## 数学代写|变分法代写variational methods代考|ME333

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法variational methods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。我们的专家在代写变分法variational methods代写方面经验极为丰富，各种代写变分法variational methods相关的作业也就用不着说。

• 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|变分法代写variational methods代考|Sufficient Statistics

When working with statistical models, it is necessary to recover some, or even all, of its parameters from a set of randomly drawn samples $x_1, x_2, \cdots, x_n$. Assuming these observations are iid and sampled from a PDF $f(x ; \theta)$, estimating the parameter $\theta$ is the goal of many statisticians and engineers, imposing a real challenge sometimes. A rather common approach is to capture and summarize some information from the observations and use it to estimate the parameter $\boldsymbol{\theta}$ instead of using the observations themselves. This strategy is known as data reduction, whereas engineers and computer scientists call it feature extraction.

The problem of the data reduction is the loss of information. How do we guarantee that with the statistics we computed from the observations, call it $T(\mathbf{X})$, we are not discarding information to estimate $\theta$ ? The answer to this question is provided by the Sufficiency Principle, which defines $T(\mathbf{X})$ as a sufficient statistics when, for any two samples $\mathbf{x}_1$ and $\mathbf{x}_2$, where $T\left(\mathbf{x}_1\right)=T\left(\mathbf{x}_2\right)$, the estimation of $\theta$ yields the same results despite observing $\mathbf{X}=\mathbf{x}_1$ or $\mathbf{X}=\mathbf{x}_2$.

Computing the sufficient statistics for a parameter is a rather difficult task in most scenarios, but a rather simple way to do that is by using the Fisher-Neyman Theorem (also known as factorization theorem).

Theorem $2.2$ (Fisher-Neyman Factorization Theorem) Let $x_1, x_2, \cdots, x_n$ be random observations from a discrete distribution with $P D F f(\mathbf{x} ; \boldsymbol{\theta})$, and $\mathbf{x}=$ $\left[x_1, x_2, \cdots, x_n\right]^t . T(\mathbf{X})$ is a sufficient statistics if, and only if, there exist functions $g(T(\mathbf{x}) ; \boldsymbol{\theta})$ and $h(\mathbf{x})$, such that $h(\mathbf{x}) \geq 0$ and, for all sample points $\mathbf{x}$ and all values of $\boldsymbol{\theta}$, the distribution $f(\mathbf{x} ; \boldsymbol{\theta})$ can be factorized as follows:
$$f(\mathbf{x} ; \boldsymbol{\theta})-g(T(\mathbf{x}) ; \boldsymbol{\theta}) h(\mathbf{x}) .$$
For example, for a one-dimensional Poisson distribution with unknown mean $\theta$, if we draw a sample $\mathbf{x}$ formed by $n$ iid observations, it is possible to write the probability function as:
$$f(\mathbf{x} ; \theta)=\prod_{i=1}^n f\left(x_i ; \theta\right)=\prod_{i=1}^n \frac{e^{-\theta} \theta^{x_i}}{x_{i} !}=\frac{e^{-n \theta} \theta \sum_{i=1}^n x_i}{\prod_{i=1}^n x_{i} !}=g(T(\mathbf{x}) ; \theta) h(\mathbf{x})$$
where $g(T(\mathbf{x}) ; \theta)=e^{-n \theta} \theta \sum_{i=1}^n x_i$ and $h(\mathbf{x})=\left(\prod_{i=1}^n x_{i} !\right)^{-1}$. So, from the factorization theorem, $T(\mathbf{x})=\sum_{i=1}^n x_i$ is a sufficient statistics for $\theta$.

## 数学代写|变分法代写variational methods代考|Fisher Information

The Fisher information measures the variance in the distribution $f(x ; \theta)$ inflicted by changes in the parameter space $\Theta$. Intuitively, it quantifies the amount of information about $\theta$ that lies in the random variable $X$.

For the $k$-dimensional parameter space $\boldsymbol{\Theta}$ and random variable $\mathbf{X}$ with PDF $f(\mathbf{x} ; \boldsymbol{\theta})$, the elements of the Fisher information matrix are
$$I_{i, j}(\boldsymbol{\theta})=\operatorname{Cov}\left(\frac{\partial}{\partial \theta_i} \log f(\mathbf{X} ; \boldsymbol{\theta}), \frac{\partial}{\partial \theta_j} \log f(\mathbf{X} ; \boldsymbol{\theta})\right),$$
where $\operatorname{Cov}(\cdot, \cdot)$ is the covariance function.
The vector $\frac{\partial}{\partial \theta} \log f(\mathbf{X} ; \boldsymbol{\theta})$ is called the score function and indicates the sensitivity of the likelihood to the parameter $\boldsymbol{\theta}$. When a likelihood $L(\boldsymbol{\theta} \mid \mathbf{x})$, corresponding to the PDF $f(\mathbf{x} ; \boldsymbol{\theta})$, is very sensitive to variations in $\boldsymbol{\theta}$ it is easier to find strong candidates to the true parameter value: even small changes in $\theta$ are enough to cause the likely observations to be considerably different. However, since the score function has mean equal to zero [9], a high $I_{i, j}(\theta)$ implies that the score function is generally high and then, $\mathbf{X}$ distinguishes well the plausible values of $\boldsymbol{\theta}$. We state “generally” because, being the covariance of the score, the Fisher information is an expectation over all possible values of $\mathbf{x}$.

The Fisher information encodes the curvature of the parameter space and plays an important role in optimization. In Chap. 4 we shall see one method that relies on the Fisher information and in Appendix A.4 we show that the Fisher matrix is the negative of the expected value of the Hessian of the log-likelihood.

# 变分法代写

## 数学代写|变分法代写variational methods代考|Fisher Information

Fisher 信息衡量分布的方差 $f(x ; \theta)$ 由参数空间的变化引起 $\Theta$. 直观上，它量化了有关的信息量 $\theta$ 在于随机变量 $X$

$$I_{i, j}(\boldsymbol{\theta})=\operatorname{Cov}\left(\frac{\partial}{\partial \theta_i} \log f(\mathbf{X} ; \boldsymbol{\theta}), \frac{\partial}{\partial \theta_j} \log f(\mathbf{X} ; \boldsymbol{\theta})\right),$$

Fisher 信息对参数空间的曲率进行编码，在优化中起着重要作用。在第一章 在图 4 中，我们将看到一种依赖 Fisher 信息的方法，在附录 A.4 中，我们证明了 Fisher 矩阵是对数似然 Hessian 矩阵期望值的负值。

## 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|变分法代写variational methods代考|МATH687

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法variational methods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。我们的专家在代写变分法variational methods代写方面经验极为丰富，各种代写变分法variational methods相关的作业也就用不着说。

• 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|变分法代写variational methods代考|De Finetti’s Representation Theorem

Independence is a strong assumption. Instead we can resort to the infinite exchangeability property. Exchangeability is the minimal assumption of symmetry. A finite sequence of random variables $\left(\mathbf{X}_1, \mathbf{X}_2, \ldots, \mathbf{X}_n\right)$ is said to be exchangeable if any permutation of its elements has the same probability distribution [7]. Consequently, the order of the sequence is not relevant to determine the joint distribution nor any marginal. In particular, all marginal distributions are equal. An infinite sequence of random variables is infinitely exchangeable if every finite subsequence is exchangeable. Note that this is more general than the iid property.

De Finetti’s representation theorem [7] states that if the sequence of random variables is infinitely exchangeable, then there exists $p(\mathbf{z})$ for which the joint distribution can be written as
$$p\left(\mathbf{x}_1, \mathbf{x}_2, \ldots\right)=\int \prod_i p\left(\mathbf{x}_i \mid \mathbf{Z}\right) p(\mathbf{z}) d \mathbf{z}$$
We can thus see the joint distribution of an infinitely exchangeable sequence of random variables as representing a process in which a random parameter is drawn from some (prior) distribution and then all observables in question are iid conditioned on that parameter.

The representation theorem shows how statistical models emerge in a Bayesian context: under the hypothesis of exchangeability of ${\mathbf{X}}_{i=1}^{\infty}$, a much weaker assumption than iid, there exists a parameter such that, given its value, the observables are conditionally iid. The theorem is a powerful motivation for Bayesian parametric models even though it does not say anything about $p(\mathbf{z})$.

In practical problems, even if we deal with unordered data, the number of random variables is always finite. Therefore, the infinite exchangeability assumption may be impractical or wrong, but the result still holds approximately true for large $n$ [5].

## 数学代写|变分法代写variational methods代考|The Likelihood Function

The likelihood function $L(\theta \mid \mathbf{x})$ measures how well the parameter $\theta$ explains the observations $\mathbf{x}$ of the random variable $\mathbf{X}$. Thus, it measures the model’s ability to fit the observed data for different values of $\theta$. The definition is similar to the PDF $f(\mathbf{x} ; \theta)$, following
$$L(\theta \mid \mathbf{x})=f(\mathbf{x} ; \theta) .$$
The higher its value, the more likely the given value of $\theta$, indicating that $\mathbf{x}$ had higher probability of being observed over other realizations of $\mathbf{X}$.

Despite the similarity in the definition, the likelihood considers $\mathbf{x}$ as known and fixed while $\theta$ as the unknown variable. On the other hand, the PDF considers $\theta$ as fixed and $\mathbf{x}$ as the variable. Hence, the likelihood function is not a PDF and, as a consequence, does not necessarily sum to one.

We can understand the role of the likelihood term $L(\theta \mid \mathbf{x})$ more practically by means of an example. Let $f(\cdot ; w)$ be a regression model parameterized by $w$ that predicts a scalar value $\hat{y}$, such that $\hat{y}=f(x)$. Suppose a probabilistic model that assumes a given level of noise and we thus place an observation noise model on top of the output, such that the observed output is corrupted by a known process $g(\cdot)$, say an additive Gaussian noise with variance $\sigma^2$. So, now our model does not output the correct value $f(x ; w)$, instead it outputs a value that fluctuates around it according to a Gaussian distribution with variance $\sigma^2$. The log-likelihood then has the form
\begin{aligned} \log L(w \mid y, x) &=\log \mathcal{N}\left(y ; f(x ; w), \sigma^2\right) \ &=-\frac{1}{2} \log \left(2 \pi \sigma^2\right)-\frac{1}{2 \sigma^2}(y-f(x ; w))^2 \end{aligned}
What we wish is maximize $L(w \mid y, x)$, which means minimize $(y-f(x ; w))^2$. So, the prediction and the noise model could in principle be anything.

# 变分法代写

## 数学代写|变分法代写variational methods代考|The Likelihood Function

$$L(\theta \mid \mathbf{x})=f(\mathbf{x} ; \theta) .$$

$$\log L(w \mid y, x)=\log \mathcal{N}\left(y ; f(x ; w), \sigma^2\right) \quad=-\frac{1}{2} \log \left(2 \pi \sigma^2\right)-\frac{1}{2 \sigma^2}(y-f(x ; w))^2$$

## 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|变分法代写variational methods代考|МATH4315

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法variational methods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。我们的专家在代写变分法variational methods代写方面经验极为丰富，各种代写变分法variational methods相关的作业也就用不着说。

• 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|变分法代写variational methods代考|Parametric Models

A parametric model $\mathcal{P}$ is a family of distributions $f$ that can be indexed by a finite number of parameters. Let $\boldsymbol{\theta}$ be an element of the parameter space $\Theta$ and $\mathbf{X}$ a random variable, we define the set of possible distribution of the parametric model as
$$\mathcal{P}_{\Theta}={f(\mathbf{x} ; \boldsymbol{\theta}): \boldsymbol{\theta} \in \Theta} .$$
A simple, yet clear example is the uniform distribution $\mathcal{U}(a, b)$ defined by
$$f(x ; a, b)=\left{\begin{array}{l} 1 /(a-b), \text { if } x \in[a, b] \ 0, \text { otherwise } . \end{array}\right.$$
Note that each pair of parameters ${a, b}$ defines a different distribution that follows the same functional form.

We can also generate families of distributions by modifying an original base PDF, hence named standard PDF, in a predefined manner. Concisely, we can either shift, scale, or shift-and-scale the standard distribution.

Theorem 2.1 Let $f(x)$ be a PDF and $\mu$ and $\sigma>0$ constants. Then, the following functions are also a PDF:
$$g(x ; \mu, \sigma)=\frac{1}{\sigma} f\left(\frac{x-\mu}{\sigma}\right) .$$
Hence, introducing the scale $\sigma$ and/or the location $\mu$ parameters in the PDF and tweaking their values lead to new PDFs. Examples of families generated from these procedures include many of the well-known distributions.

## 数学代写|变分法代写variational methods代考|Nonparametric Models

Nonparametric models assume an infinite dimensional parameter space $\boldsymbol{\Theta}$, instead of a finite one. We interpret $\theta$ as a realization from a stochastic process, what defines a probability distribution over $\Theta$ and further allows us to understand $\theta$ as a random function.

A well-known example is given by infinite mixture models [6], which can have a countably infinite number of components, and uses a Dirichlet Process to define a distribution of distributions [9]. The model allows the number of latent components to grow as necessary to accommodate the data, which is a typical characteristic of nonparametric models.

Given observed data $\mathbf{x}$, how should we model the distribution $p(\mathbf{x})$ so that it reflects the true real-world population? This distribution may be arbitrarily complex and to readily assume the data points $\mathbf{x}_i$ to be independent and identically distributed (iid) seems rather naive. After all, they cannot be completely independent, as there must be an underlying reason for them to exist the way they do, even if unknown or latent. We represent this hidden cause by the variable $\mathbf{Z}$, thus obtaining the joint distribution $p(\mathbf{x}, \mathbf{z})$. Naturally, by marginalizing over $\mathbf{z}$ we obtain
$$p(\mathbf{x})=\int p(\mathbf{x}, \mathbf{z}) d \mathbf{z}=\int p(\mathbf{x} \mid \mathbf{Z}) p(\mathbf{z}) d \mathbf{z} .$$
Note that we use latent variables and unknown model parameters interchangeably. For Bayesians, there is no fundamental difference between model parameters and latent variables, as they are all random variables whose values we wish to infer. For example, if our observables $\mathbf{X}_i$ are Bernoulli random variables, then $\mathbf{Z}$ corresponds to the probability of success $p \in(0,1)$.

# 变分法代写

## 数学代写|变分法代写variational methods代考|Parametric Models

$$\mathcal{P}_{\Theta}=f(\mathbf{x} ; \boldsymbol{\theta}): \boldsymbol{\theta} \in \Theta$$

$$g(x ; \mu, \sigma)=\frac{1}{\sigma} f\left(\frac{x-\mu}{\sigma}\right) .$$

## 数学代写|变分法代写variational methods代考|Nonparametric Models

$$p(\mathbf{x})=\int p(\mathbf{x}, \mathbf{z}) d \mathbf{z}=\int p(\mathbf{x} \mid \mathbf{Z}) p(\mathbf{z}) d \mathbf{z} .$$

## 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|变分法代写Calculus of Variations代考|MA431

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法Calculus of Variations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。我们的专家在代写变分法Calculus of Variations代写方面经验极为丰富，各种代写变分法Calculus of Variations相关的作业也就用不着说。

• 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|变分法代写Calculus of Variations代考|Directional Derivative and Gradient

A quantity with magnitude and direction is called the vector. The magnitude of the vector is called the length of vector or modulus of vector. The modulus of vector $\boldsymbol{a}$ is expressed by $|\boldsymbol{a}|$. A vector that its modulus equals 1 is called the unit vector or vector of unit length. A vector that its modulus equals 0 is called the zero vector, it is written as $\mathbf{0}$.

The first partial derivative $\frac{\partial \varphi}{\partial x}$, $\frac{\partial \varphi}{\partial y}$ and $\frac{\partial \varphi}{\partial z}$ of the function $\varphi=\varphi(M)=\varphi(x, y, z)$ denote respectively the rate of change on three specific directions along the $x, y$ and $z$ axis at point $M$. However, in many problems, the rate of changed of the function $\varphi=\varphi(x, y, z)$ along the other direction also has practical significance, so it is necessary to study its derivative in the other direction.

Suppose that $M_0$ is a determined point of the function $\varphi(M)$, a straight line $L$ is drawn through the point, a moving point $M$ near $M_0$ is chosen on the straight line, the distance from point $M_0$ to point $M$ is $\overline{M_0 M}$, when $M \rightarrow M_0$, if the limit of ratio
$$\frac{\varphi(M)-\varphi\left(M_0\right)}{\overline{M_0 M}}$$ exists, then it is called the directional derivative of the function $\varphi(M)$ along direction $L$ at point $M_0$, and it is written as
$$\frac{\partial \varphi\left(M_0\right)}{\partial L}=\lim _{M_0 M \rightarrow 0} \frac{\varphi(M)-\varphi\left(M_0\right)}{\overline{M_0 M}}$$
Thus it can be seen that the directional derivative is the rate of change of the function $\varphi(M)$ to the distance along a certain direction at a given point. If $\frac{\partial \varphi}{\partial L}>0$, the value of the function $\varphi$ increases along the direction $L$; If $\frac{\partial \varphi}{\partial L}<0$, the value of the function $\varphi$ the decreases $L$; If $\frac{\partial \varphi}{\partial L}=0$, the value of the function $\varphi$ does not change along the direction $L$.

Infinitely multiple directions can be chosen through point $M_0$, every direction corresponds with the directional derivative. In the rectangular coordinate system, the given formula by the following theorem can calculate the direction derivative.

## 数学代写|变分法代写Calculus of Variations代考|The Gauss Theorem and Green’s Formulae

Theorem 1.3.2 Suppose that a vector $a$ within a closed surface $S$ has a continuous first order partial derivative, then the flux of $a$ on $S$ equals the integral of the divergence of the vector with respect to the volume $V$ surrounded by $S$, that is
$$\oiint_S \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}=\oiint_S \boldsymbol{a} \cdot \boldsymbol{n} \mathrm{d} S=\iiint_V \operatorname{div} \boldsymbol{a} \mathrm{dV}=\iiint_V \nabla \cdot \boldsymbol{a} \mathrm{dV}$$
where, $\boldsymbol{n}$ is the unit outward normal vector on $S ; \nabla$ is the Hamiltonian operator.
Equation (1.3.45) is called the Gauss-Ostrogradsky theorem, it is called the Gauss theorem for short, it is also called the Gauss formula in the form of divergence or divergence theorem. It has established the relationship between the triple integral of a continuous function in the spatial domain $V$ and the surface integral on the boundary surface $S$. The Gauss formula in the theoretical research and practical work has been widely used, it is a powerful mathematical tool, many formulae are derived from the Gauss formula.

Proof The volume $V$ surrounded by the closed surface $S$ is divided into $n$ volume element $\Delta V_1, \Delta V_2, \ldots, \Delta V_n$ surrounded by closed surface $\Delta S_1, \Delta S_2, \ldots, \Delta S_n$. taking out the $k$ th surface $\Delta S_k$ and the volume $\Delta V_k$ surrounded by it, by the definition of divergence, there is the following relation
$$\operatorname{div} \boldsymbol{a}=\frac{\mathscr{D}{\Delta S_k} \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}}{\Delta V_k}+\varepsilon_k$$ namely $$\Delta V_k \operatorname{div} \boldsymbol{a}=\oiint{\Delta S_k} \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}+\varepsilon_k \Delta V_k$$
where, the divergence is the value at some point $M$ in the elemental volume; $\varepsilon_k$ is small enough, and when $\Delta V_k \rightarrow 0, \varepsilon_k \rightarrow 0$. Summing the above expression to $k$ from 1 to $n$, we obtain
$$\sum_{k=1}^n \Delta V_k \operatorname{div} \boldsymbol{a}=\sum_{k=1}^n \oiint_{\Delta S_k} \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}+\sum_{k=1}^n \varepsilon_k \Delta V_k$$

## 数学代写|变分法代写Calculus of Variations代考|Directional Derivative and Gradient

$$\frac{\varphi(M)-\varphi\left(M_0\right)}{\overline{M_0 M}}$$

$$\frac{\partial \varphi\left(M_0\right)}{\partial L}=\lim _{M_0 M \rightarrow 0} \frac{\varphi(M)-\varphi\left(M_0\right)}{\overline{M_0 M}}$$

## 数学代写|变分法代写Calculus of Variations代考|The Gauss Theorem and Green’s Formulae

$$\backslash \text { oiint }S \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}=\backslash \text { oiint }_S \boldsymbol{a} \cdot \boldsymbol{n} \mathrm{d} S=\iiint_V \operatorname{div} \boldsymbol{a} \mathrm{dV}=\iiint_V \nabla \cdot \boldsymbol{a} \mathrm{dV}$$ 在哪里， $\boldsymbol{n}$ 是单位外法向量 $S ; \nabla$ 是哈密顿算子。 方程(1.3.45)称为Gauss-Ostrogradsky定理，简称高斯定理，也称为散度或散度定理形式的高斯公式。它建立了空 间域中连续函数的三重积分之间的关系 $V$ 和边界面上的曲面积分 $S$. 高斯公式在理论研究和实际工作中得到了广泛 的应用，它是一种强大的数学工具，很多公式都是从高斯公式推导出来的。 证明卷 $V$ 被封闭曲面包围 $S$ 分为 $n$ 体积元 $\Delta V_1, \Delta V_2, \ldots, \Delta V_n$ 被封闭面包围 $\Delta S_1, \Delta S_2, \ldots, \Delta S_n$. 取出 $k$ 表面 $\Delta S_k$ 和音量 $\Delta V_k$ 被它包围，根据散度的定义，有如下关系 $$\operatorname{div} \boldsymbol{a}=\frac{\mathscr{D} \Delta S_k \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}}{\Delta V_k}+\varepsilon_k$$ 即 $$\Delta V_k \operatorname{div} \boldsymbol{a}=\backslash \text { oiint } \Delta S_k \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}+\varepsilon_k \Delta V_k$$ 其中，散度是某个点的值 $M$ 在元素体积中； $\varepsilon_k$ 足够小，当 $\Delta V_k \rightarrow 0, \varepsilon_k \rightarrow 0$. 将上述表达式求和为 $k$ 从 1 到 $n$ ， 我们获得 $$\sum{k=1}^n \Delta V_k \operatorname{div} \boldsymbol{a}=\sum_{k=1}^n \backslash \text { oiint }{\Delta S_k} \boldsymbol{a} \cdot \mathrm{d} \boldsymbol{S}+\sum{k=1}^n \varepsilon_k \Delta V_k$$

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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。

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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。

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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。

## 数学代写|变分法代写Calculus of Variations代考|МATH11179

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法Calculus of Variations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。我们的专家在代写变分法Calculus of Variations代写方面经验极为丰富，各种代写变分法Calculus of Variations相关的作业也就用不着说。

• 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|变分法代写Calculus of Variations代考|Integrals with Parameters

Suppose that a function $f(x, y)$ is the bounded function in the rectangular domain $D[a \leq x \leq b, c \leq y \leq d]$, for any fixed $y_0$ in $[c, d]$, the function $f\left(x, y_0\right)$ is the function of $x$, if it is integrable in $[a, b]$, then the integral $\int_a^b f\left(x, y_0\right) \mathrm{d} x$ uniquely identifies a number, this number is related to $y_0$, when $y_0$ changes in $[c, d]$, the obtained integral value in general is different, it can be expressed as
$$\varphi(y)=\int_a^b f(x, y) \mathrm{d} x$$
where, $\varphi(y)$ is a function of $y$, its domain is $[c, d]$, usually $y$ is called the parameter. It is considered a constant in the process of integral, and the integral $\int_a^b f(x, y) \mathrm{d} x$ is called the integral with parameter.

Below the properties of the continuity, derivability and integrability etc. of integral with parameter are discussed, these properties can be expressed with some theorems.
Theorem 1.2.1 (The continuity) Suppose that a function $f(x, y)$ is continuous in the closed domain $D[a, b ; c, d]$, then the function
$$\varphi(y)=\int_a^b f(x, y) \mathrm{d} x$$
is continuous in the closed interval $[c, d]$. This property can also be rewritten into
$$\lim {y \rightarrow y_0} \int_a^b f(x, y) \mathrm{d} x=\int_a^b \lim {y \rightarrow y_0} f(x, y) \mathrm{d} x$$ namely the order of operations of the limit and integral can be exchanged. This property is called the finding limit under the integral sign or taking limit under the integral sign.

## 数学代写|变分法代写Calculus of Variations代考|Fundamentals of the Theory of Field

A field is a form of relationship between physical quantities in the real world and space and time, it is a form of material existence. If at each point in a space region, it corresponds to a certain value of a physical quantity, then this space region is called the field of the physical quantity existing in it. A physical quantity distribution in the field can be expressed as a function of spatial position, this function is called the point function of the physical quantity. Of course, a physical quantity in the field may also change with time, therefore a point function can also be related to time. If a physical quantity has the property of the quantity, then the field formed by the physical quantity is called the scalar field. If a physical quantity has the property of the vector, then the field formed by the physical quantity is called the vector field. If a physical quantity has the property of the tensor, then the field formed by the physical quantity is called the tensor ficld. In the field of physical quantity, a function that its value is a quantity is called the scalar function, a function that its value is a vector is called the vector function, a function that its value is a tensor is called the tensor function. The point function, scalar function, vector function and tensor function can all be called the function for short.

## 数学代写|变分法代写Calculus of Variations代考|Integrals with Parameters

$$\varphi(y)=\int_a^b f(x, y) \mathrm{d} x$$

$$\varphi(y)=\int_a^b f(x, y) \mathrm{d} x$$

$$\lim y \rightarrow y_0 \int_a^b f(x, y) \mathrm{d} x=\int_a^b \lim y \rightarrow y_0 f(x, y) \mathrm{d} x$$

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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。

## 数学代写|变分法代写Calculus of Variations代考|MS-E1991

statistics-lab™ 为您的留学生涯保驾护航 在代写变分法Calculus of Variations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。我们的专家在代写变分法Calculus of Variations代写方面经验极为丰富，各种代写变分法Calculus of Variations相关的作业也就用不着说。

• 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|变分法代写Calculus of Variations代考|Case of a Function of One Variable

Theorem 1.1.1 If a function $f(x)$ has the continuous derivative of order $n+1$ in a certain open interval $(a, b)$ at point $x_0$, then when $x$ is within the open interval $(a, b)$, the function $f(x)$ can be expressed as
$$f(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right)+\frac{f^{\prime \prime}\left(x_0\right)}{2 !}\left(x-x_0\right)^2+\cdots+\frac{f^{(n)}\left(x_0\right)}{n !}\left(x-x_0\right)^n+R_n$$
where
$$R_n=\frac{f^{(n+1)}(\xi)}{(n+1) !}\left(x-x_0\right)^{n+1}$$
where, $\xi$ is a value between $x_0$ and $x$.
Theorem 1.1.1 is called the Taylor mean value theorem or Taylor theorem of a function of one variable. Equation (1.1.1) is called the Taylor formula of $f(x)$ expansion to $n$th order at point $x_0$ by the power of $\left(x-x_0\right)$ or is called the Tayler series expansion. Equation (1.1.2) is called the Lagrange’s remainder. When $x \rightarrow x_0$, $R_n$ is a higher order infinitesimal than $\left|x-x_0\right|^n$, or $R_n$ is a higher infinitesimal of order $n-1$ than $\left|x-x_0\right|$.

Theorem 1.1.2 Suppose that a derivable function $f(x)$ has an extremal value at a point $x_0$ in the interval of definition, then there must be $f^{\prime}\left(x_0\right)=0$. The theorem is called the extremal theorem of function of one variable.

## 数学代写|变分法代写Calculus of Variations代考|Cases of Functions of Several Variables

The Taylor mean value theorem of a function of one variable can be extended to the case of function of several variables. Below the Taylor formula of binary function which has Lagrange’s remainder will be discussed first.

Theorem 1.1.3 Suppose that a function $f(x, y)$ is continuous in a convex neighborhood at point $\left(x_0, y_0\right)$ and has until the continuous partial derivative of order $n$ $+1$, and let $x=x_0+\Delta x, y=y_0+\Delta y$ be an arbitrary point within the neighborhood, then there is always a $\theta(0<\theta<1)$, to make the following Taylor formula of order $n$
\begin{aligned} f(x, y)=& f\left(x_0, y_0\right)+\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right) f\left(x_0, y_0\right)+\frac{1}{2 !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^2 f\left(x_0, y_0\right) \ &+\cdots+\frac{1}{k !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^k f\left(x_0, y_0\right)+\cdots+\frac{1}{n !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^n f\left(x_0, y_0\right)+R_n \end{aligned}
hold, where, the general term is
$$\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^k f\left(x_0, y_0\right)=\sum_{r=0}^k C_k^r(\Delta x)^r(\Delta y)^{k-r} \frac{\partial^k f\left(x_0, y_0\right)}{\partial x^r \partial y^{k-r}}$$
namely according to Newton binomial theorem, to expand it into the summer of $k+1$ terms, here, $C_k^r=\frac{k !}{r !(k-r) !}$ is a combination number to take $r$ elements from $k$ elements. The remainder is
$$R_n=\frac{1}{(n+1) !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^{n+1} f\left(x_0+\theta \Delta x, y_0+\theta \Delta y\right)$$
where, $R_n$ is called the $n$th order Lagrange(‘s) remainder of $f(x, y)$ at point $\left(x_0, y_0\right)$.

Theorem $1.1 .3$ is called the Taylor mean value theorem of function of two variables.

## 数学代写|变分法代写Calculus of Variations代考|Case of a Function of One Variable

$$f(x)=f\left(x_0\right)+f^{\prime}\left(x_0\right)\left(x-x_0\right)+\frac{f^{\prime \prime}\left(x_0\right)}{2 !}\left(x-x_0\right)^2+\cdots+\frac{f^{(n)}\left(x_0\right)}{n !}\left(x-x_0\right)^n+R_n$$

$$R_n=\frac{f^{(n+1)}(\xi)}{(n+1) !}\left(x-x_0\right)^{n+1}$$

## 数学代写|变分法代写Calculus of Variations代考|Cases of Functions of Several Variables

$$f(x, y)=f\left(x_0, y_0\right)+\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right) f\left(x_0, y_0\right)+\frac{1}{2 !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^2 f\left(x_0, y_0\right) \quad+\cdots$$

$$\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^k f\left(x_0, y_0\right)=\sum_{r=0}^k C_k^r(\Delta x)^r(\Delta y)^{k-r} \frac{\partial^k f\left(x_0, y_0\right)}{\partial x^r \partial y^{k-r}}$$

$$R_n=\frac{1}{(n+1) !}\left(\Delta x \frac{\partial}{\partial x}+\Delta y \frac{\partial}{\partial y}\right)^{n+1} f\left(x_0+\theta \Delta x, y_0+\theta \Delta y\right)$$

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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

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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。