distribution closest to the distribution of $X$ in relative entropy, that is, by finding the $D$-adic $\mathbf{r}\left(r_i=D^{-l_i} / \sum_j D^{-l_j}\right)$ minimizing
$$L-H_D=D(\mathbf{p} | \mathbf{r})-\log \left(\sum D^{-l_i}\right) \geq 0$$
The choice of word lengths $l_i=\log _D \frac{1}{p_i}$ yields $L=H$. Since $\log _D \frac{1}{p_i}$ may not equal an integer, we round it up to give integer word-length assignments,
$$l_i=\left\lceil\log _D \frac{1}{p_i}\right\rceil,$$

where $\lceil x\rceil$ is the smallest integer $\geq x$. These lengths satisfy the Kraft inequality since
$$\sum D^{-\left\lceil\log \frac{1}{p_i}\right\rceil} \leq \sum D^{-\log \frac{1}{p_i}}=\sum p_i=1$$
This choice of codeword lengths satisfies
$$\log _D \frac{1}{p_i} \leq l_i<\log _D \frac{1}{p_i}+1 .$$
Multiplying by $p_i$ and summing over $i$, we obtain
$$H_D(X) \leq L<H_D(X)+1$$
Since an optimal code can only be better than this code, we have the following theorem.

## 数学代写|信息论作业代写information theory代考|KRAFT INEQUALITY FOR UNIQUELY DECODABLE CODES

We have proved that any instantaneous code must satisfy the Kraft inequality. The class of uniquely decodable codes is larger than the class of instantaneous codes, so one expects to achieve a lower expected codeword length if $L$ is minimized over all uniquely decodable codes. In this section we prove that the class of uniquely decodable codes does not offer any further possibilities for the set of codeword lengths than do instantaneous codes. We now give Karush’s elegant proof of the following theorem.
Theorem 5.5.1 (McMillan) The codeword lengths of any uniquely decodable D-ary code must satisfy the Kraft inequality
$$\sum D^{-l_i} \leq 1$$
Conversely, given a set of codeword lengths that satisfy this inequality, it is possible to construct a uniquely decodable code with these codeword lengths.

Proof: Consider $C^k$, the $k$ th extension of the code (i.e., the code formed by the concatenation of $k$ repetitions of the given uniquely decodable code $C)$. By the definition of unique decodability, the $k$ th extension of the code is nonsingular. Since there are only $D^n$ different $D$-ary strings of length $n$, unique decodability implies that the number of code sequences of length $n$ in the $k$ th extension of the code must be no greater than $D^n$. We now use this observation to prove the Kraft inequality.

Let the codeword lengths of the symbols $x \in \mathcal{X}$ be denoted by $l(x)$. For the extension code, the length of the code sequence is
$$l\left(x_1, x_2, \ldots, x_k\right)=\sum_{i=1}^k l\left(x_i\right) .$$
The inequality that we wish to prove is
$$\sum_{x \in \mathcal{X}} D^{-l(x)} \leq 1$$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|BOUNDS ON THE OPTIMAL CODE LENGTH

$$H(X) \leq L<H(X)+1$$

$$L-H_D=D(\mathbf{p} | \mathbf{r})-\log \left(\sum D^{-l_i}\right) \geq 0$$

$$l_i=\left\lceil\log _D \frac{1}{p_i}\right\rceil,$$

$$\sum D^{-\left\lceil\log \frac{1}{p_i}\right\rceil} \leq \sum D^{-\log \frac{1}{p_i}}=\sum p_i=1$$

$$\log _D \frac{1}{p_i} \leq l_i<\log _D \frac{1}{p_i}+1 .$$

$$H_D(X) \leq L<H_D(X)+1$$

## 数学代写|信息论作业代写information theory代考|KRAFT INEQUALITY FOR UNIQUELY DECODABLE CODES

$$\sum D^{-l_i} \leq 1$$

$$l\left(x_1, x_2, \ldots, x_k\right)=\sum_{i=1}^k l\left(x_i\right) .$$

$$\sum_{x \in \mathcal{X}} D^{-l(x)} \leq 1$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|HIGH-PROBABILITY SETS AND THE TYPICAL SET

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

## 数学代写|信息论作业代写information theory代考|HIGH-PROBABILITY SETS AND THE TYPICAL SET

From the definition of $A_\epsilon^{(n)}$, it is clear that $A_\epsilon^{(n)}$ is a fairly small set that contains most of the probability. But from the definition, it is not clear whether it is the smallest such set. We will prove that the typical set has essentially the same number of elements as the smallest set, to first order in the exponent.

Definition For each $n=1,2, \ldots$, let $B_\delta^{(n)} \subset \mathcal{X}^n$ be the smallest set with
$$\operatorname{Pr}\left{B_\delta^{(n)}\right} \geq 1-\delta$$

We argue that $B_\delta^{(n)}$ must have significant intersection with $A_\epsilon^{(n)}$ and therefore must have about as many elements. In Problem 3.3.11, we outline the proof of the following theorem.

Theorem 3.3.1 Let $X_1, X_2, \ldots, X_n$ be i.i.d. $\sim p(x)$. For $\delta<\frac{1}{2}$ and any $\delta^{\prime}>0$, if $\operatorname{Pr}\left{B_\delta^{(n)}\right}>1-\delta$, then
$$\frac{1}{n} \log \left|B_\delta^{(n)}\right|>H-\delta^{\prime} \quad \text { for } n \text { sufficiently large. }$$
Thus, $B_\delta^{(n)}$ must have at least $2^{n H}$ elements, to first order in the exponent. But $A_\epsilon^{(n)}$ has $2^{n(H \pm \epsilon)}$ elements. Therefore, $A_\epsilon^{(n)}$ is about the same size as the smallest high-probability set.

We will now define some new notation to express equality to first order in the exponent.

## 数学代写|信息论作业代写information theory代考|MARKOV CHAINS

A stochastic process $\left{X_i\right}$ is an indexed sequence of random variables. In general, there can be an arbitrary dependence among the random variables. The process is characterized by the joint probability mass functions $\operatorname{Pr}\left{\left(X_1, X_2, \ldots, X_n\right)=\left(x_1, x_2, \ldots, x_n\right)\right}=p\left(x_1, x_2, \ldots, x_n\right),\left(x_1, x_2, \ldots\right.$, $\left.x_n\right) \in \mathcal{X}^n$ for $n=1,2, \ldots$.

Definition A stochastic process is said to be stationary if the joint distribution of any subset of the sequence of random variables is invariant with respect to shifts in the time index; that is,
\begin{aligned} & \operatorname{Pr}\left{X_1=x_1, X_2=x_2, \ldots, X_n=x_n\right} \ & \quad=\operatorname{Pr}\left{X_{1+l}=x_1, X_{2+l}=x_2, \ldots, X_{n+l}=x_n\right} \end{aligned}
for every $n$ and every shift $l$ and for all $x_1, x_2, \ldots, x_n \in \mathcal{X}$.

A simple example of a stochastic process with dependence is one in which each random variable depends only on the one preceding it and is conditionally independent of all the other preceding random variables. Such a process is said to be Markov.

Definition A discrete stochastic process $X_1, X_2, \ldots$ is said to be a Markov chain or a Markov process if for $n=1,2, \ldots$,
\begin{aligned} \operatorname{Pr}\left(X_{n+1}\right. & \left.=x_{n+1} \mid X_n=x_n, X_{n-1}=x_{n-1}, \ldots, X_1=x_1\right) \ & =\operatorname{Pr}\left(X_{n+1}=x_{n+1} \mid X_n=x_n\right) \end{aligned}
for all $x_1, x_2, \ldots, x_n, x_{n+1} \in \mathcal{X}$.
In this case, the joint probability mass function of the random variables can be written as
$$p\left(x_1, x_2, \ldots, x_n\right)=p\left(x_1\right) p\left(x_2 \mid x_1\right) p\left(x_3 \mid x_2\right) \cdots p\left(x_n \mid x_{n-1}\right)$$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|HIGH-PROBABILITY SETS AND THE TYPICAL SET

$$\operatorname{Pr}\left{B_\delta^{(n)}\right} \geq 1-\delta$$

$$\frac{1}{n} \log \left|B_\delta^{(n)}\right|>H-\delta^{\prime} \quad \text { for } n \text { sufficiently large. }$$

## 数学代写|信息论作业代写information theory代考|MARKOV CHAINS

\begin{aligned} & \operatorname{Pr}\left{X_1=x_1, X_2=x_2, \ldots, X_n=x_n\right} \ & \quad=\operatorname{Pr}\left{X_{1+l}=x_1, X_{2+l}=x_2, \ldots, X_{n+l}=x_n\right} \end{aligned}

\begin{aligned} \operatorname{Pr}\left(X_{n+1}\right. & \left.=x_{n+1} \mid X_n=x_n, X_{n-1}=x_{n-1}, \ldots, X_1=x_1\right) \ & =\operatorname{Pr}\left(X_{n+1}=x_{n+1} \mid X_n=x_n\right) \end{aligned}

$$p\left(x_1, x_2, \ldots, x_n\right)=p\left(x_1\right) p\left(x_2 \mid x_1\right) p\left(x_3 \mid x_2\right) \cdots p\left(x_n \mid x_{n-1}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|交换代数代写commutative algebra代考|Local Criteria for Flatness

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

## 数学代写|交换代数代写commutative algebra代考|Local Criteria for Flatness

In this section we give criteria for flatness over local rings, which can be checked with a computer. We shall weaken the condition $\operatorname{Tor}_1^A(A / I, M)=0$ for all $I \subset A$ to $\operatorname{Tor}_1^A(A / \mathfrak{m}, M)=0, \mathfrak{m}$ the maximal ideal. We shall see that we can compute $\operatorname{Tor}_1^A(A / \mathfrak{m}, M)$ and, therefore, check flatness.

Proposition 7.4.1. Let $M$ be an A-module. The following conditions are equivalent:
(1) $M$ is a flat $A-\operatorname{module}$.
(2) $M_P$ is a flat $A_P-$ module for all prime ideals $P$.
(3) $M_P$ is a flat $A_P$-module for all maximal ideals $P$.
Proof. (1) implies (2), by Exercise 7.3.1. That (2) implies (3) is trivial. Finally, to prove that (3) implies (1), let $I \subset A$ be an ideal. Then
$$\left(I \otimes_A M\right)P=I_P \otimes{A_P} M_P \longrightarrow M_P$$
is injective by assumption. This is true for all maximal ideals and, therefore, $I \otimes_A M \rightarrow M$ is injective by Corollary 2.1.39.

The following theorem can easily be proved if $M$ is a finitely generated $A$ module, by using Nakayama’s lemma. However, its importance is just the fact that we need only a much weaker finiteness assumption, which turns out to be extremely useful in applications.

## 数学代写|交换代数代写commutative algebra代考|Flatness and Standard Bases

In this section we show that standard bases can be characterized in terms of flatness. More precisely, for a suitable weighted ordering $>w$ “approximating” the given ordering $>$, the ring $K[x]{>w} / I$ and the ring $K[x]{>w} / L(I)$ are fibres in a flat family $K[t] \rightarrow K[x, t]{>w} / J$, where $K[x]{>w} / L(I)$ is the special fibre and $K[x]{>w} / I$ is the general fibre. Even more, all fibres different from the special fibre are isomorphic to $K[x]{>w} / I$. In this family, each component of maximal dimension maps surjectively to the target, in particular, the family is faithfully flat. These properties are the background and the reason for many applications. Let $K$ be a field, and let $x=\left(x_1, \ldots, x_n\right)$ and $t$ be variables. Theorem 7.5.1. Let $>$ be any monomial ordering on $\operatorname{Mon}(x), F \subset \operatorname{Mon}(x)$ a finite subset, and $I^{\prime} \subset K[x]{>}$an ideal. Then there exist a weighted degree ordering $>w$ on $\operatorname{Mon}(x, t)$, which is global in $t$ and coincides with $>$ on $F$, such that the following holds: let $J \subset K[x, t]{>w}$ be the ideal generated by the weighted homogenization $I^h$ of $I:=I^{\prime} \cap K[x]$ (w.r.t. the weights $w$ and with homogenizing variable $t$ ), then the following holds: (1) $B:=K[x, t]{>w} / J$ is a flat $K[t]-$-algebra. If $I K[x]{>} \subsetneq K[x]{>}$then, for any maximal ideal $\mathfrak{m} \subset K[t]$, there exists a maximal ideal $M \subset B$ such that $M \cap K[t]=\mathfrak{m}$ and $\operatorname{dim}(B)=\operatorname{dim}\left(B_M\right)$. In particular, in this case $B$ is faithfully flat over $K[t]$. (2) $L(J)=L{>}(I) K[x, t]$.
(3) $L_{>}\left(\left.J\right|{t=\lambda}\right)=L{>}(I)$ for all $\lambda \in K$.
(4) $\left.J\right|{t=0}=L{>}(I) K[x]{>_w}$ and $\left.J\right|{t=1}=I K[x]{\rangle_w}$. Moreover, the fibre $B \otimes{K[t]} K[t] /\langle t-\lambda\rangle \cong K[x]{>_w} /\left(\left.J\right|{t=\lambda}\right)$ is isomorphic to $K[x]{>_w} / I K[x]{>_w}$, for all $\lambda \neq 0$.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Local Criteria for Flatness

(1) $M$是一个平面$A-\operatorname{module}$。
(2) $M_P$是一个平面$A_P-$模，适用于所有素理想$P$。
(3) $M_P$是所有极大理想$P$的平面$A_P$ -模。

$$\left(I \otimes_A M\right)P=I_P \otimes{A_P} M_P \longrightarrow M_P$$

## 数学代写|交换代数代写commutative algebra代考|Flatness and Standard Bases

(3) $L_{>}\left(\left.J\right|{t=\lambda}\right)=L{>}(I)$为所有$\lambda \in K$。
(4) $\left.J\right|{t=0}=L{>}(I) K[x]{>_w}$和$\left.J\right|{t=1}=I K[x]{\rangle_w}$。此外，对于所有$\lambda \neq 0$，纤维$B \otimes{K[t]} K[t] /\langle t-\lambda\rangle \cong K[x]{>_w} /\left(\left.J\right|{t=\lambda}\right)$与$K[x]{>_w} / I K[x]{>_w}$是同构的。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|交换代数代写commutative algebra代考|Characterization of the Dimension of Local Rings

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

## 数学代写|交换代数代写commutative algebra代考|Characterization of the Dimension of Local Rings

Let $A$ be a Noetherian local ring, $\mathfrak{m}$ its maximal ideal and assume, as before, for simplicity that $K=A / \mathfrak{m} \subset A$.

In this section, we shall prove that the dimension of a local ring is equal to the degree of the Hilbert-Samuel polynomial and equal to the least number of generators of an $\mathfrak{m}$-primary ideal. In particular, we shall define and study regular local rings.
Definition 5.6.1. We introduce the following non-negative integers:

$\delta(A):=$ the minimal number of generators of an $\mathfrak{m}$-primary ideal of $A$,

$d(A):=\operatorname{deg}\left(\operatorname{HSP}_{A, \mathfrak{m}}\right)$,

$\operatorname{edim}(A):=$ the embedding dimension of $A$, defined as minimal number of generators for $\mathfrak{m}$. Hence, $\operatorname{edim}(A)=\operatorname{dim}K\left(\mathfrak{m} / \mathfrak{m}^2\right)$, by Nakayama’s Lemma. Theorem 5.6.2. Let $(A, \mathfrak{m})$ be a Noetherian local ring, then, with the above notation, $\delta(A)=d(A)=\operatorname{dim}(A)$. We first prove the following proposition: Proposition 5.6.3. Let $(A, \mathfrak{m})$ be a Noetherian local ring, let $M$ be a finitely generated $A$-module, and let $Q$ an $\mathfrak{m}$-primary ideal. Then (1) $\operatorname{deg}\left(\operatorname{HSP}{M, Q}\right)=\operatorname{deg}\left(\operatorname{HSP}{M, \mathrm{~m}}\right)$; Moreover, let $x \in A$ be a non-zerodivisor for $M$, then (2) $\operatorname{deg}\left(\operatorname{HSP}{M / x M, Q}\right) \leq \operatorname{deg}\left(\operatorname{HSP}_{M, Q}\right)-1$.

## 数学代写|交换代数代写commutative algebra代考|Singular Locus

The aim of this section is to describe the singular locus and prove that the non-normal locus is contained in the singular locus. This means that regular local rings are normal. The proof of this result is, in general, difficult and uses the following result of Serre: the localization of a regular local ring in a prime ideal is again regular. In this section, we shall prove this result for rings of type $K\left[x_1, \ldots, x_n\right]_P /\left\langle f_1, \ldots, f_m\right\rangle, P$ a prime ideal, using a generalization of the Jacobian criterion. A proof for the general case is given in Chapter 7.
Another way to prove that regular rings are normal is used in [66] proving that regular rings are factorial.

Theorem 5.7.1 (General Jacobian criterion). Let $I=\left\langle f_1, \ldots, f_m\right\rangle \subset$ $K\left[x_1, \ldots, x_n\right]$ be an ideal and $P$ an associated prime ideal of $I$. Moreover, let $Q \supset P$ be a prime ideal such that the quotient field of $K\left[x_1, \ldots, x_n\right] / Q$ is separable over $K$. Then
$$\operatorname{rank}\left(\frac{\partial f_i}{\partial x_j} \bmod Q\right) \leq \operatorname{ht}(P)$$
and $K\left[x_1, \ldots, x_n\right]_Q / I_Q$ is a regular local ring if and only if
$$\operatorname{rank}\left(\frac{\partial f_i}{\partial x_j} \bmod Q\right)=\mathrm{ht}(P)$$
Remark 5.7.2. If $Q=\mathfrak{m}$ is a maximal ideal and $K$ is algebraically closed, then we obtain the Jacobian criterion proved in the previous section.

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Characterization of the Dimension of Local Rings

5.6.1.定义我们引入以下非负整数:

$\delta(A):=$ 生成器的最小数量$\mathfrak{m}$ -初级理想$A$，

$d(A):=\operatorname{deg}\left(\operatorname{HSP}_{A, \mathfrak{m}}\right)$，

$\operatorname{edim}(A):=$$A的嵌入维数，定义为\mathfrak{m}的最小生成器数。因此，根据中山引理\operatorname{edim}(A)=\operatorname{dim}K\left(\mathfrak{m} / \mathfrak{m}^2\right)。定理5.6.2。设(A, \mathfrak{m})是一个诺瑟局部环，那么，用上面的符号，\delta(A)=d(A)=\operatorname{dim}(A)。我们首先证明以下命题:命题5.6.3。设(A, \mathfrak{m})为一个诺瑟局部环，设M为一个有限生成的A -模块，设Q为一个\mathfrak{m} -初级理想。然后(1)\operatorname{deg}\left(\operatorname{HSP}{M, Q}\right)=\operatorname{deg}\left(\operatorname{HSP}{M, \mathrm{~m}}\right);此外，设x \in A为M的非零因子，则(2)\operatorname{deg}\left(\operatorname{HSP}{M / x M, Q}\right) \leq \operatorname{deg}\left(\operatorname{HSP}_{M, Q}\right)-1。 ## 数学代写|交换代数代写commutative algebra代考|Singular Locus 本节的目的是描述奇异轨迹，并证明奇异轨迹中包含非正常轨迹。这意味着规则的局部环是正常的。一般来说，这个结果的证明是困难的，并且使用了Serre的以下结果:素理想中正则局部环的局部化又是正则的。在本节中，我们将利用雅可比准则的推广，证明对于类型为K\left[x_1, \ldots, x_n\right]_P /\left\langle f_1, \ldots, f_m\right\rangle, P a素理想的环的这个结果。第7章给出了一般情况的证明。 证明正则环是正规环的另一种方法是在[66]中证明正则环是阶乘。 定理5.7.1(一般雅可比准则)。假设I=\left\langle f_1, \ldots, f_m\right\rangle \subset$$K\left[x_1, \ldots, x_n\right]$是一个理想，$P$是与$I$相关的原理想。此外，设$Q \supset P$为素理想，使得$K\left[x_1, \ldots, x_n\right] / Q$的商域在$K$上是可分离的。然后
$$\operatorname{rank}\left(\frac{\partial f_i}{\partial x_j} \bmod Q\right) \leq \operatorname{ht}(P)$$
$K\left[x_1, \ldots, x_n\right]_Q / I_Q$是正则局部环当且仅当
$$\operatorname{rank}\left(\frac{\partial f_i}{\partial x_j} \bmod Q\right)=\mathrm{ht}(P)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|交换代数代写commutative algebra代考|Triangular Sets

statistics-lab™ 为您的留学生涯保驾护航 在代写交换代数commutative algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写交换代数commutative algebra代写方面经验极为丰富，各种代写交换代数commutative algebra相关的作业也就用不着说。

## 数学代写|交换代数代写commutative algebra代考|Triangular Sets

In this chapter we introduce another method, triangular sets, in order to show how to decompose a zero-dimensional ideal in $K\left[x_1, \ldots, x_n\right]$ into socalled triangular ideals, ideals generated by a lexicographical Gröbner basis of $n$ elements. This is a basic tool for symbolic pre-processing to solve zerodimensional systems of polynomial equations.
In this chapter we fix the lexicographical ordering $1 \mathrm{p}$.
Definition 4.7.1. A set of polynomials $F=\left{f_1, \ldots, f_n\right} \subset K\left[x_1, \ldots, x_n\right]$ is called a triangular set if for each $i$
(1) $f_i \in K\left[x_{n-i+1}, \ldots, x_n\right]$,

(2) $\operatorname{LM}\left(f_i\right)=x_{n-i+1}^{m_i}$, for some $m_i>0$.
Hence, $f_1$ depends only on $x_n, f_2$ on $x_{n-1}, x_n$ and so on, until $f_n$ which depends on all variables.

A list of triangular sets $F_1, \ldots, F_s$ is called a triangular decomposition of the zero-dimensional ideal $I$ if
$$\sqrt{I}=\sqrt{\left\langle F_1\right\rangle} \cap \ldots \cap \sqrt{\left\langle F_s\right\rangle} .$$
Remark 4.7.2. If $F$ is a triangular set then Exercise 1.7.1 implies that $F$ is a Gröbner basis of $\langle F\rangle$.

## 数学代写|交换代数代写commutative algebra代考|The Hilbert Function and the Hilbert Polvnomial

The Hilbert function of a graded module associates to an integer $n$ the dimension of the $n$-th graded part of the given module. For sufficiently large $n$, the values of this function are given by a polynomial, the Hilbert polynomial. To show this, we use the Hilbert-Poincaré series, a formal power series in $t$ with coefficients being the values of the Hilbert function. This power series turns out to be a rational function.
Let $K$ be a field.
Definition 5.1.1. Let $A=\bigoplus_{\nu>0} A_\nu$ be a Noetherian graded $K$-algebra (cf. Definition 2.2.1), and let $M=\bigoplus_{\nu \in \mathbb{Z}} M_\nu$ be a finitely generated graded $A-$ module. The Hilbert function $H_M: \mathbb{Z} \rightarrow \mathbb{Z}$ of $M$ is defined by
$$H_M(n):=\operatorname{dim}K\left(M_n\right),$$ and the Hilbert-Poincaré series $\mathrm{HP}_M$ of $M$ is defined by $$\operatorname{HP}_M(t):=\sum{\nu \in \mathbb{Z}} H_M(\nu) \cdot t^\nu \in \mathbb{Z}[[t]]\left[t^{-1}\right] .$$
By definition, $H_M$ (and, hence, $\mathrm{HP}_M$ ) depend only on the graded structure of $M$. Hence, if $\varphi: B \rightarrow A$ is a graded $K$-algebra map, then it does not matter whether we consider $M$ as $A$-module or as $B$-module. In particular, since $A / \operatorname{Ann}_A(M)$ is a graded $A$-algebra (cf. Exercise 2.2.3), we may always consider $M$ as $A / \operatorname{Ann}_A(M)$-module when computing the Hilbert function (or Hilbert-Poincaré series).

# 交换代数代考

## 数学代写|交换代数代写commutative algebra代考|Triangular Sets

4.7.1.定义一个多项式集$F=\left{f_1, \ldots, f_n\right} \subset K\left[x_1, \ldots, x_n\right]$称为三角集，如果对于每个$i$
(1) $f_i \in K\left[x_{n-i+1}, \ldots, x_n\right]$;

(2) $\operatorname{LM}\left(f_i\right)=x_{n-i+1}^{m_i}$，对于一些$m_i>0$。

$$\sqrt{I}=\sqrt{\left\langle F_1\right\rangle} \cap \ldots \cap \sqrt{\left\langle F_s\right\rangle} .$$

## 数学代写|交换代数代写commutative algebra代考|The Hilbert Function and the Hilbert Polvnomial

5.1.1.定义让 $A=\bigoplus_{\nu>0} A_\nu$ 我是诺埃尔等级的 $K$-代数(参见定义2.2.1)和let $M=\bigoplus_{\nu \in \mathbb{Z}} M_\nu$ 是一个有限生成的分级 $A-$ 模块。希尔伯特函数 $H_M: \mathbb{Z} \rightarrow \mathbb{Z}$ 的 $M$ 定义为
$$H_M(n):=\operatorname{dim}K\left(M_n\right),$$ 以及希尔伯特-庞卡罗系列 $\mathrm{HP}_M$ 的 $M$ 定义为 $$\operatorname{HP}_M(t):=\sum{\nu \in \mathbb{Z}} H_M(\nu) \cdot t^\nu \in \mathbb{Z}[[t]]\left[t^{-1}\right] .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|Refinements and Paracompactness

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

## 数学代写|拓扑学代写Topology代考|Refinements and Paracompactness

The notion of being locally finite extends in a natural way to arbitrary families of subsets.

Definition 7.10 A family $\mathcal{A}$ of subsets in a space $X$ is locally finite if every point $x \in X$ admits a neighbourhood $V \in \mathcal{I}(x)$ such that $V \cap A \neq \emptyset$ for at most finitely many $A \in \mathcal{A}$.

Since any neighbourhood contains an open set, and an open set intersects a subset $A$ if and only if it intersects the closure, a family $\left{A_i \mid i \in I\right}$ is locally finite if and only if $\left{\overline{A_i} \mid i \in I\right}$ is locally finite.
Lemma 7.11 For any locally finite family $\left{A_i\right}$ of subsets,
$$\overline{\cup_i A_i}=\bigcup_i \overline{A_i}$$
In particular the union of a locally finite family of closed sets is closed.
Proof The relation $\cup_i \overline{A_i} \subset \overline{\cup_i A_i}$ is always true because the closed set $\overline{\cup_i A_i}$ contains $A_i$, and so $\overline{A_i}$, for every $i$. There remains to prove that if $\left{A_i\right}$ is locally finite, then $\cup_i \overline{A_i}$ is closed. We can find an open cover $X=\cup_j U_j$ such that $U_j$ intersects finitely many sets $A_i$, whence $\left(\cup_i \overline{A_i}\right) \cap U_j=\cup_i\left(U_j \cap \overline{A_i}\right)$ is closed in $U_j$. To conclude, recall that any open cover is an identification cover.

## 数学代写|拓扑学代写Topology代考|Topological Manifolds

Definition 7.20 A space $M$ is called an n-dimensional topological manifold if:

1. $M$ is Hausdorff;
2. every point in $M$ has an open neighbourhood homeomorphic to an open set of $\mathbb{R}^n$;
3. every connected component of $M$ is second countable.
Example 7.21 Any open set in $\mathbb{R}^n$ is an $n$-dimensional topological manifold, and any open subset in $\mathbb{C}^n$ is a topological manifold of dimension $2 n$.

Example 7.22 The sphere $S^n$ is a topological manifold of dimension $n$ : each point $x$ lies in the open set $S^n-{-x}$, which is homeomorphic to $\mathbb{R}^n$ under stereographic projection.

Example 7.23 The real projective space $\mathbb{P}^n(\mathbb{R})$ is an $n$-dimensional topological manifold: every point lies in the complement of some hyperplane $H$, and $\mathbb{P}^n(\mathbb{R})-H$ is an open set homeomorphic to $\mathbb{R}^n$.

Example 7.24 The complex projective space $\mathbb{P}^n(\mathbb{C})$ is a $2 n$-dimensional topological manifold: any point lies in the complement of some hyperplane $H$ and $\mathbb{P}^n(\mathbb{C})-H$ is open and homeomorphic to $\mathbb{C}^n$.

Remark 7.25 The three conditions of Definition 7.20 are independent from one another, in that any two do not imply the third one. The space described in Exercise 5.8 is connected and second countable, any of its points has a neighbourhood homeomorphic to $\mathbb{R}$, but it is not Hausdorff. Exercise 6.6 provides an instance of a connected Hausdorff space that is locally homeomorphic to $\mathbb{R}^2$ but not second countable.

In many textbooks condition 3. is replaced by paracompactness, and in the rest of this section we set out to prove that the two definitions are equivalent.

# 拓扑学代考

## 学代写|拓扑学代写Topology代考|Refinements and Paracompactness

$$\overline{\cup_i A_i}=\bigcup_i \overline{A_i}$$

## 数学代写|拓扑学代写Topology代考|Topological Manifolds

$M$ 是豪斯多夫;

$M$上的每一点都有一个开邻域同胚于$\mathbb{R}^n$的开集;

$M$的每个连接组件都是秒数的。
$\mathbb{R}^n$中的任何开集都是一个$n$维的拓扑流形，$\mathbb{C}^n$中的任何开子集都是一个$2 n$维的拓扑流形。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|Baire’s Theorem

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

## 数学代写|拓扑学代写Topology代考|Baire’s Theorem

Definition 6.37 A subset in a topological space is said to be nowhere dense if its closure has empty interior, and meagre if it is contained in the union of countably many nowhere-dense subsets.

Being nowhere dense or meagre is not an intrinsic property, in other words it also depends on the ambient space $X$. For example, the point ${0}$ is nowhere dense in $\mathbb{R}$ but not in $\mathbb{Z}$ (it’s not even meagre in the integers); so it makes sense to speak of nowheredense and meagre subspaces, whereas a nowhere-dense or meagre topological space alone is meaningless.

We may, rather punningly, distinguish meagre sets in two categories: truly thin and slender subsets, and ‘false lean’ ones. The former have empty interior, while the second sort do not albeit still being meagre. A Baire space is a space that has no subsets of the second type:

Definition 6.38 A topological space $X$ is a Baire space if each meagre subset has empty interior.

To check such a property it obviously suffices to show that countable unions of nowhere-dense closed sets have non-empty interior, or equivalently, countable intersections of open dense sets are dense.

Example 6.39 The empty set is a Baire space. Any non-empty discrete set is a Baire space: the only nowhere-dense subset is $\emptyset$.

The space $\mathbb{Q}$ is not a Baire space: every point is nowhere dense and closed, and is the countable union of its elements.

## 数学代写|拓扑学代写Topology代考|Completions

Definition 6.43 Let $(X, d)$ and $(\widehat{X}, \hat{d})$ be metric spaces. A map $\Phi: X \rightarrow \widehat{X}$ is called a completion of $(X, d)$ if:

1. $\Phi$ is an isometry: $\hat{d}(\Phi(x), \Phi(y))=d(x, y)$ for all $x, y \in X$;
2. $(\widehat{X}, \hat{d})$ is a complete metric space;
3. $\Phi(X)$ is dense in $\widehat{X}$.
Example 6.44 The inclusion maps $(\mathbb{Q}, d) \subset(\mathbb{R}, d)$ and (] $0,1[, d) \subset([0,1], d)$ are completions ( $d$ is the Euclidean distance).

In this section we shall prove the existence, uniqueness and the main features of completions.

Lemma 6.45 Let $\left{a_n\right},\left{b_n\right}$ be Cauchy sequences in a metric space $(X, d)$. The limit
$$\lim _{n \rightarrow \infty} d\left(a_n, b_n\right) \in[0,+\infty[$$
exists and is finite.

Proof The quadrangle inequality (Exercise 3.32) implies
$$\left|d\left(a_n, b_n\right)-d\left(a_m, b_m\right)\right| \leq d\left(a_n, a_m\right)+d\left(b_n, b_m\right)$$
and so the real sequence $d\left(a_n, b_n\right)$ is Cauchy.
Given a metric space $(X, d)$ we denote by $\mathfrak{c}(X, d)$ the set of all Cauchy sequences in it. Consider on $\mathfrak{c}(X, d)$ the equivalence relation
$$\left{a_n\right} \sim\left{b_n\right} \quad \text { if and only if } \lim _{n \rightarrow \infty} d\left(a_n, b_n\right)=0$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Baire’s Theorem

6.38如果拓扑空间$X$的每个微子集都有空的内部，则该拓扑空间$X$是一个贝尔空间。

## 数学代写|拓扑学代写Topology代考|Completions

6.43设$(X, d)$和$(\widehat{X}， \hat{d})$为度量空间。映射$\Phi: X \rightarrow \widehat{X}$称为$(X, d)$的补全:

\φ是一个等距:美元美元的帽子\ d{}(\φ(x) \φ(y)) = d (x, y)  x, y \ x美元;

$(\widehat{X}， \hat{d})$是完全度量空间;

$\Phi(X)$在$\widehat{X}$中是密集的。


\lim _{n \right \ inty} d\left(a_n, b_n\right) \in[0，+\ inty]


$$\left|d\left(a_n, b_n\right)-d\left(a_m, b_m\right)\right| \leq d\left(a_n, a_m\right)+d\left(b_n, b_m\right)$$

$$\left{a_n\right} \sim\left{b_n\right} \quad \text { if and only if } \lim _{n \rightarrow \infty} d\left(a_n, b_n\right)=0$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|拓扑学代写Topology代考|Locally Compact Spaces

statistics-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

## 数学代写|拓扑学代写Topology代考|Locally Compact Spaces

Definition 5.23 A topological space is locally compact if every point has a compact neighbourhood.

Example 5.24 Open sets in $\mathbb{R}^n$ are locally compact: if $U \subset \mathbb{R}^n$ is open and $x \in U$ we can find $r>0$ satisfying $B(x, r) \subset U$; then the closed ball $\overline{B(x, R)}$, for any $0<R<r$, is a compact neighbourhood of $x$ in $U$.

Straight from the definition we have that any compact space is locally compact. Furthermore,

Proposition 5.25 Every closed subspace in a locally compact space is locally compact. The product of two locally compact spaces is locally compact.

Proof Let $Y$ be closed in $X$ locally compact. Any $y \in Y$ has a compact neighbourhood $U \subset X$. The intersection $Y \cap U$ is a neighbourhood of $y$ in $Y$ and is also compact because closed in $U$.

If $X, Y$ are locally compact and $(x, y) \in X \times Y$, the product $U \times V$ of two compact neighbourhoods of $x \in U \subset X$ and $y \in V \subset Y$ is a compact neighbourhood of $(x, y)$.

On a par with compactness, local compactness is especially useful side by side with the Hausdorff property.

## 数学代写|拓扑学代写Topology代考|The Fundamental Theorem of Algebra

We have all the ingredients to present the most classical proof of the fundamental theorem of algebra, which relies on results from point-set topology. Corollary 15.26 will offer a different proof involving homotopy theory.

Lemma 5.28 Let $p(z)$ be a polynomial of positive degree with complex coefficients, and suppose $p(0) \neq 0$. Then there exists $z \in \mathbb{C}$ such that $|p(z)|<|p(0)|$.

Proof Call $k>0$ the multiplicity of 0 as root of the polynomial $p(z)-p(0)$, so that
$$p(z)=p(0)-z^k\left(b_0+b_1 z+\cdots+b_r z^r\right), \quad \text { with } \quad b_0 \neq 0 .$$
Suppose $c$ is a $k$ th root of $\frac{p(0)}{b_0}$ and consider the continuous map
$$g:[0,1] \rightarrow \mathbb{R}, \quad g(t)=|p(c t)|=\left|p(0)-t^k p(0)-t^{k+1} \sum_{i=1}^r b_i c^{k+i} t^{i-1}\right| .$$
By the triangle inequality
$$g(t) \leq|p(0)|\left(1-t^k\right)+t^{k+1}\left|\sum_{i=1}^r b_i c^{k+i} t^{i-1}\right| .$$
Since by assumption $|p(0)|>0$, for any positive and sufficiently small $t$ we have $g(t)<g(0)$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Locally Compact Spaces

$\mathbb{R}^n$中的开集是局部紧的:如果$U \subset \mathbb{R}^n$是开的，$x \in U$可以找到$r>0$满足$B(x, r) \subset U$;那么封闭的球$\overline{B(x, R)}$，对于任何$0<R<r$，都是$U$中$x$的紧邻。

## 数学代写|拓扑学代写Topology代考|The Fundamental Theorem of Algebra

$$p(z)=p(0)-z^k\left(b_0+b_1 z+\cdots+b_r z^r\right), \quad \text { with } \quad b_0 \neq 0 .$$

$$g:[0,1] \rightarrow \mathbb{R}, \quad g(t)=|p(c t)|=\left|p(0)-t^k p(0)-t^{k+1} \sum_{i=1}^r b_i c^{k+i} t^{i-1}\right| .$$

$$g(t) \leq|p(0)|\left(1-t^k\right)+t^{k+1}\left|\sum_{i=1}^r b_i c^{k+i} t^{i-1}\right| .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|图论作业代写Graph Theory代考|Edge Coloring

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

## 数学代写|图论作业代写Graph Theory代考|Edge Coloring

In addition to coloring the regions of a map and coloring the vertices of a graph, it is also of interest to color the edges of a graph. An edge coloring of a nonempty graph $G$ is an assignment of colors to the edges of $G$, one color to each edge, such that adjacent edges are assigned different colors. The minimum number of colors that can be used to color the edges of $G$ is called the chromatic index (or sometimes the edge chromatic number) and is denoted by $\chi^{\prime}(G)$. An edge coloring that uses $k$ colors is a $k$-edge coloring. In Figure 10.15, a 4-edge coloring of a graph $G$ is given.

Let $G$ be a graph containing a vertex $v$ with $\operatorname{deg} v=k \geq 1$. Then there are $k$ edges incident with $v$. Any edge coloring must assign $k$ distinct colors to the edges incident with $v$ and $\operatorname{so} \chi^{\prime}(G) \geq \operatorname{deg} v=k$. In particular,
$$\chi^{\prime}(G) \geq \Delta(G)$$
for every nonempty graph $G$.

## 数学代写|图论作业代写Graph Theory代考|Excursion: The Heawood Map Coloring Theorem

We mentioned that during an 11-year period in the 19th century (1879-1890), the Four Color Theorem was considered to have been verified by Alfred Bray Kempe. However, all this changed in 1890 when Percy John Heawood wrote that he had discovered an error Kempe had made in the way he interchanged colors in what were to be called Kempe chains. It was not accidental that Heawood had read Kempe’s paper. When Arthur Cayley asked, at a meeting of the London Mathematical Society in 1878, for the status of the Four Color Conjecture, Henry Smith was presiding over the meeting. Smith was a Professor of Geometry at Oxford University who would mention this conjecture during his lectures. Soon afterwards, Heawood became a student of Smith and Heawood became interested in this problem after hearing about it from Smith.

In his paper, Heawood produced a counterexample (see Figure 10.22), not to the statement Kempe was trying to prove (the Four Color Theorem) but to the proof Kempe had given. Indeed, Kempe’s proof was quite ingenious and Heawood was able to use Kempe’s technique to show that every map could be colored with five or fewer colors. We’ve seen that this is equivalent to showing that every planar graph can be colored with five or fewer colors.

Proof. Assume, to the contrary, that this statement is false. Then among all planar graphs that are not 5-colorable, let $G$ be the one of smallest order. Since $G$ is not 5-colorable, the order of $G$ is necessarily 6 or more.

# 图论代考

## 数学代写|图论作业代写Graph Theory代考|Edge Coloring

$$\chi^{\prime}(G) \geq \Delta(G)$$

﻿

﻿

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## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

﻿

## 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。

﻿

The graphs above are incomplete. These figures only show a vertex with degree four (vertex E), its nearest neighbors (A, B, C, and D), and segments of A-C Kempe chains. The entire graphs would also contain several other vertices (especially, more colored the same as B or D) and enough edges to be MPG’s. The left figure has A connected to $C$ in a single section of an A-C Kempe chain (meaning that the vertices of this chain are colored the same as A and C). The left figure shows that this A-C Kempe chain prevents B from connecting to $\mathrm{D}$ with a single section of a B-D Kempe chain. The middle figure has A and C in separate sections of A-C Kempe chains. In this case, B could connect to D with a single section of a B-D Kempe chain. However, since the A and C of the vertex with degree four lie on separate sections, the color of C’s chain can be reversed so that in the vertex with degree four, C is effectively recolored to match A’s color, as shown in the right figure. Similarly, D’s section could be reversed in the left figure so that D is effectively recolored to match B’s color.

Kempe also attempted to demonstrate that vertices with degree five are fourcolorable in his attempt to prove the four-color theorem [Ref. 2], but his argument for vertices with degree five was shown by Heawood in 1890 to be insufficient [Ref. 3]. Let’s explore what happens if we attempt to apply our reasoning for vertices with degree four to a vertex with degree five.

## 数学代写|图论作业代写Graph Theory代考|The previous diagrams

The previous diagrams show that when the two color reversals are performed one at a time in the crossed-chain graph, the first color reversal may break the other chain, allowing the second color reversal to affect the colors of one of F’s neighbors. When we performed the $2-4$ reversal to change B from 2 to 4 , this broke the 1-4 chain. When we then performed the 2-3 reversal to change E from 3, this caused C to change from 3 to 2 . As a result, F remains connected to four different colors; this wasn’t reversed to three as expected.
Unfortunately, you can’t perform both reversals “at the same time” for the following reason. Let’s attempt to perform both reversals “at the same time.” In this crossed-chain diagram, when we swap 2 and 4 on B’s side of the 1-3 chain, one of the 4’s in the 1-4 chain may change into a 2, and when we swap 2 and 3 on E’s side of the 1-4 chain, one of the 3’s in the 1-3 chain may change into a 2 . This is shown in the following figure: one 2 in each chain is shaded gray. Recall that these figures are incomplete; they focus on one vertex (F), its neighbors (A thru E), and Kempe chains. Other vertices and edges are not shown.

Note how one of the 3’s changed into 2 on the left. This can happen when we reverse $\mathrm{C}$ and $\mathrm{E}$ (which were originally 3 and 2 ) on E’s side of the 1-4 chain. Note also how one of the 4’s changed into 2 on the right. This can happen when we reverse B and D (which were originally 2 and 4) outside of the 1-3 chain. Now we see where a problem can occur when attempting to swap the colors of two chains at the same time. If these two 2’s happen to be connected by an edge like the dashed edge shown above, if we perform the double reversal at the same time, this causes two vertices of the same color to share an edge, which isn’t allowed. We’ll revisit Kempe’s strategy for coloring a vertex with degree five in Chapter $25 .$

## 数学代写|图论作业代写Graph Theory代考|The shading of one section of the B-R

• MPG 是三角测量的。它由具有三个边和三个顶点的面组成。
• 每个面的三个顶点必须是三种不同的颜色。
• 每条边由两个相邻的三角形共享，形成一个四边形。
• 每个四边形将有 3 或 4 种不同的颜色。如果与共享边相对的两个顶点恰好是相同的颜色，则它有 3 种颜色。
• 对于每个四边形，四个顶点中的至少 1 个顶点和最多 3 个顶点具有任何颜色对的颜色。例如，具有 R、G、B 和G有 1 个顶点R−是和3个顶点乙−G，或者您可以将其视为 1 个顶点乙−是和3个顶点G−R，或者您可以将其视为 BR 的 2 个顶点和 GY 的 2 个顶点。在后一种情况下，2G’ 不是同一链的连续颜色。
• 当您将更多三角形组合在一起（四边形仅组合两个）并考虑可能的颜色时，您将看到 Kempe 的部分

• 画一张R顶点和一个是由边连接的顶点。
• 如果一个新顶点连接到这些顶点中的每一个，它必须是乙或者G.
• 如果一个新顶点连接到 R 而不是是，可能是是,乙， 或者G.
• 如果一个新的顶点连接到是但不是R，可能是R,乙， 或者G.
• RY 链要么继续增长，要么被 B 包围，G.
• 如果你关注 B 和 G，你会为它的链条得出类似的结论。
• 如果一条链条完全被其对应物包围，则链条的新部分可能会出现在其对应物的另一侧。
Kempe 证明了所有具有四阶的顶点（那些恰好连接到其他四个顶点的顶点）都是四色的 [Ref. 2]。例如，考虑下面的中心顶点。

## 数学代写|图论作业代写Graph Theory代考|In the previous figure

• A 和 C 或者是 AC Kempe 链的同一部分的一部分，或者它们各自位于 AC Kempe 链的不同部分。（如果一种和C例如，是红色和黄色的，则 AC 链是红黄色链。） – 如果一种和C每个位于 AC Kempe 链的不同部分，其中一个部分的颜色可以反转，这有效地重新着色 C 以匹配 A 的颜色。如果 A 和 C 是 AC Kempe 链的同一部分的一部分，则 B 和 D每个都必须位于 BD Kempe 链的不同部分，因为 AC Kempe 链将阻止任何 BD Kempe 链从 B 到达 D。（如果乙和D是蓝色和绿色，例如，那么一种BD Kempe 链是蓝绿色链。）在这种情况下，由于 B 和 D 分别位于 BD Kempe 链的不同部分，因此 BD Kempe 链的其中一个部分的颜色可以反转，这有效地重新着色 D 以匹配 B颜色。– 因此，可以使 C 与 A 具有相同的颜色或使 D 具有与 A 相同的颜色乙通过反转 Kempe 链的分离部分。

Kempe 还试图证明五阶顶点是可四色的，以证明四色定理 [Ref. 2]，但 Heawood 在 1890 年证明他关于五次顶点的论点是不充分的 [Ref. 3]。让我们探讨一下如果我们尝试将我们对度数为四的顶点的推理应用于度数为五的顶点会发生什么。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。