## 数学代写|勒贝格积分代写Lebesgue Integration代考|Math720

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration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|勒贝格积分代写Lebesgue Integration代考|Continuity and Differentiability

The fourth big question asks for the relationship between continuity and differentiability. We know that a function that is differentiable at a given value of $x$ must also be continuous at that value, and it is clear that the converse does not hold. The function $f(x)=|x|$ is continuous but not differentiable at $x=0$. But how nondifferentiable can a continuous function be?

Throughout the first half of the nineteenth century, it was generally believed that a continuous function would be differentiable at most points. ${ }^4$ Mathematicians recognized that a function might have finitely many values at which it failed to have a derivative. There might even be a sparse infinite set of points at which a continuous function was not differentiable, but the mathematical community was honestly surprised when, in 1875 , Gaston Darboux and Paul du-Bois Reymond ${ }^5$ published examples of continuous functions that are not differentiable at any value.
The question then shifted to what additional assumptions beyond continuity would ensure differentiability. Monotonicity was a natural candidate. Weierstrass constructed a strictly increasing continuous function that is not differentiable at any algebraic number, that is to say, at any number that is the root of a polynomial with rational coefficients. It is not differentiable at $1 / 2$ or $\sqrt{2}$ or $\sqrt[3]{5}-2 \sqrt[21]{35}$. Weierstrass’s function is differentiable at $\pi$. Can we find a continuous, increasing function that is not differentiable at any value? The surprising answer is No. In fact, in a sense that later will be made precise, a continuous, monotonic function is differentiable at “most” values of $x$. There are very important subtleties lurking behind this fourth question.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Term-by-term Integration

Returning to Fourier series, we saw that the heuristic justification relied on interchanging summation and integration, integrating an infinite series of functions by integrating each summand. This works for finite summations. It is not hard to find infinite series for which term-by-term integration leads to a divergent series or, even worse, a series that converges to the wrong value.

Weierstrass had shown that if the series converges uniformly, then term-by-term integration is valid. The problem with this result is that the most interesting series, especially Fourier series, often do not converge uniformly and yet term-by-term integration is valid. Uniform convergence is sufficient, but it is very far from necessary. As we shall see, finding useful conditions under which term-by-term integration is valid is very difficult so long as we cling to the Riemann integral. As Lebesgue would show in the opening years of the twentieth century, his definition of the integral yields a simple, elegant solution, the Lebesgue dominated convergence theorem.

1.1.1. Find the Fourier expansions for $f_1(x)=x$ and $f_2(x)=x^2$ over $[-\pi, \pi]$.
1.1.2. For the functions $f_1$ and $f_2$ defined in Exercise 1.1.1, differentiate each summand in the Fourier series for $f_2$. Do you get the summands in the Fourier series for $2 f_1$ ? Differentiate each summand in the Fourier series for $f_1$. Do you get the summand in the Fourier series for $f_1^{\prime}(x)$ ?
1.1.3. Using the Fourier series expansion for $x^2$ (Exercise 1.1.1) evaluated at $x=\pi$, show that
$$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6} .$$
1.1.4. Show that if $k$ is an integer $\geq 1$, then
$$\int_{-\pi}^\pi \cos (k x) d x=\int_{-\pi}^\pi \sin (k x) d x=0 .$$

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Term-by-term Integration

Weierstrass 已经表明，如果级数一致收敛，则逐项积分是有效的。这个结果的问题在于，最有趣的级 数，尤其是傅立叶级数，通常不会一致收敛，但逐项积分是有效的。均匀收敛就足够了，但远非必要。 正如我们将要看到的，只要我们坚持黎曼积分，就很难找到使逐项积分有效的有用条件。正如勒贝格在 20 世纪初所表明的那样，他对积分的定义产生了一个简单、优雅的解决方案，即勒贝格支配的收敛定 理。
1.1.1. 求傅立叶展开式 $f_1(x)=x$ 和 $f_2(x)=x^2$ 超过 $[-\pi, \pi]$.
1.1.2. 对于函数 $f_1$ 和 $f_2$ 在练习 1.1.1 中定义，区分傅立叶级数中的每个被加数 $f_2$. 你得到傅立叶级数中的 被加数了吗 $2 f_1$ ? 区分傅里叶级数中的每个被加数 $f_1$. 你得到傅立叶级数中的被加数了吗 $f_1^{\prime}(x)$ ?
1.1.3. 使用傅里叶级数展开 $x^2$ (练习 1.1.1) 评估于 $x=\pi$ ， 显示
$$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$
1.1.4. 证明如果 $k$ 是一个整数 $\geq 1$ ， 然后
$$\int_{-\pi}^\pi \cos (k x) d x=\int_{-\pi}^\pi \sin (k x) d x=0$$

## 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|勒贝格积分代写Lebesgue Integration代考|MATH6210

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration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|勒贝格积分代写Lebesgue Integration代考|The Fundamental Theorem of Calculus

The fundamental theorem of calculus is, in essence, simply a statement of the equivalence of the two means of understanding integration, as the inverse process of differentiation and as a limit of sums of products. The precise theorems to which this designation refers today arise from the assumption that integration is defined as a limiting process. They then clarify the precise relationship between integration and differentiation. The actual statements that we shall use are given by the following theorems.

Theorem 1.1 (FTC, evaluation). If $f$ is the derivative of $F$ at every point on $[a, b]$, then under suitable hypotheses we have that
$$\int_a^b f(t) d t=F(b)-F(a) .$$
Theorem $1.2$ (FTC, antiderivative). If $f$ is integrable on the interval $[a, b]$, then under suitable hypotheses we have that
$$\frac{d}{d x} \int_a^x f(t) d t=f(x) .$$
The first of these theorems tells us how we can use any antiderivative to obtain a simple evaluation of a definite integral. The second shows that the definite integral can be used to create an antiderivative, the definite integral of $f$ from $a$ to $x$ is a function of $x$ whose derivative is $f$. Both of these statements would be meaningless if we had defined the integral as the antiderivative. Their meaning and importance comes from the assumption that $\int_a^b f(t) d t$ is defined as a limit of summations.
In both cases, I have not specified the hypotheses under which these theorems hold. There are two reasons for this. One is that much of the interesting story that is to be told about the creation of analysis in the late nineteenth century revolves around finding necessary and sufficient conditions under which the conclusions hold. When working with Riemann’s definition of the integal, the answer is complicated. The second reason is that the hypotheses that are needed depend on the way we choose to define the integral. For Lebesgue’s definition, the hypotheses are quite different.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|A Brief History of Theorems 1.1 and 1.22

The earliest reference to Theorem $1.1$ of which I am aware is Siméon Denis Poisson’s 1820 Suite du Mémoire sur les Intégrales Définies. There he refers to it as “the fundamental proposition of the theory of definite integrals.” Poisson’s work is worth some digression because it illustrates the importance of how we define the definite integral and the difficulties encountered when it is defined as the difference of the values of an antiderivative at the endpoints.

Siméon Denis Poisson (1781-1840) studied and then taught at the École Polytechnique. He succeeded to Fourier’s professorship in mathematics when Fourier departed for Grenoble to become prefect of the department of Isère. It was Poisson who wrote up the rejection of Fourier’s Theory of the Propoagation of Heat in Solid Bodies in 1808. When, in 1815, Poisson published his own article on the flow of heat, Fourier pointed out its many flaws and the extent to which Poisson had rediscovered Fourier’s own work.

Poisson, as a colleague of Cauchy at the École Polytechnique, almost certainly was aware of Cauchy’s definition of the definite integral even though Cauchy had not yet published it. But the relationship between Poisson and Cauchy was far from amicable, and it would have been surprising had Poisson chosen to embrace his colleague’s approach. Poisson defines the definite integral as the difference of the values of the antiderivative. It would seem there is nothing to prove. What Poisson does prove is that if $F$ has a Taylor series expansion and $F^{\prime}=f$, then
$$F(b)-F(a)=\lim {n \rightarrow \infty} \sum{j=1}^n t f(a+(j-1) t), \quad \text { where } t=\frac{b-a}{n} .$$
Poisson begins with the observation that for $1 \leq j \leq n$ and $t=(b-a) / n$, there is a $k \geq 1$ and a collection of functions $R_j$ such that
$$F(a+j t)=F(a+(j-1) t)+t f(a+(j-1) t)+t^{1+k} R_j(t),$$
and therefore
\begin{aligned} F(b)-F(a) & =\sum_{j=1}^n[F(a+j t)-F(a+(j-1) t)] \ & =\sum_{j=1}^n t f(a+(j-1) t)+t^{1+k} \sum_{j=1}^n R_j(t) . \end{aligned}

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Fundamental Theorem of Calculus

$$\int_a^b f(t) d t=F(b)-F(a)$$

$$\frac{d}{d x} \int_a^x f(t) d t=f(x)$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|A Brief History of Theorems 1.1 and 1.22

Siméon Denis Poisson (1781-1840) 在巴黎综合理工学院学习并任教。当傅立叶前往格勒诺布尔成为 伊泽尔省省长时，他继承了傅立叶的数学教授职位。正是泊松在 1808 年写下了拒绝傅立叶的固体热传播 理论的文章。在 1815 年，泊松发表了他自己关于热流的文章时，傅立叶指出了它的许多缺陷以及泊松在 多大程度上重新发现了傅里叶自己的工作。

$$F(b)-F(a)=\lim n \rightarrow \infty \sum j=1^n t f(a+(j-1) t), \quad \text { where } t=\frac{b-a}{n} .$$

$$F(a+j t)=F(a+(j-1) t)+t f(a+(j-1) t)+t^{1+k} R_j(t),$$

$$F(b)-F(a)=\sum_{j=1}^n[F(a+j t)-F(a+(j-1) t)] \quad=\sum_{j=1}^n t f(a+(j-1) t)+t^{1+k} \sum_{j=1}^n R_j(t)$$

## 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|勒贝格积分代写Lebesgue Integration代考|MAT00013H

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration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|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

Fourier’s method for expanding an arbitrary function $F$ defined on $[-\pi, \pi]$ into a trigonometric series is to use integration to calculate coefficients:
\begin{aligned} & a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \ & b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) . \end{aligned}
The Fourier expansion is then given by
$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$
The heuristic argument for the validity of this procedure is that if $F$ really can be expanded in a series of the form given in Equation (1.3), then
\begin{aligned} \int_{-\pi}^\pi & F(x) \cos (n x) d x \ = & \int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x \ = & \int_{-\pi}^\pi \frac{a_0}{2} \cos (n x) d x+\sum_{k=1}^{\infty} \int_{-\pi}^\pi a_k \cos (k x) \cos (n x) d x \ & +\sum_{k=1}^{\infty} \int_{-\pi}^\pi b_k \sin (k x) \cos (n x) d x \end{aligned}
Since $n$ and $k$ are integers, all of the integrals are zero except for the one involving $a_n$. These integrals are easily evaluated:
$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

Fourier and Cauchy were among the first to fully realize the inadequacy of defining integration as the inverse process of differentiation. It is too restrictive. Fourier wanted to apply his methods to arbitrary functions. Not all functions have antiderivatives that can be expressed in terms of standard functions. Fourier tried defining the definite integral of a nonnegative function as the area between the graph of the function and the $x$-axis, but that begs the question of what we mean by area. Cauchy embraced Leibniz’s understanding as a limit of products, and he found a way to avoid infinitesimals.

To define $\int_a^b f(x) d x$, Cauchy worked with finite approximating sums. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we consider
$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$
If we can force all of these approximating sums to be as close to each as other as we wish simply by limiting the size of the difference between consecutive values in the partition, then these summations have a limiting value that is designated as the value of the definite integral, and the function $f$ is said to be integrable over $[a, b]$.

Equipped with this definition, Cauchy succeeded in proving that any continuous or piecewise continuous function is integrable. The class of functions to which Fourier’s analysis could be applied was suddenly greatly expanded.

When Riemann turned to the study of trigonometric series, he wanted to know the limits of Cauchy’s approach to integration. Was there an easy test that could be used to determine whether or not a function could be integrated? Cauchy had chosen to evaluate the function at the left-hand endpoint of the interval simply for convenience. As Riemann thought about how far this definition could be pushed, he realized that his analysis would be simpler if the definition were stated in a slightly more complicated but essentially equivalent manner. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we assign a tag to each interval, a number $x_j^$ contained in that interval, and consider all sums of the form $$\sum_{k=1}^n f\left(x_k^\right)\left(x_k-x_{k-1}\right) \text {. }$$

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

$$a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \quad b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) .$$

$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x=\int_{-\pi}^\pi \frac{a_0}{2}$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$

## 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|表示论代写Representation theory代考|MAST90017

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation 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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Matrix Reduction Algorithm

There is a version of the algorithm PAIRPERSISTENCE that uses only matrix operations. First notice the following:

• The boundary operator $\partial_p: \mathbf{C}p \rightarrow \mathbf{C}{p-1}$ can be represented by a boundary matrix $D_p$ where the columns correspond to the $p$-simplices and rows correspond to $(p-1)$-simplices.
• It represents the transformation of a basis of $\mathrm{C}p$ given by the set of $p$-simplices to a basis of $C{p-1}$ given by the set of $(p-1)$-simplices:
$$D_p[i, j]= \begin{cases}1 & \text { if } \sigma_i \in \partial_p \sigma_j, \ 0 & \text { otherwise. }\end{cases}$$
• One can combine all boundary matrices into a single matrix $D$ that represents all linear maps $\bigoplus_p \partial_p-\bigoplus_p\left(\mathrm{C}p \rightarrow \mathrm{C}{p-1}\right)$, that is, transformation of a basis of all chain groups together to a basis of itself, but with a shift to a one lower dimension:
$$D[i, j]= \begin{cases}1 & \text { if } \sigma_i \in \partial_* \sigma_j, \ 0 & \text { otherwise. }\end{cases}$$
Definition 3.12. (Filtered boundary matrix) Let $\mathcal{F}: \varnothing=K_0 \hookrightarrow K_1 \hookrightarrow \cdots$ $\hookrightarrow K_n=K$ be a filtration induced by an ordering of simplices $\left(\sigma_1, \sigma_2, \ldots, \sigma_n\right)$ in $K$. Let $D$ denote the boundary matrix for simplices in $K$ that respects the ordering of the simplices in the filtration, that is, the simplex $\sigma_i$ in the filtration occupies column $i$ and row $i$ in $D$. We call $D$ the filtered boundary matrix for $\mathcal{F}$.

Given any matrix $A$, let row ${ }_A\lfloor i\rfloor$ and $\operatorname{col}_A\lfloor j\rfloor$ denote the $i$ th row and $j$ th column of $A$, respectively. We abuse notation slightly to let $\operatorname{col}_A[j]$ denote also the chain $\left{\sigma_i \mid A[i, j]=1\right}$, which is the collection of simplices corresponding to $1 \mathrm{~s}$ in the column $\operatorname{col}_A[j]$.

Definition 3.13. (Reduced matrix) Let $\operatorname{low}_A[j]$ denote the row index of the last 1 in the $j$ th column of $A$, which we call the low-row index of the column $j$. It is undefined for empty columns (marked with $-1$ in Algorithm 3). The matrix $A$ is reduced (or is in reduced form) if low $[j] \neq \operatorname{low}_A\left[j^{\prime}\right]$ for any $j \neq j^{\prime}$; that is, no two columns share the same low-row indices.

## 数学代写|表示论代写Representation theory代考|Efficient Implementation

The matrix reduction algorithm considers a column from left to right and reduces it by left-to-right additions. As we have observed, every addition to a column with index $j$ pushes $\operatorname{low}_D[j]$ upward. In the case that $\sigma_j$ is a positive simplex, the entire column is zeroed out. In general, positive simplices incur more cost than negative ones because $\operatorname{low}_D[\cdot]$ needs to be pushed all the way up for zeroing out the entire column. However, they do not participate in any future left-to-right column additions. Therefore, if it is known beforehand that the simplex $\sigma_j$ will be a positive simplex, then the costly step of zeroing out the column $j$ can be avoided.

Chen and Kerber [94] observed the following simple fact. If we process the input filtration backward in dimension, that is, process the boundary matrices $D_p, p=1, \ldots, d$, in decreasing order of dimensions, then a persistence pair $\left(\sigma^{p-1}, \sigma^p\right)$ is detected from $D_p$ before processing the column for $\sigma^{p-1}$ in $D_{p-1}$. Fortunately, we already know that $\sigma^{p-1}$ has to be a positive simplex because it cannot pair with a negative simplex $\sigma^p$ otherwise. So, we can simply ignore the column of $\sigma^{p-1}$ while processing $D_{p-1}$. We call it clearing out column $p-1$. In practice, this saves a considerable amount of computation in cases where a lot of positive simplices occur such as in Rips filtrations. Algorithm 4: ClearPersistence implements this idea.

We cannot take advantage of the clearing for the last dimension in the filtration. If $d$ is the highest dimension of the simplices in the input filtration, the matrix $D_d$ has to be processed for all columns because the pairings for the positive $d$-simplices are not available.

If the number of $d$-simplices is large compared to the number of simplices of lower dimensions, the incurred cost of processing their columns can still be high. For example, in a Rips filtration restricted to a certain dimension $d$, the number of $d$-simplices becomes usually much larger than the number of, say,

1-simplices. In those cases, the clearing can be more cost-effective if it can be applied forward.

In this respect, the following observation becomes helpful. Let $D_p^$ denote the anti-transpose of the matrix $D_p$, defined by the transpose of $D_p$ with the columns and rows being ordered in reverse. This means that if $D_p$ has row and column indices $1, \ldots, m$ and $1, \ldots, n$, respectively, then $D_p^(i, j)=D_p(n+$ $1-j, m+1-i)$. We call it the twisted matrix of $D_p$. Figure $3.13$ shows the twisted matrix $D^$ of the matrix $D$ in Figure $3.12$ where the rows and columns are marked with the indices of the original matrix. The following proposition guaranteés thăt wé cañ computê thê persistencee pairs in $D_P$ from the matrix $D_p^$

# 表示论代考

## 数学代写|表示论代写Representation theory代考|Matrix Reduction Algorithm

• 边界运算符 $\partial_p: \mathbf{C} p \rightarrow \mathbf{C} p-1$ 可以用边界矩阵表示 $D_p$ 其中列对应于 $p$-单纯形和行对应 $(p-1)$ 简单的
• 它代表了基础的转变 $\mathrm{C} p$ 由一组给出 $p$-单纯形的基础 $C p-1$ 由一组给出 $(p-1)$-简单的:
$$D_p[i, j]=\left{1 \quad \text { if } \sigma_i \in \partial_p \sigma_j, 0 \quad\right. \text { otherwise. }$$
• 可以将所有边界矩阵组合成一个矩阵 $D$ 表示所有线性映射 $\bigoplus_p \partial_p-\bigoplus_p(\mathrm{C} p \rightarrow \mathrm{C} p-1)$ ，也就 是说，将所有链组的基础一起转换为自身的基础，但转移到一个较低的维度:
$$D[i, j]=\left{1 \quad \text { if } \sigma_i \in \partial_* \sigma_j, 0 \quad\right. \text { otherwise. }$$
定义 3.12。 (过滤后的边界矩阵) 让 $\mathcal{F}: \varnothing=K_0 \hookrightarrow K_1 \hookrightarrow \cdots \hookrightarrow K_n=K$ 是由单纯形的排 序引起的过滤 $\left(\sigma_1, \sigma_2, \ldots, \sigma_n\right)$ 在 $K$. 让 $D$ 表示单纯形的边界矩阵 $K$ 尊重过滤中单纯形的顺序，即 单纯形 $\sigma_i$ 在过滤占列 $i$ 和行 $i$ 在 $D$. 我们称之为 $D$ 过滤后的边界矩阵 $\mathcal{F}$.
给定任何矩阵 $A$ ， 让行 $A\lfloor i\rfloor$ 和 $\operatorname{col}A\lfloor j\rfloor$ 表示 $i$ 行和 $j$ 第 列 $A$ ，分别。我们稍微滥用符号让 $\operatorname{col}_A[j]$ 也表示链 \eft{\sigma_i \mid A[i, j]=1\right } } \text { , 这是对应于 } 1 \text { s在专栏中 } \operatorname { c o l } { A } [ j ] \text { . }
定义 3.13。(简化矩阵) 让 $\operatorname{low}_A[j]$ 表示最后一个 1 的行索引 $j$ 第列 $A$ ，我们称之为列的低行索引 $j$. 它对 于空列是末定义的 (标有 $-1$ 在算法 3) 中。矩阵 $A$ 如果低，则减少 (或减少形式) $[j] \neq \operatorname{low}_A\left[j^{\prime}\right]$ 对于 任何 $j \neq j^{\prime}$ ；也就是说，没有两列共享相同的低行索引。

## 数学代写|表示论代写Representation theory代考|Efficient Implementation

Chen 和 Kerber [94] 观察到以下简单事实。如果我们对输入过滤进行维度逆向处理，即对边界矩阵进行 处理 $D_p, p=1, \ldots, d$ ，按维度降序排列，然后是持久性对 $\left(\sigma^{p-1}, \sigma^p\right)$ 从检测到 $D_p$ 在处理列之前 $\sigma^{p-1}$ 在 $D_{p-1}$. 幸运的是，我们已经知道 $\sigma^{p-1}$ 必须是正单纯形，因为它不能与负单纯形配对 $\sigma^p$ 除此以外。所 以，我们可以简单地忽略列 $\sigma^{p-1}$ 加工时 $D_{p-1}$. 我们称之为清除列 $p-1$. 实际上，在出现大量正单纯形的 情况下（例如在 Rips 过滤中），这可以节省大量计算。算法 4：ClearPersistence 实现了这个想法。

1-单纯形。在这些情况下，如果可以向前应用清算，则清算可能更具成本效益。

## 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|表示论代写Representation theory代考|MATH4314

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation 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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Stability of Persistence Diagrams

A persistence diagram $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$, as a set of points in the extended plane $\overline{\mathbb{R}}^2$, summarizes certain topological information of a simplicial complex (space) in relation to the function $f$ that induces the filtration $\mathcal{F}_f$. However, this is not useful in practice unless we can be certain that a slight change in $f$ does not change this diagram dramatically. In practice $f$ is seldom measured accurately, and if its persistence diagram can be approximated from a slightly perturbed version, it becomes useful. Fortunately, persistence diagrams are stable. To formulate this stability, we need a notion of distance between persistence diagrams.

Let $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$ be two persistence diagrams for two functions $f$ and $g$. We want to consider bijections between points from $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$. However, they may have different cardinality for off-diagonal points. Recall that persistence diagrams include the points on the diagonal $\Delta$ each with infinite multiplicity. This addition allows us to borrow points from the diagonal when necessary to define the bijections. Note that we are considering only filtrations of finite complexes which also make each homology group finite.

Definition 3.9. (Bottleneck distance) Let $\Pi=\left{\pi: \operatorname{Dgm}p\left(\mathcal{F}_f\right) \rightarrow\right.$ $\left.\operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right}$ denote the set of all bijections. Consider the distance between two points $x=\left(x_1, x_2\right)$ and $y=\left(y_1, y_2\right)$ in $L{\infty}$-norm $|x-y|_{\infty}=$ $\max \left{\left|x_1-x_2\right|,\left|y_1-y_2\right|\right}$ with the assumption that $\infty-\infty=0$. The bottleneck distance between the two diagrams (see Figure $3.10$ ) is
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf {\pi \in \Pi} \sup {x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)}|x-\pi(x)|{\infty} .$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

Let $A$ and $B$ be the nondiagonal points in two persistence diagrams $\operatorname{Dgm}_p\left(\mathcal{F}_f\right)$ and $\operatorname{Dgm}_p\left(\mathcal{F}_g\right)$, respectively. For a point $a \in A$, let $\bar{a}$ denote the nearest point of $a$ on the diagonal. Define $\bar{b}$ for every point $b \in B$ similarly. Let $\bar{A}={\bar{a}}$ and $\bar{B}={\bar{b}}$. Let $\tilde{A}=A \cup \bar{B}$ and $\tilde{B}=B \cup \bar{A}$. We want to bijectively match points in $\tilde{A}$ and $\tilde{B}$. Let $\Pi={\pi}$ denote such a matching. It follows from the definition that
$$\mathrm{d}b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\min {\pi \in \Pi} \sup {a \in \tilde{A}, \pi(a) \in \tilde{B}}|a-\pi(a)|{\infty} .$$
Then, the bottleneck distance we want to compute must be $L_{\infty}$ distance $\max \left{\left|x_a-x_b\right|,\left|y_a-y_b\right|\right}$ for two points $a \in \tilde{A}$ and $b \in \tilde{B}$. We do a binary search on all such possible $O\left(n^2\right)$ distances where $|\tilde{A}|=|\tilde{B}|=n$. Let $\delta_0, \delta_1, \ldots, \delta_{n^{\prime}}$ be the sorted sequence of these distances in a nondecreasing order.

Given a $\delta=\delta_i \geq 0$ where $i$ is the median of the index in the hinary search interval $[\ell, u]$, we construct a bipartite graph $G=(\tilde{A} \cup \tilde{B}, E)$ where an edge $e=(a, b){{a \in \tilde{A}, b \in \tilde{B}}}$ is in $E$ if and only if either both $a \in \bar{A}$ and $b \in \bar{B}$ (weight $(e)=0$ ) or $|a-b|{\infty} \leq \delta$ (weight $(e)=|a-b|_{\infty}$ ). A complete matching in $G$ is a set of $n$ edges so that every vertex in $\tilde{A}$ and $\tilde{B}$ is incident to exactly one edge in the set. To determine if $G$ has a complete matching, one can use an $O\left(n^{2.5}\right)$ algorithm of Hopcroft and Karp [198] for complete matching in a bipartite graph. However, exploiting the geometric embedding of the points in the persistence diagrams, we can apply an $O\left(n^{1.5}\right)$ time algorithm of Efrat et al. [154] for the purpose. If such an algorithm affirms that a complete matching exists, we do the following: If $\ell=u$ we output $\delta$, otherwise we set $u=i$ and repeat. If no matching exists, we set $\ell=i$ and repeat. Observe that matching has to exist for some value of $\delta$, in particular for $\delta_{n^{\prime}}$ and thus the binary search always succeeds. Algorithm 1: BoTTLENECK lays out the pseudocode for this matching. The algorithm runs in $O\left(n^{1.5} \log n\right)$ time accounting for the $O(\log n)$ probes for binary search each applying an $O\left(n^{1.5}\right)$ time matching algorithm. However, to achieve this complexity, we have to avoid sorting $n^{\prime}=O\left(n^2\right)$ values taking $O\left(n^2 \log n\right)$ time. Again, using the geometric embedding of the points, one can perform the binary probes without incurring the cost for sorting. For details and an efficient implementation of this algorithm, seee [209].

## 数学代写|表示论代写Representation theory代考|Stability of Persistence Diagrams

$$\mathrm{d} b\left(\operatorname{Dgm}_p\left(\mathcal{F}_f\right), \operatorname{Dgm}_p\left(\mathcal{F}_g\right)\right)=\inf \pi \in \Pi \sup x \in \operatorname{Dgm}_p\left(\mathcal{F}_f\right)|x-\pi(x)| \infty$$

## 数学代写|表示论代写Representation theory代考|Computing Bottleneck Distances

$(e)=0$ ) 要么 $|a-b| \infty \leq \delta$ (重量 $(e)=|a-b|{\infty}$ ). 一个完整的匹配 $G$ 是一组 $n$ 边使得每个顶点在 $\tilde{A}$ 和 $\tilde{B}$ 恰好与集合中的一条边相关。确定是否 $G$ 有一个完整的匹配，一个可以使用 $O\left(n^{2.5}\right)$ Hopcroft 和 Karp [198] 的算法用于二分图中的完全匹配。然而，利用持久性图中点的几何嵌入，我们可以应用 $O\left(n^{1.5}\right)$ Efrat 等人的时间算法。[154] 的目的。如果这样的算法确认存在完全匹配，我们将执行以下操 作: 如果 $\ell=u$ 我们输出 $\delta$ ，否则我们设置 $u=i$ 并重复。如果不存在匹配项，我们设置 $\ell=i$ 并重复。观 察到对于某些值必须存在匹配 $\delta$ ，特别是对于 $\delta{n^{\prime}}$ 因此二分查找总是成功的。算法 1: BOTTLENECK 列出 了此匹配的伪代码。该算法运行于 $O\left(n^{1.5} \log n\right)$ 时间占 $O(\log n)$ 用于二进制搜索的探针每个应用一个 $O\left(n^{1.5}\right)$ 时间匹配算法。然而，为了实现这种复杂性，我们必须避免排序 $n^{\prime}=O\left(n^2\right)$ 取值
$O\left(n^2 \log n\right)$ 时间。同样，使用点的几何嵌入，可以在不产生排序成本的情况下执行二元探测。有关此 算法的详细信息和有效实现，请参阅 [209]。

## 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|表示论代写Representation theory代考|MTH4107

statistics-lab™ 为您的留学生涯保驾护航 在代写表示论Representation 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。我们的专家在代写表示论Representation theory代写方面经验极为丰富，各种代写表示论Representation theory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|表示论代写Representation theory代考|Persistence

In both cases of space and simplicial filtration $\mathcal{F}$, we arrive at a homology module:
$$\mathrm{H}p \mathcal{F}: 0=\mathrm{H}_p\left(X_0\right) \rightarrow \mathrm{H}_p\left(X_1\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_i\right) \rightarrow \stackrel{h_p^{h_j}}{ } \rightarrow \mathrm{H}_p\left(X_j\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_n\right)=\mathrm{H}_p(X),$$ where $X_i=\mathbb{T}{a_i}$ if $\mathcal{F}$ is a space filtration of a topological space $X=\mathbb{T}$ or $X_i=K_i$ if $\mathcal{F}$ is a simplicial filtration of a simplicial complex $X=K$. Persistent homology groups for a homology module are algebraic structures capturing the survival of the homology classes through this sequence. In general, we will call homology modules persistence modules in Section $3.4$ recognizing that we can replace homology groups with vector spaces.

Definition 3.4. (Persistent Betti number) The $p$-th persistent homology groups are the images of the homomorphisms: $\mathrm{H}_p^{i, j}=\operatorname{im} h_p^{i, j}$, for $0 \leq i \leq$ $j \leq n$. The $p$-th persistent Betti numbers are the dimensions $\beta_p^{i, j}=\operatorname{dim} \mathrm{H}_p^{i, j}$ of the vector spaces $\mathrm{H}_p^{i, j}$.

The $p$-th persistent homology groups contain the important information of when a homology class is born or when it dies. The issue of birth and death of a class becomes more subtle because when a new class is born, many other classes that are the sum of this new class and any other existing class are also born. Similarly, when a class ceases to exist, many other classes may cease to exist along with it. Therefore, we need a mechanism to pair births and deaths canonically. Figure $3.7$ illustrates the birth and death of a class, though the pairing of birth and death events is more complicated as stated in Fact 3.3.
Observe that the nontrivial elements of $p$-th persistent homology groups $\mathrm{H}_p^{i, j}$ consist of classes that survive from $X_i$ to $X_j$, that is, the classes which do not get “quotiented out” by the boundaries in $X_j$. So, one can observe the following.

## 数学代写|表示论代写Representation theory代考|Persistence Diagram

Fact $3.3$ provides a qualitative characterization of the pairing of births and deaths of classes. Now we give a quantitative characterization which helps to draw a visual representation of this pairing called a persistence diagram; see Figure 3.8(a). Consider the extended plane $\overline{\mathbb{R}}^2:=(\mathbb{R} \cup{\pm \infty})^2$ on which we represent the birth at $a_i$ paired with the death at $a_j$ as a point $\left(a_i, a_j\right)$. This pairing uses a persistence pairing function $\mu_p^{i, j}$ defined below. Strictly positive values of this function correspond to multiplicities of points in the persistence diagram (Definition 3.8). In what follows, to account for classes that never die, we extend the induced module in Eq. (3.3) on the right end by assuming that $\mathrm{H}p\left(X{n+1}\right)=0$.
Definition 3.6. For $0<i<j \leq n+1$, define
$$\mu_p^{i, j}=\left(\beta_p^{i, j-1}-\beta_p^{i, j}\right)-\left(\beta_p^{i-1, j-1}-\beta_p^{i-1, j}\right) .$$
The first difference on the right-hand side counts the number of independent classes that are born at or before $X_i$ and die entering $X_j$. The second difference counts the number of independent classes that are born at or before $X_{i-1}$ and die entering $X_j$. The difference between the two differences thus counts the number of independent classes that are born at $X_i$ and die entering $X_j$. When $j-n+1, \mu_p^{i, n+1}$ counts the number of independent classes that are born at $X_i$ and die entering $X_{n+1}$. They remain alive till the end in the original filtration without extension, or we say that they never die. To emphasize that classes which exist in $X_n$ actually never die, we equate $n+1$ with $\infty$ and take $a_{n+1}=$ $a_{\infty}=\infty$. Observe that, with this assumption, we have $\beta^{i, n+1}=\beta^{i, \infty}=0$ for every $i \leq n$.

## 数学代写|表示论代写Representation theory代考|Persistence

$$\mathrm{H} p \mathcal{F}: 0=\mathrm{H}_p\left(X_0\right) \rightarrow \mathrm{H}_p\left(X_1\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_i\right) \rightarrow h_p^{h_j} \rightarrow \mathrm{H}_p\left(X_j\right) \rightarrow \cdots \rightarrow \mathrm{H}_p\left(X_n\right)=\mathrm{H}_p$$

## 数学代写|表示论代写Representation theory代考|Persistence Diagram

$$\mu_p^{i, j}=\left(\beta_p^{i, j-1}-\beta_p^{i, j}\right)-\left(\beta_p^{i-1, j-1}-\beta_p^{i-1, j}\right) .$$

## 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|MATH3061

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。

• 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|拓扑学代写Topology代考|Functions on Smooth Manifolds

In previous sections, we introduced topological spaces, including the special case of (smooth) manifolds. Very often, a space can be equipped with continuous functions defined on it. In this section, we focus on real-valued functions of the form $f: X \rightarrow \mathbb{R}$ defined on a topological space $X$, also called scalar functions; see Figure 1.8(a) for the graph of a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. Scalar functions appear commonly in practice that describe space/data of interest (e.g., the elevation function defined on the surface of the Earth). We are interested in the topological structures behind scalar functions. In this section, we limit our discussion to nicely behaved scalar functions (called Morse functions) defined on smooth manifolds. Their topological structures are characterized by the so-called critical points which we will introduce below. Later in the book we will also discuss scalar functions on simplicial complex domains, as well as more complex maps defined on a space $X$, for example, a multivariate function $f: X \rightarrow \mathbb{R}^d$

In what follows, for simplicity of presentation, we assume that we consider smooth ( $C^{\infty}$-continuous) functions and smooth manifolds embedded in $\mathbb{R}^d$, even though often we only require the functions (resp. manifolds) to be $C^2$ continuous (resp. $C^2$-smooth).

To provide intuition, let us start with a smooth scalar function defined on the real line, $f: \mathbb{R} \rightarrow \mathbb{R}$; the graph of such a function is shown in Figure 1.8(b). Recall that the derivative of a function at a point $x \in \mathbb{R}$ is defined as
$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t} .$$ The value $D f(x)$ gives the rate of change of the value of $f$ at $x$. This can be visualized as the slope of the tangent line of the graph of $f$ at $(x, f(x))$. The critical points of $f$ are the set of points $x$ such that $D f(x)=0$. For a function defined on the real line, there are two types of critical points in the generic case: maxima and minima, as marked in Figure 1.8(b).

## 数学代写|拓扑学代写Topology代考|Morse Functions and Morse Lemma

From the first-order derivatives of a function we can determine critical points. We can learn more about the “type” of the critical points by inspecting the second-order derivatives of $f$.

A critical point $x$ of $f$ is nondegenerate if its Hessian matrix, Hessian $(x)$, is nonsingular (has nonzero determinant); otherwise, it is a degenerate critical point.

For example, consider $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^3-3 x y^2$. The origin $(0,0)$ is a degenerate critical point often referred to as a “monkey saddle:” see Figure 1.9(d), where the graph of the function around $(0,0)$ goes up and down three times (instead of twice as for a nondegenerate saddle shown in Figure 1.9b). It turns out that, as a consequence of the Morse Lemma below, nondegenerate critical points are always isolated whereas the degenerate ones may not be so. A simple example is $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^2$, where all points on the $y$-axis are degenerate critical points. The local neighborhood of nondegenerate critical points can be completely characterized by the following Morse Lemma.

Proposition 1.2. (Morse Lemma) Given a smooth function $f: M \rightarrow \mathbb{R}$ defined on a smooth $m$-manifold $M$, let $p$ be a nondegenerate critical point of $f$. Then there is a local coordinate system in a neighborhood $U(p)$ of $p$ so that (i) the coordinate of $p$ is $(0,0, \ldots, 0)$, and (ii) locally for every point $x=\left(x_1, x_2, \ldots, x_m\right)$ in neighborhood $U(p)$,
$f(x)=f(p)-x_1^2-\cdots-x_s^2+x_{s+1}^2 \cdots+x_m^2, \quad$ for some $s \in[0, m]$.
The number s of minus signs in the above quadratic representation of $f(x)$ is called the index of the critical point $p$.

A critical point $x$ of $f$ is nondegenerate if its Hessian matrix, Hessian $(x)$, is nonsingular (has nonzero determinant); otherwise, it is a degenerate critical point.

For example, consider $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^3-3 x y^2$. The origin $(0,0)$ is a degenerate critical point often referred to as a “monkey saddle:” see Figure 1.9(d), where the graph of the function around $(0,0)$ goes up and down three times (instead of twice as for a nondegenerate saddle shown in Figure 1.9b). It turns out that, as a consequence of the Morse Lemma below, nondegenerate critical points are always isolated whereas the degenerate ones may not be so. A simple example is $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $f(x, y)=x^2$, where all points on the $y$-axis are degenerate critical points. The local neighborhood of nondegenerate critical points can be completely characterized by the following Morse Lemma.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Functions on Smooth Manifolds

$$D f(x)=\frac{d}{d x} f(x)=\lim _{t \rightarrow 0} \frac{f(x+t)-f(x)}{t} .$$

## 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|MTH3130

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。

• 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|拓扑学代写Topology代考|Maps, Homeomorphisms, and Homotopies

The equivalence of two topological spaces is determined by how the points that comprise them are connected. For example, the surface of a cube can be deformed into a sphere without cutting or gluing it because they are connected the same way. They have the same topology. This notion of topological equivalence can be formalized via functions that send the points of one space to points of the other while preserving the connectivity.

This preservation of connectivity is achieved by preserving the open sets. A function from one space to another that preserves the open sets is called a continuous function or a map. Continuity is a vehicle to define topological equivalence, because a continuous function can send many points to a single point in the target space, or send no points to a given point in the target space. If the former does not happen, that is, when the function is injective, we call it an embedding of the domain into the target space. True equivalence is given by a homeomorphism, a bijective function from one space to another which has continuity as well as a continuous inverse. This ensures that open sets are preserved in both directions.

Definition 1.15. (Continuous function; Map) A function $f: \mathbb{T} \rightarrow \mathbb{U}$ from the topological space $\mathbb{T}$ to another topological space $\mathbb{U}$ is continuous if for every open set $Q \subseteq \mathbb{U}, f^{-1}(Q)$ is open. Continuous functions are also called maps.
Definition 1.16. (Embedding) A map $g: \mathbb{T} \rightarrow \mathbb{U}$ is an embedding of $\mathbb{V}$ into $\mathbb{U}$ if $g$ is injective.

A topological space can be embedded into a Euclidean space by assigning coordinates to its points so that the assignment is continuous and injective. For example, drawing a triangle on paper is an embedding of $\mathbb{S}^1$ into $\mathbb{R}^2$. There are topological spaces that cannot be embedded into a Euclidean space, or even into a metric space – these spaces cannot be represented by any metric.

Next we define a homeomorphism that connects two spaces that have essentially the same topology.

Definition 1.17. (Homeomorphism) Let $\mathbb{T}$ and $\mathbb{U}$ be topological spaces. A homeomorphism is a bijective map $h: \mathbb{T} \rightarrow \mathbb{U}$ whose inverse is continuous too.

Two topological spaces are homeomorphic if there exists a homeomorphism between them.

## 数学代写|拓扑学代写Topology代考|Manifolds

A manifold is a topological space that is locally connected in a particular way. A 1-manifold has this local connectivity looking like a segment. A 2manifold (with boundary) has the local connectivity looking like a complete or partial disk. In layman’s terms, a 2-manifold has the structure of a piece of paper or rubber sheet, possibly with the houndaries glued together to form a closed surface – a category that includes disks, spheres, tori, and Möbius bands.

Definition 1.22. (Manifold) A topological space $M$ is an m-manifold, or simply a manifold, if every point $x \in M$ has a neighborhood homeomorphic to $\mathbb{B}_o^m$ or $\mathbb{H}^m$. The dimension of $M$ is $m$.

Every manifold can be partitioned into boundary and interior points. Observe that these words mean very different things for a manifold than they do for a metric space or topological space.

Definition 1.23. (Boundary; Interior) The interior Int $M$ of an $m$-manifold $M$ is the set of points in $M$ that have a neighborhood homeomorphic to $\mathbb{B}_o^m$. The boundary $\mathrm{Bd} M$ of $M$ is the set of points $M \backslash \operatorname{Int} M$. The boundary $\operatorname{Bd} M$, if not empty, consists of the points that have a neighborhood homeomorphic to $\mathbb{H}^m$. If $\mathrm{Bd} M$ is the empty set, we say that $M$ is without boundary.

A single point, a 0 -ball, is a 0 -manifold without boundary according to this definition. The closed disk $\mathbb{B}^2$ is a 2-manifold whose interior is the open disk $\mathbb{B}_o^2$ and whose boundary is the circle $\mathbb{S}^1$. The open disk $\mathbb{B}_o^2$ is a 2-manifold whose interior is $\mathbb{B}_o^2$ and whose boundary is the empty set. This highlights an important difference between Definitions $1.13$ and $1.23$ of “boundary”: when $\mathbb{B}_o^2$ is viewed as a point set in the space $\mathbb{R}^2$, its boundary is $\mathbb{S}^1$ according to Definition 1.13; but viewed as a manifold, its boundary is empty according to Definition 1.23. The boundary of a manifold is always included in the manifold.

The open disk $\mathbb{B}_o^2$, the Euclidean space $\mathbb{R}^2$, the sphere $\mathbb{S}^2$, and the torus are all connected 2-manifolds without boundary. The first two are homeomorphic to each other, but the last two are not. The sphere and the torus in $\mathbb{R}^3$ are compact (bounded and closed with respect to $\mathbb{R}^3$ ) whereas $\mathbb{B}_o^2$ and $\mathbb{R}^2$ are not.

A $d$-manifold, $d \geq 2$, can have orientations whose formal definition we skip here. Informally, we say that a 2-manifold $M$ is non-orientable if, starting from a point $p$, one can walk on one side of $M$ and end up on the opposite side of $M$ upon returning to $p$. Otherwise, $M$ is orientable. Spheres and balls are orientable, whereas the Möbius band in Figure 1.7(a) is a non-orientable 2-manifold with boundary.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Manifolds

1.13；但作为流形来看，根据定义 1.23，它的边界是空的。流形的边界总是包含在流形中。

## 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|MATH3402

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。

• 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|拓扑学代写Topology代考|Topological Space

The basic object in a topological space is a ground set whose elements are called points. A topology on these points specifies how they are connected by listing what points constitute a neighborhood – the so-called open set.

The expression “rubber-sheet topology” commonly associated with the term “topology” exemplifies this idea of connectivity of neighborhoods. If we bend and stretch a sheet of rubber, it changes shape but always preserves the neighborhoods in terms of the points and how they are connected.

We first introduce basic notions from point set topology. These notions are prerequisites for more sophisticated topological ideas – manifolds, homeomorphism, isotopy, and other maps – used later to study algorithms for topological data analysis. Homeomorphisms, for example, offer a rigorous way to state that an operation preserves the topology of a domain, and isotopy offers a rigorous way to state that the domain can be deformed intoo aa shape without ever colliding with itself.

Perhaps it is more intuitive to understand the concept of topology in the presence of a metric because then we can use the metric balls such as Euclidean balls in a Euclidean space to define neighborhoods – the open sets. Topological spaces provide a way to abstract out this idea without a metric or point coordinates, so they are more general than metric spaces. In place of a metric, we encode the connectivity of a point set by supplying a list of all of the open sets. This list is called a system of subsets of the point set. The point set and its system together describe a topological space.

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

Metric spaces are a special type of topological space commonly encountered in practice. Such a space admits a metric that specifies the scalar distance between every pair of points satisfying certain axioms.

Definition 1.8. (Metric space) A metric space is a pair ( $\mathbb{T}, d)$ where $\mathbb{T}$ is a set and $d$ is a distance function $d: \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ satisfying the following properties:

• $\mathrm{d}(p, q)=0$ if and only if $p=q$ for all $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ for all $p, q \in \mathbb{\pi}$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ for all $p, q, r \in \mathbb{T}$.
It can be shown that the three axioms above imply that $\mathrm{d}(p, q) \geq 0$ for every pair $p, q \in \mathbb{T}$. In a metric space $\mathbb{T}$, an open metric ball with center $c$ and radius $r$ is defined to be the point set $B_o(c, r)={p \in \mathbb{T}: \mathrm{d}(p, c)<r}$. Metric balls define a topology on a metric space.

Definition 1.9. (Metric space topology) Given a metric space $\mathbb{T}$, all metric balls $\left{B_o(c, r) \mid c \in \mathbb{T}\right.$ and $\left.0<r \leq \infty\right}$ and their union constituting the open sets define a topology on $\mathbb{T}$.

All definitions for general topological spaces apply to metric spaces with the above defined topology. However, we give alternative definitions using the concept of limit points which may be more intuitive.

As we have mentioned already, the heart of topology is the question of what it means for a set of points to be connected. After all, two distinct points cannot be adjacent to each other; they can only be connected to one another by passing through uncountably many intermediate points. The idea of limit points helps express this concept more concretely, specifically in the case of metric spaces. We use the notation $\mathrm{d}(\cdot, \cdot)$ to express minimum distances between point sets $P, Q \subseteq \mathbb{T}:$
\begin{aligned} & \mathrm{d}(p, Q)=\inf {\mathrm{d}(p, q): q \in Q} \ & \mathrm{d}(P, Q)=\inf {\mathrm{d}(p, q): p \in P, q \in Q} \end{aligned}

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Metric Space Topology

• $\mathrm{d}(p, q)=0$ 当且仅当 $p=q$ 对所有人 $p \in \mathbb{T}$;
• $\mathrm{d}(p, q)=\mathrm{d}(q, p)$ 对所有人 $p, q \in \pi$;
• $\mathrm{d}(p, q) \leq \mathrm{d}(p, r)+\mathrm{d}(r, q)$ 对所有人 $p, q, r \in \mathbb{T}$.
可以证明，上面的三个公理意味着 $\mathrm{d}(p, q) \geq 0$ 每对 $p, q \in \mathbb{T}$. 在度量空间 $\mathbb{T}$, 一个中心为空心的公 制球 $c$ 和半径 $r$ 被定义为点集 $B_o(c, r)=p \in \mathbb{T}: \mathrm{d}(p, c)<r$. 度量球定义度量空间上的拓扑。
定义 1.9。 (度量空间拓扑) 给定一个度量空间 $\mathbb{T}$ ，所有公制球 个拓扑T⿺丄⺊.
一般拓扑空间的所有定义都适用于具有上述定义拓扑的度量空间。但是，我们使用可能更直观的极限点 概念给出替代定义。
正如我们已经提到的，拓扑的核心问题是连接一组点意味着什么。毕竟，两个不同的点不能彼此相邻； 它们只能通过无数个中间点才能相互连接。极限点的概念有助于更具体地表达这个概念，特别是在度量 空间的情况下。我们使用符号 $\mathrm{d}(\cdot, \cdot)$ 表达点集之间的最小距离 $P, Q \subseteq \mathbb{T}$ :
$$\mathrm{d}(p, Q)=\inf \mathrm{d}(p, q): q \in Q \quad \mathrm{~d}(P, Q)=\inf \mathrm{d}(p, q): p \in P, q \in Q$$

## 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。

## 计算机代写|机器学习代写machine learning代考|COMP4702

statistics-lab™ 为您的留学生涯保驾护航 在代写机器学习 machine learning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。我们的专家在代写机器学习 machine learning代写方面经验极为丰富，各种代写机器学习 machine learning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

## 计算机代写|机器学习代写machine learning代考|Distance and Inner-Product Random Kernel Matrices

The most widely used kernel model in machine learning applications is the heat kernel $\mathbf{K}=\left{\exp \left(-\left|\mathbf{x}i-\mathbf{x}_j\right|^2 / 2 \sigma^2\right)\right}{i, j=1}^n$, for some $\sigma>0$. It is thus natural to start the large-dimensional analysis of kernel random matrices by focusing on this model.
As mentioned in the previous sections, for the Gaussian mixture model above, as the dimension $p$ increases, $\sigma^2$ needs to scale as $O(p)$, so say $\sigma^2=\tilde{\sigma}^2 p$ for some $\tilde{\sigma}^2=O(1)$, to avoid evaluating the exponential at increasingly large values for $p$ large. As such, the prototypical kernel of present interest is
$$\mathbf{K}=\left{f\left(\frac{1}{p}\left|\mathbf{x}i-\mathbf{x}_j\right|^2\right)\right}{i, j-1}^n,$$
for $f$ a sufficiently smooth function (specifically, $f(t)=\exp \left(-t / 2 \tilde{\sigma}^2\right)$ for the heat kernel). As we will see though, it is much desirable not to restrict ourselves to $f(t)=\exp \left(-t / 2 \tilde{\sigma}^2\right)$ so to better appreciate the impact of the nonlinear kernel function $f$ on the (asymptotic) structural behavior of the kernel matrix $\mathbf{K}$.

## 计算机代写|机器学习代写machine learning代考|Euclidean Random Matrices with Equal Covariances

In order to get a first picture of the large-dimensional behavior of $\mathbf{K}$, let us first develop the distance $\left|\mathbf{x}_i-\mathbf{x}_j\right|^2 / p$ for $\mathbf{x}_i \in \mathcal{C}_a$ and $\mathbf{x}_j \in \mathcal{C}_b$, with $i \neq j$.

For simplicity, let us assume for the moment $\mathbf{C}_1=\cdots=\mathbf{C}_k=\mathbf{I}_p$ and recall the notation $\mathbf{x}_i=\boldsymbol{\mu}_a+\mathbf{z}_i$. We have, for $i \neq j$ that “entry-wise,”
\begin{aligned} \frac{1}{p}\left|\mathbf{x}_i-\mathbf{x}_j\right|^2= & \frac{1}{p}\left|\boldsymbol{\mu}_a-\boldsymbol{\mu}_b\right|^2+\frac{2}{p}\left(\boldsymbol{\mu}_a-\boldsymbol{\mu}_b\right)^{\top}\left(\mathbf{z}_i-\mathbf{z}_j\right) \ & +\frac{1}{p}\left|\mathbf{z}_i\right|^2+\frac{1}{p}\left|\mathbf{z}_j\right|^2-\frac{2}{p} \mathbf{z}_i^{\top} \mathbf{z}_j . \end{aligned}
For $\left|\mathbf{x}_i\right|$ of order $O(\sqrt{p})$, if $\left|\mu_a\right|=O(\sqrt{p})$ for all $a \in{1, \ldots, k}$ (which would be natural), then $\left|\mu_a-\mu_b\right|^2 / p$ is a priori of order $O(1)$ while, by the central limit theorem, $\left|\mathbf{z}_i\right|^2 / p=1+O\left(p^{-1 / 2}\right)$. Also, again by the central limit theorem, $\mathbf{z}_i^{\top} \mathbf{z}_j / p=$ $O\left(p^{-1 / 2}\right)$ and $\left(\mu_a-\mu_b\right)^{\top}\left(\mathbf{z}_i-\mathbf{z}_j\right) / p=O\left(p^{-1 / 2}\right)$

As a consequence, for $p$ large, the distance $\left|\mathbf{x}i-\mathbf{x}_j\right|^2 / p$ is dominated by $| \boldsymbol{\mu}_a-$ $\boldsymbol{\mu}_b |^2 / p+2$ and easily discriminates classes from the pairwise observations of $\mathbf{x}_i, \mathbf{x}_j$, making the classification asymptotically trivial (without having to resort to any kernel method). It is thus of interest consider the situations where the class distances are less significant to understand how the choices of kernel come into play in such more practical scenario. To this end, we now demand that $$\left|\mu_a-\mu_b\right|=O(1),$$ which is also the minimal distance rate that can be discriminated from a mere Bayesian inference analysis, as thoroughly discussed in Section 1.1.3. Since the kernel function $f(\cdot)$ operates only on the distances $\left|\mathbf{x}_i-\mathbf{x}_j\right|$, we may even request (up to centering all data by, say, the constant vector $\frac{1}{n} \sum{a=1}^k n_a \mu_a$ ) for simplicity that $\left|\mu_a\right|=O(1)$ for each $a$.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Euclidean Random Matrices with Equal Covariances

$$\frac{1}{p}\left|\mathbf{x}_i-\mathbf{x}_j\right|^2=\frac{1}{p}\left|\boldsymbol{\mu}_a-\boldsymbol{\mu}_b\right|^2+\frac{2}{p}\left(\boldsymbol{\mu}_a-\boldsymbol{\mu}_b\right)^{\top}\left(\mathbf{z}_i-\mathbf{z}_j\right) \quad+\frac{1}{p}\left|\mathbf{z}_i\right|^2+\frac{1}{p}\left|\mathbf{z}_j\right|^2-\frac{2}{p} \mathbf{z}_i^{\top} \mathbf{z}_j$$

$$\left|\mu_a-\mu_b\right|=O(1)$$

## 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。