分类：拓扑学代写

澳洲代写｜BFW3540 ｜Modelling in finance金融建模蒙纳士大学

Posted on 2023年10月16日2024年1月7日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Modelling in finance金融建模澳洲代写代考和辅导服务！

课程介绍：

Topics include the development and application of financial spreadsheets, Excel and Visual Basic programming in financial modelling, modelling company financial statements, fixed income securities analysis, asset allocation and portfolio analysis, optimization using Solver, Interest rate models, option pricing models, numerical methods and risk management models.

澳洲代写｜BFW3540 ｜Modelling in finance金融建模蒙纳士大学

Fourier Transform傅立叶变换案例

In this section, we recall the Fourier transform definition, both for notational reasons and for the reader’s convenience.
The Fourier transform, for $f \in \mathcal{S}\left(\mathbb{R}^m\right)$, is denoted here as
$$
\widehat{f}(\xi):=\mathcal{F}f:=\int_{\mathbb{R}^m} \mathrm{e}^{i \xi x} f(x) \mathrm{d} x,
$$
where $\mathcal{S}\left(\mathbb{R}^m\right)$ is the Schwartz space of $\mathcal{C}^{\infty}\left(\mathbb{R}^m\right)$ functions of rapid decrease, see [RS75]. This is not the usual definition found in the mathematical literature. However, it is standard in probability, see [Chu01] and in the finance literature, see [CT04].

The Fourier transform is a linear bijection from $\mathcal{S}\left(\mathbb{R}^m\right)$ onto $\mathcal{S}\left(\mathbb{R}^m\right)$, whose inverse is given by the Fourier inversion formula

\begin{equation}
f(x)=\mathcal{F}^{-1}\widehat{f}=\frac{1}{(2 \pi)^m} \int_{\mathbb{R}^m} \mathrm{e}^{-i \xi x} \widehat{f}(\xi) \mathrm{d} \xi
\end{equation}

We also recall the Fourier transform for $f \in \mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$, where $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ is the space of tempered distributions, which is the dual of $\mathcal{S}\left(\mathbb{R}^m\right)$, the Fourier transform can be defined as
$$
(\mathcal{F}[f], \varphi)=(2 \pi)^m\left(f, \mathcal{F}^{-1}[\varphi]\right) \quad \varphi \in \mathcal{S}\left(\mathbb{R}^m\right),
$$
see [RR04]. This definition makes the Fourier transform in $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$ an extension of the Fourier transform in $\mathcal{S}\left(\mathbb{R}^m\right)$. The Fourier transform for $L^1\left(\mathbb{R}^m\right)$ and $L^2\left(\mathbb{R}^m\right)$ are restrictions of the Fourier transform for $\mathcal{S}^{\prime}\left(\mathbb{R}^m\right)$.

The Fourier transform has several useful properties. Some of them are reviewed below with the purpose of calling attention to the notation used here:

$\mathcal{F}f(x-a)=\mathrm{e}^{i a \xi} \widehat{f}(\xi)$
$D^\alpha \widehat{f}(\xi)=\mathcal{F}\left(i x)^\alpha f\right$
$(-i \xi)^\alpha \widehat{f}(\xi)=\mathcal{F}\leftD^\alpha f\right$
Some specific distributions are often used in this thesis. To present the notation, we give a brief overview of them. First, consider the Cauchy principal value
$$
\begin{aligned}
1 / x: \mathcal{S}(\mathbb{R}) & \rightarrow \mathbb{R} \
f & \rightarrow(1 / x, f):=f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x,
\end{aligned}
$$
where
$$
f_{-\infty}^{\infty} \frac{f(x)}{x} \mathrm{~d} x:=\lim {\epsilon \downarrow 0}\left(\int\epsilon^{\infty} \frac{f(x)}{x} \mathrm{~d} x+\int_{-\infty}^{-\epsilon} \frac{f(x)}{x} \mathrm{~d} x\right) .
$$
This defines a distribution in $\mathcal{S}^{\prime}(\mathbb{R})$.

Probability and Stochastic Processes概率与随机过程案例

In this section, we present a brief overview of the topics on probability and stochastic processes used herein. References on the subject are [CW90] and [Sat99].

In this thesis, the triple $(\Omega, \mathcal{F}, \mathbb{P})$ denotes a complete probability space, where $\Omega$ is a set of points $\omega, \mathcal{F}$ is a $\sigma$-algebra containing all $\mathbb{P}$-null sets, and $\mathbb{P}$ is a probability measure. When we say that $X$ is a random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, we mean that $X$ is real-valued function on $\Omega$, measurable with respect to $\mathcal{F}$.
The characteristic function of a random variable is defined as
$$
\varphi(z)=\mathbb{E}\left[\mathrm{e}^{i z X}\right] .
$$
For properties of the characteristic function and a review of probability theory we refer to [CW90].

The filtered complete probability space is denoted by $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$, where, as in [Pro04], we write $\mathbb{F}$ for the filtration $\left(\mathcal{F}t\right){0 \leq t \leq \infty}$ and we assume that $\mathcal{F}_0$ contains all the $\mathbb{P}$-null sets. We use $\stackrel{\mathbb{P}}{\longrightarrow}$ to denote convergence in probability, see [CW90] and the French acronyms càdlàg (continu à droite, limité à gauche) is used to define the right continuous, left limited process, see [Pro04].

The main class of stochastic processes we are interested in this work are the Levy processes, see [Sat99] for a comprehensive treatment of the subject. We briefly review the definition of a Levy process
Definition: A Levy process is a càdlàg stochastic process, $\left(X_t\right)_{t \geq 0}$, on $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ taking values in $\mathbb{R}$ and with the following properties:

Independent increments. That is, given $t_0 \leq \ldots \leq t_N$, and defined $Y_n:=X_{t_n}-X_{t_{n-1}}$ we have $\left{Y_n\right}_{n=1}^N$ independent;
Stationary increments. That is, the distribution of $X_{t+s}-X_t$ does not depend on $t$;
Stochastic continuity. That is,
$$
X_{t+h} \underset{h \downarrow 0}{\stackrel{\mathbb{P}}{\longrightarrow}} X_t
$$
An important stochastic process is the Brownian motion.

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

澳洲代写｜MTH3130 ｜Topology: The mathematics of shape拓扑学形状数学蒙纳士大学

Posted on 2023年10月16日2023年10月16日 by statistics-lab

statistics-lab^TM为您提供蒙纳士大学（Monash University）Topology: The mathematics of shape拓扑学形状数学澳洲代写代考和辅导服务！

课程介绍：

From point-set topology to manifolds: sets, topological spaces, basis of topology, and properties of spaces such as compact, connected, and Hausdorff. Maps between spaces and their properties, including continuity, homeomorphism, and homotopy.

Constructing spaces via subspace, product, identification, and cell complexes. Manifolds. Additional topics from algebraic and low-dimensional topology may include fundamental group and Seifert-van Kampen theorem, classification of surfaces, and topics in knot theory. Throughout, examples of spaces will include Euclidean spaces, surfaces (real projective plane, Klein bottle, Mobius strip), complexes, function spaces, and others.

澳洲代写｜MTH3320｜Computational linear algebra计算线性代数蒙纳士大学

Topology拓扑学问题集

问题 1.

Show that if $f: X \rightarrow Y$ induces an isomorphism in homology with coefficients in the prime fields $\mathbb{F}_p$ (for all primes $p$ ) and $\mathbb{Q}$, then it induces an isomorphism in homology with coefficients in $\mathbb{Z}$. (Hint: 18 (c).)

问题 2.

Let $A \subseteq X$ and $B \subseteq Y$ be subsets. Construct a natural chain map
$$
S_(X, A) \otimes S_(Y, B) \rightarrow S_(X \times Y, A \times Y \cup X \times B) $$ that is a homology isomorphism if $A$ and $B$ are open. (Hint: Problem 26., or its proof, might be useful.) So there is a natural “relative cross product” map $$ H_(X, A ; R) \otimes_R H_(Y, B ; R) \rightarrow H_(X \times Y, A \times Y \cup X \times B ; R)
$$
that is an isomorphism if $A$ and $B$ are open, $R$ is a PID, and either $H_(X, A ; R)$ or $H_(Y, B ; R)$ is free over $R$.

问题 3.

(a) What is the $k$ th Betti number of $\left(S^1\right)^n$ ?
(b) Define an equivalence relation on $\mathbb{R}^n$ by saying that two vectors are equivalent if they differ by a vector with entries in $\mathbb{Z}$. Identify the quotient space of $\mathbb{R}^n$ by this equivalence relation with the product space $\left(S^1\right)^n$. Let $M$ be an $n \times n$ matrix with entries in $\mathbb{Z}$. It defines a linear map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ in the usual way. Show that this map descends to a self-map of $\left(S^1\right)^n$. Compute the effect of this map on $H_n\left(\left(S^1\right)^n\right)$.

澳洲代写｜ETC3250｜Introduction to machine learning机器学习入门蒙纳士大学

Attribute	Detail
Course Code	ECC2610
Course Title	Game theory and strategic thinking
Coordinating Unit	Introductory microeconomics
Semester	Second semester
Mode	On-campus
Delivery Location	Clayton
Number of Units	Not provided in the text
Pre-Requisites	ECB1101, ECC1000, ECF1100, ECS1101, ECW1101
Lecturers	Associate Professor Paola Labrecciosa

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|MATH421

Posted on 2023年8月24日2023年8月24日 by statistics-lab

如果你也在怎样代写拓扑学Topology 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。拓扑学Topology很早就成为一般数学语言的一部分。它教导一个人清楚而准确地谈论与连续性概念有关的事情。它的存在不仅是为了解释克莱因瓶到底是什么。这也是一种将几何图像引入任何数学领域的方法，无论该领域乍一看与几何距离有多远。

拓扑学Topology作为一个活跃的研究领域，已基本完成。作为一种通用的数学语言，它的长期使用已经完善了它的定义和定理体系。如今，研究一般拓扑的确更像是学习一门语言，而不是学习数学:人们必须学习许多新单词，而大多数定理的证明却极其简单。但是定理的数量是巨大的。这并不奇怪，因为它们扮演着规范词汇使用的规则角色。

拓扑学Topology代写，免费提交作业要求，满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕博写手团队，所有订单可靠准时，保证 100% 原创。最高质量的拓扑学Topology作业代写，服务覆盖北美、欧洲、澳洲等国家。在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。由于作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此拓扑学Topology作业代写的价格不固定。通常在各个科目专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

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

数学代写|拓扑学代写Topology代考|Continuity and Convergence of Nets

One can, of course, define convergence for sequences in topological spaces as for metric spaces.

Definition 3.2.1. Let $(X, \mathcal{T})$ be a topological space. A sequence $\left(x_n\right)_{n=1}^{\infty}$ in $X$ is said to converge to $x \in X$ if, for each $N \in \mathcal{N}_x$, there is $n_N \in \mathbb{N}$ such that $x_n \in N$ for all $n \geq n_N$.

This definition is perfectly fine, but if one attempts to prove analogues of results for convergent sequences in metric spaces, problems show up. Proposition 2.3.4, for example, is no longer true in general topological spaces.

Example 3.2.2. Let $X$ be an uncountable set equipped with the topology of Example 3.1.2(e); that is, the open sets are $\varnothing$ and those with a countable complement. Fix a point $x_0 \in X$. Then $X \backslash\left{x_0\right}$ is not closed, so that $\overline{X \backslash\left{x_0\right}}=X$ must hold. Let $\left(x_n\right){n=1}^{\infty}$ be a sequence in $X \backslash\left{x_0\right}$, and let $U:=X \backslash\left{x_1, x_2, \ldots\right}$. Due to the nature of our topology, $U$ is open and thus is a neighborhood of $x_0$. However, $x_n \notin U$ for all $n \in \mathbb{N}$ by definition, so that $\left(x_n\right){n=1}^{\infty}$ cannot converge to $x_0$.

A less contrived example for the failure of Proposition 2.3.4 in general topological spaces is given in Exercise 11 below.

So, how are we going to define continuity on arbitrary topological spaces? Of course, we could try it via sequences as for metric spaces, but in view of Example 3.2.2, we are likely to run into unexpected difficulties. Of the four equivalent conditions of Theorem 2.3.7, the fourth one doesn’t make any explicit reference to a metric. We thus use it as the definition of continuity.

数学代写|拓扑学代写Topology代考|Compactness

Compactness for topological spaces is defined as in the metric situation.
Definition 3.3.1. Let $(X, \mathcal{T})$ be a topological space, and let $S \subset X$. An open cover for $S$ is a collection $\mathcal{U}$ of open subsets of $X$ such that $S \subset \bigcup{U: U \in \mathcal{U}}$.
Definition 3.3.2. A subset $K$ of a topological space $(X, \mathcal{T})$ is called compact if, for each open cover $\mathcal{U}$ of $K$, there are $U_1, \ldots, U_n \in \mathcal{U}$ such that $K \subset$ $U_1 \cup \cdots \cup U_n$.

Before we flesh out this definition with examples (nonmetrizable ones), we introduce yet another definition.

Definition 3.3.3. A topological space $(X, \mathcal{T})$ has the finite intersection property if, for any collection $\mathcal{F}$ of closed subsets of $X$ such that $\bigcap{F: F \in \mathcal{F}}=$ $\varnothing$, there are $F_1, \ldots, F_n \in \mathcal{F}$ such that $F_1 \cap \cdots \cap F_n=\varnothing$.
The following is straightforward (just pass to complements).
Proposition 3.3.4. Let $(X, \mathcal{T})$ be a topological space. Then the following are equivalent.
(i) $X$ is compact.
(ii) $X$ has the finite intersection property.
The reason why we introduced the finite intersection property at all is that it is sometimes easier to verify than compactness.

Example 3.3.5. Let $R$ be a commutative ring with identity. We claim that $\operatorname{Spec}(R)$ has the finite intersection property (and thus is compact). Let $\mathcal{I}$ be a family of ideals of $R$ such that
$$
\bigcap{V(I): I \in \mathcal{I}}=V\left(\sum{I: I \in \mathcal{I}}\right)=\varnothing .
$$

拓扑学代考

数学代写|拓扑学代写Topology代考|Continuity and Convergence of Nets

当然，我们可以像定义度量空间一样，定义拓扑空间中序列的收敛性。

3.2.1.定义设$(X, \mathcal{T})$为拓扑空间。如果对于每一个$N \in \mathcal{N}x$，都有一个$n_N \in \mathbb{N}$使得对于所有$n \geq n_N$，都有一个$x_n \in N$，那么我们就说$X$中的一个序列$\left(x_n\right){n=1}^{\infty}$收敛到$x \in X$。

这个定义是完美的，但是如果有人试图证明度量空间中收敛序列的类似结果，问题就出现了。例如，命题2.3.4在一般拓扑空间中不再成立。

例3.2.2设$X$是一个不可数集合，具有例3.1.2(e)的拓扑结构;也就是说，开集是$\varnothing$和具有可数补集的开集。固定一个点$x_0 \in X$。那么$X \backslash\left{x_0\right}$是不闭合的，那么$\overline{X \backslash\left{x_0\right}}=X$一定要hold住。设$\left(x_n\right){n=1}^{\infty}$是$X \backslash\left{x_0\right}$中的一个序列，设$U:=X \backslash\left{x_1, x_2, \ldots\right}$。由于我们拓扑的性质，$U$是开放的，因此是$x_0$的邻域。但是，$x_n \notin U$对于所有$n \in \mathbb{N}$的定义，使得$\left(x_n\right){n=1}^{\infty}$不能收敛到$x_0$。

下面的练习11给出了命题2.3.4在一般拓扑空间中失败的一个不那么人为的例子。

那么，我们如何定义任意拓扑空间上的连续性呢?当然，我们也可以像度量空间那样通过序列来尝试，但是考虑到例3.2.2，我们很可能会遇到意想不到的困难。在定理2.3.7的四个等价条件中，第四个条件没有明确提到度规。因此我们用它作为连续性的定义。

数学代写|拓扑学代写Topology代考|Compactness

拓扑空间的紧性定义为度量情况下的紧性。
3.3.1.定义设$(X, \mathcal{T})$为拓扑空间，设$S \subset X$。$S$的开放覆盖是$X$的开放子集的集合$\mathcal{U}$，例如$S \subset \bigcup{U: U \in \mathcal{U}}$。
3.3.2.定义拓扑空间$(X, \mathcal{T})$的子集$K$被称为紧化，如果对于$K$的每个开盖$\mathcal{U}$，存在$U_1, \ldots, U_n \in \mathcal{U}$使得$K \subset$$U_1 \cup \cdots \cup U_n$。

在用示例(不可度量的示例)充实这个定义之前，我们先介绍另一个定义。

3.3.3.定义拓扑空间$(X, \mathcal{T})$具有有限交性质，如果对于$X$的闭子集的任何集合$\mathcal{F}$使得$\bigcap{F: F \in \mathcal{F}}=$$\varnothing$，存在$F_1, \ldots, F_n \in \mathcal{F}$使得$F_1 \cap \cdots \cap F_n=\varnothing$。
下面的代码很简单(只需传递给补语)。
提案3.3.4。设$(X, \mathcal{T})$为拓扑空间。那么下面是等价的。
(i) $X$是紧凑的。
(ii) $X$具有有限交性质。
我们引入有限交性质的原因是它有时比紧性更容易验证。

例3.3.5。设$R$是一个具有恒等的交换环。我们声明$\operatorname{Spec}(R)$具有有限交性质(因此是紧的)。让$\mathcal{I}$成为$R$的理想之家，这样
$$
\bigcap{V(I): I \in \mathcal{I}}=V\left(\sum{I: I \in \mathcal{I}}\right)=\varnothing .
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|MATH622

Posted on 2023年8月24日2023年8月24日 by statistics-lab

数学代写|拓扑学代写Topology代考|Compactness for Metric Spaces

The notion of compactness is one of the most crucial in all of topology (and one of the hardest to grasp).

Definition 2.5.1. Let $(X, d)$ be a metric space, and let $S \subset X$. An open cover for $S$ is a collection $\mathcal{U}$ of open subsets of $X$ such that $S \subset \cup{U: U \in \mathcal{U}}$.
Definition 2.5.2. A subset $K$ of a metric space $(X, d)$ is called compact if, for each open cover $\mathcal{U}$ of $K$, there are $U_1, \ldots, U_n \in \mathcal{U}$ such that $K \subset U_1 \cup$ $\cdots \cup U_n$.

Definition 2.5.2 is often worded as, “A set is compact if and only if each open cover has a finite subcover.”

Examples 2.5.3. (a) Let $(X, d)$ be a metric space, and let $S \subset X$ be finite; that is, $S=\left{x_1, \ldots, x_n\right}$. Let $\mathcal{U}$ be an open cover of $X$. Then, for each $j=1, \ldots, n$, there is $U_j \in \mathcal{U}$ such that $x_j \in U_j$. It follows that $S \subset$ $U_1 \cup \cdots \cup U_n$. Hence, $S$ is compact.
(b) Let $(X, d)$ be a compact metric space, and let $\varnothing \neq K \subset X$ be compact. Fix $x_0 \in K$. Since $\left{B_r\left(x_0\right): r>0\right}$ is an open cover of $K$, there are $r_1, \ldots, r_n>0$ such that
$$
K \subset B_{r_1}\left(x_0\right) \cup \cdots \cup B_{r_n}\left(x_0\right) .
$$
With $R:=\max \left{r_1, \ldots, r_n\right}$, we see that $K \subset B_R\left(x_0\right)$, so that $\operatorname{diam}(K) \leq$ $2 R<\infty$. This means, for example, that any unbounded subset of $\mathbb{R}^n$ (or, more generally, of any normed space) cannot be compact. In particular, the only compact normed space is ${0}$.
(c) Let $X=(0,1)$ be equipped with the usual metric. For $r \in(0,1)$, let $U_r:=(r, 1)$. Then $\left{U_r: r \in(0,1)\right}$ is an open cover for $(0,1)$ which has no finite subcover.

Before we turn to more (and more interesting) examples of compact metric spaces, we establish a few hereditary properties.

数学代写|拓扑学代写Topology代考|Topological Spaces-Definitions and Examples

A topological space is supposed to be a set that has just enough structure to meaningfully speak of continuous functions on it. In view of Corollary 2.3.10, a reasonable approach would be to axiomatize the notion of an open set:
Definition 3.1.1. Let $X$ be a set. $A$ topology on $X$ is a subset $\mathcal{T}$ of $\mathfrak{P}(X)$ such that:
(a) $\varnothing, X \in \mathcal{T}$;
(b) If $\mathcal{U} \subset \mathcal{T}$ is arbitrary, then $\bigcup{U: U \in \mathcal{U}}$ lies in $\mathcal{T}$;
(c) If $U_1, U_2 \in \mathcal{T}$, then $U_1 \cap U_2 \in \mathcal{T}$.
The sets in $\mathcal{T}$ are called open. A set together with a topology is called a topological space.

We often write $(X, \mathcal{T})$ for a topological space $X$ with topology $\mathcal{T}$; sometimes, if the topology is obvious or irrelevant, we may also simply write $X$.
Examples 3.1.2. (a) Let $(X, d)$ be a metric space, and let $\mathcal{T}$ denote the collection of all subsets of $X$ that are open in the sense of Definition 2.2.3. By Proposition 2.2.5, $\mathcal{T}$ is indeed a topology. It is clear that $\mathcal{T}$ does not depend on the particular metric $d$, but only on its equivalence class: any metric on $X$ equivalent to $d$ yields the same topology. Topological spaces of this type are called metrizable.
(b) Let $X$ be any set, and let $\mathcal{T}=\mathfrak{P}(X)$. This is just a special case of the first example: equip $X$ with the discrete metric. Such topological spaces are called discrete.
(c) Let $X$ be any set, and let $\mathcal{T}={\varnothing, X}$. Such topological spaces are called chaotic.
(d) Let $X$ be any set, and let $\mathcal{T}$ consist of $\varnothing$ and all subsets of $X$ with finite complement.
(e) Let $X$ be any set, and let $\mathcal{T}$ consist of $\varnothing$ and all subsets of $X$ with countable complement.
(f) Let $(X, \mathcal{T})$ be a topological space, and let $Y \subset X$. The relative topology on $Y$ (or the topology inherited from $X$ ) is the collection
$$
\left.\mathcal{T}\right|_Y:={Y \cap U: U \in \mathcal{T}}
$$
of subsets of $Y$. It is clearly a topology on $Y$. The space $\left(Y,\left.\mathcal{T}\right|_Y\right)$ is then called a subspace of $X$.

拓扑学代考

数学代写|拓扑学代写Topology代考|Compactness for Metric Spaces

紧性的概念是所有拓扑中最重要的概念之一(也是最难掌握的概念之一)。

2.5.1.定义设$(X, d)$是度量空间，设$S \subset X$。$S$的开放覆盖是$X$的开放子集的集合$\mathcal{U}$，例如$S \subset \cup{U: U \in \mathcal{U}}$。
2.5.2.定义度量空间$(X, d)$的子集$K$被称为紧化，如果对于$K$的每个开盖$\mathcal{U}$，存在$U_1, \ldots, U_n \in \mathcal{U}$使得$K \subset U_1 \cup$$\cdots \cup U_n$。

定义2.5.2通常被表述为:“一个集合是紧的当且仅当每个开盖都有一个有限的子盖。”

例2.5.3。(a)设$(X, d)$为度量空间，$S \subset X$为有限空间;也就是$S=\left{x_1, \ldots, x_n\right}$。让$\mathcal{U}$成为$X$的公开封面。然后，对于每个$j=1, \ldots, n$，有$U_j \in \mathcal{U}$使得$x_j \in U_j$。由此可知$S \subset$$U_1 \cup \cdots \cup U_n$。因此，$S$是紧凑的。
(b)设$(X, d)$为紧致度量空间，设$\varnothing \neq K \subset X$为紧致度量空间。修复$x_0 \in K$。因为$\left{B_r\left(x_0\right): r>0\right}$是$K$的公开封面，所以有$r_1, \ldots, r_n>0$
$$
K \subset B_{r_1}\left(x_0\right) \cup \cdots \cup B_{r_n}\left(x_0\right) .
$$
对于$R:=\max \left{r_1, \ldots, r_n\right}$，我们看到$K \subset B_R\left(x_0\right)$，所以是$\operatorname{diam}(K) \leq$$2 R<\infty$。这意味着，例如，$\mathbb{R}^n$的任何无界子集(或者更一般地说，任何赋范空间)都不能是紧的。特别地，唯一紧致赋范空间是${0}$。
(c)让$X=(0,1)$配备通常的公制。对于$r \in(0,1)$，让$U_r:=(r, 1)$。那么$\left{U_r: r \in(0,1)\right}$是$(0,1)$的开盖，它没有有限子盖。

在我们讨论更多(和更有趣的)紧化度量空间的例子之前，我们先建立一些遗传性质。

数学代写|拓扑学代写Topology代考|Topological Spaces-Definitions and Examples

一个拓扑空间应该是一个集合，它有足够的结构来有意义地谈论它上面的连续函数。根据推论2.3.10，一个合理的方法是公理化开集的概念:
3.1.1.定义设$X$为集合。$X$上的$A$拓扑是$\mathfrak{P}(X)$的一个子集$\mathcal{T}$，这样:
(a) $\varnothing, X \in \mathcal{T}$;
(b)如果$\mathcal{U} \subset \mathcal{T}$是任意的，那么$\bigcup{U: U \in \mathcal{U}}$在$\mathcal{T}$;
(c)如果$U_1, U_2 \in \mathcal{T}$，那么$U_1 \cap U_2 \in \mathcal{T}$。
$\mathcal{T}$中的集合称为open。一个集合和一个拓扑一起称为拓扑空间。

我们经常写信 $(X, \mathcal{T})$ 对于拓扑空间 $X$ 有拓扑结构 $\mathcal{T}$;有时，如果拓扑结构很明显或不相关，我们也可以简单地写 $X$．
例3.1.2。(a)让 $(X, d)$ 是一个度量空间，让 $\mathcal{T}$ 表示的所有子集的集合 $X$ 在定义2.2.3的意义上是开放的。根据提案2.2.5， $\mathcal{T}$ 确实是一个拓扑。很明显 $\mathcal{T}$ 不依赖于特定的度量 $d$，但只在它的等价类上:上的任何度规 $X$ 相当于 $d$ 产生相同的拓扑。这种类型的拓扑空间称为可度量的。
(b)让 $X$ 是任意集合，让 $\mathcal{T}=\mathfrak{P}(X)$．这只是第一个例子的特例:装备 $X$ 用离散度规。这样的拓扑空间被称为离散的。
(c)让 $X$ 是任意集合，让 $\mathcal{T}={\varnothing, X}$．这样的拓扑空间称为混沌空间。
(d)让 $X$ 是任意集合，让 $\mathcal{T}$ 包括 $\varnothing$ 的所有子集 $X$ 有限补。
(e)让 $X$ 是任意集合，让 $\mathcal{T}$ 包括 $\varnothing$ 的所有子集 $X$ 带有可数补语。
(f)让 $(X, \mathcal{T})$ 是一个拓扑空间，令 $Y \subset X$．上的相对拓扑 $Y$ 的拓扑结构 $X$ )是集合
$$
\left.\mathcal{T}\right|_Y:={Y \cap U: U \in \mathcal{T}}
$$
的子集的 $Y$．这显然是一个拓扑结构 $Y$．空间 $\left(Y,\left.\mathcal{T}\right|_Y\right)$ 它是的子空间吗 $X$．

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|M-623

Posted on 2023年8月24日2023年8月24日 by statistics-lab

数学代写|拓扑学代写Topology代考|Open and Closed Sets

We start with the definition of an open ball in a metric space:
Definition 2.2.1. Let $(X, d)$ be a metric space, let $x_0 \in X$, and let $r>0$. The open ball centered at $x_0$ with radius $r$ is defined as
$$
B_r\left(x_0\right):=\left{x \in X: d\left(x, x_0\right)<r\right} .
$$
Of course, in Euclidean 2- or 3-space, this definition coincides with the usual intuitive one. Nevertheless, even though open balls are defined with the intuitive notions of Euclidean space in mind, matters can turn out to be surprisingly counterintuitive:

Examples 2.2.2. (a) Let $(X, d)$ be a discrete metric space, let $x_0 \in X$, and let $r>0$. Then
$$
B_r\left(x_0\right)=\left{\begin{array}{cc}
\left{x_0\right}, & r<1, \ X, & r \geq 1, \end{array}\right. $$ holds; that is, each open ball is a singleton subset or the whole space. (b) Let $(X, d)$ be any metric space, let $p \in X$, and let $d_p$ be the corresponding French railroad metric. To tell open balls in $(X, d)$ and $\left(X, d_p\right)$ apart, we write $B_r\left(x_0 ; d\right)$ and $B_r\left(x_0 ; d_p\right)$, respectively, for $x_0 \in X$ and $r>0$. Let $x_0 \in X$, and let $r>0$. Since, for $x \in X$ with $x \neq x_0$, we have
$$
d_p\left(x, x_0\right)=d(x, p)+d\left(p, x_0\right)<r \quad \Longleftrightarrow d(x, p)<r-d\left(p, x_0\right),
$$
the following dichotomy holds.
$$
B_r\left(x_0 ; d_p\right)=\left{\begin{array}{cl}
\left{x_0\right}, & \text { if } r \leq d\left(p, x_0\right), \
B_{r-d\left(p, x_0\right)}(p ; d) \cup\left{x_0\right}, & \text { otherwise. }
\end{array}\right.
$$
Like the notion of an open ball, the notion of an open set extends from Euclidean space to arbitrary metric spaces.

数学代写|拓扑学代写Topology代考|Convergence and Continuity

The notion of convergence in $\mathbb{R}^n$ carries over to metric spaces almost verbatim.

Definition 2.3.1. Let $(X, d)$ be a metric space. A sequence $\left(x_n\right){n=1}^{\infty}$ in $X$ is said to converge to $x \in X$ if, for each $\epsilon>0$, there is $n\epsilon \in \mathbb{N}$ such that $d\left(x_n, x\right)<\epsilon$ for all $n \geq n_\epsilon$. We then say that $x$ is the limit of $\left(x_n\right){n=1}^{\infty}$ and write $x=\lim {n \rightarrow \infty} x_n$ or $x_n \rightarrow x$.

It is straightforward to verify that a sequence $\left(x_n\right)_{n=1}^{\infty}$ in a metric space converges to $x$ if and only if, for each $N \in \mathcal{N}_x$, there is $n_N \in \mathbb{N}$ such that $x_n \in N$ for all $n \geq n_N$.

Examples 2.3.2. (a) Let $(X, d)$ be a discrete metric space, and let $\left(x_n\right){n=1}^{\infty}$ be a sequence in $X$ that converges to $x \in X$. Then there is $n_1 \in \mathbb{N}$ such that $d\left(x_n, x\right)<1$ for $n \geq n_1$; that is, $x_n=x$ for $n \geq n_1$. Hence, every convergent sequence in a discrete metric space is eventually constant. (b) Let $C([0,1], \mathbb{F})$ be equipped with the metric induced by $|\cdot|{\infty}$ (Example $2.1 .2(\mathrm{c}))$. We claim that a sequence $\left(f_n\right){n=1}^{\infty}$ in $C([0,1], \mathbb{F})$ converges to $f \in C([0,1], \mathbb{F})$ with respect to that metric if and only if it converges (to $f$ ) uniformly on $[0,1]$. Suppose first that $\left|f_n-f\right|{\infty} \rightarrow 0$, and let $\epsilon>0$. Then there is $n_\epsilon \in \mathbb{N}$ such that
$$
\left|f_n(t)-f(t)\right| \leq\left|f_n-f\right|_{\infty}<\epsilon \quad\left(n \geq n_\epsilon, t \in[0,1]\right),
$$
so that $f_n \rightarrow f$ uniformly on $[0,1]$. Conversely, let $\left(f_n\right){n=1}^{\infty}$ converge to $f$ uniformly on $[0,1]$, and let $\epsilon>0$. By the definition of uniform convergence, there is $n\epsilon \in \mathbb{N}$ such that
$$
\left|f_n(t)-f(t)\right|<\frac{\epsilon}{2} \quad\left(n \geq n_\epsilon, t \in[0,1]\right)
$$
and consequently,
$$
\left|f_n-f\right|_{\infty}=\sup \left{\left|f_n(t)-f(t)\right|: t \in[0,1]\right} \leq \frac{\epsilon}{2}<\epsilon \quad\left(n \geq n_\epsilon\right) .
$$

拓扑学代考

数学代写|拓扑学代写Topology代考|Open and Closed Sets

我们从度量空间中空球的定义开始:
2.2.1.定义设$(X, d)$是一个度量空间，设$x_0 \in X$，设$r>0$。以$x_0$为中心，半径$r$的开球定义为
$$
B_r\left(x_0\right):=\left{x \in X: d\left(x, x_0\right)<r\right} .
$$
当然，在欧几里得2-或3-空间中，这个定义与通常直观的定义是一致的。然而，即使开放球是用欧几里得空间的直观概念来定义的，事情也可能出人意料地违反直觉:

例2.2.2。(a)设$(X, d)$为离散度量空间，设$x_0 \in X$，设$r>0$。然后
$$
B_r\left(x_0\right)=\left{\begin{array}{cc}
\left{x_0\right}, & r<1, \ X, & r \geq 1, \end{array}\right. $$持有;也就是说，每个开放的球都是一个单独的子集或整个空间。(b)设$(X, d)$为任意公制空间，设$p \in X$，设$d_p$为相应的法国铁路公制。为了区分$(X, d)$和$\left(X, d_p\right)$中的开放式球，我们分别为$x_0 \in X$和$r>0$写$B_r\left(x_0 ; d\right)$和$B_r\left(x_0 ; d_p\right)$。让$x_0 \in X$，让$r>0$。因为，对于$x \in X$和$x \neq x_0$，我们有
$$
d_p\left(x, x_0\right)=d(x, p)+d\left(p, x_0\right)<r \quad \Longleftrightarrow d(x, p)<r-d\left(p, x_0\right),
$$
下面的二分法成立。
$$
B_r\left(x_0 ; d_p\right)=\left{\begin{array}{cl}
\left{x_0\right}, & \text { if } r \leq d\left(p, x_0\right), \
B_{r-d\left(p, x_0\right)}(p ; d) \cup\left{x_0\right}, & \text { otherwise. }
\end{array}\right.
$$
像开放球的概念一样，开放集的概念从欧几里德空间扩展到任意度量空间。

数学代写|拓扑学代写Topology代考|Convergence and Continuity

$\mathbb{R}^n$中收敛的概念几乎一字不差地延续到度量空间。

2.3.1.定义设$(X, d)$是一个度量空间。如果对于每一个$\epsilon>0$，都有一个$n\epsilon \in \mathbb{N}$使得对于所有$n \geq n_\epsilon$，都有一个$d\left(x_n, x\right)<\epsilon$，那么我们就说$X$中的一个序列$\left(x_n\right){n=1}^{\infty}$收敛到$x \in X$。然后我们说$x$是$\left(x_n\right){n=1}^{\infty}$的极限，并写$x=\lim {n \rightarrow \infty} x_n$或$x_n \rightarrow x$。

很容易验证度量空间中的序列$\left(x_n\right)_{n=1}^{\infty}$收敛到$x$当且仅当对于每个$N \in \mathcal{N}_x$，存在$n_N \in \mathbb{N}$使得$x_n \in N$对于所有$n \geq n_N$。

例2.3.2。(a)设$(X, d)$为离散度量空间，设$\left(x_n\right){n=1}^{\infty}$为$X$中的一个收敛于$x \in X$的序列。然后是$n_1 \in \mathbb{N}$，比如$d\left(x_n, x\right)<1$代表$n \geq n_1$;也就是说，$n \geq n_1$等于$x_n=x$。因此，离散度量空间中的每一个收敛序列最终都是常数。(b)设$C([0,1], \mathbb{F})$为由$|\cdot|{\infty}$(例$2.1 .2(\mathrm{c}))$)导出的度量。我们声称，一个序列$\left(f_n\right){n=1}^{\infty}$在$C([0,1], \mathbb{F})$上相对于该度规收敛于$f \in C([0,1], \mathbb{F})$当且仅当它在$[0,1]$上一致收敛(到$f$)。首先假设$\left|f_n-f\right|{\infty} \rightarrow 0$，然后让$\epsilon>0$。然后是$n_\epsilon \in \mathbb{N}$
$$
\left|f_n(t)-f(t)\right| \leq\left|f_n-f\right|{\infty}<\epsilon \quad\left(n \geq n\epsilon, t \in[0,1]\right),
$$
所以$f_n \rightarrow f$均匀地落在$[0,1]$上。反之，设$\left(f_n\right){n=1}^{\infty}$在$[0,1]$上均匀收敛于$f$，设$\epsilon>0$。根据一致收敛的定义，有$n\epsilon \in \mathbb{N}$使得
$$
\left|f_n(t)-f(t)\right|<\frac{\epsilon}{2} \quad\left(n \geq n_\epsilon, t \in[0,1]\right)
$$
因此，
$$
\left|f_n-f\right|{\infty}=\sup \left{\left|f_n(t)-f(t)\right|: t \in[0,1]\right} \leq \frac{\epsilon}{2}<\epsilon \quad\left(n \geq n\epsilon\right) .
$$

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广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

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MATLAB代写

R语言代写	问卷设计与分析代写
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MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
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数学代写|拓扑学代写Topology代考|MTH436

Posted on 2023年8月9日2023年8月28日 by statistics-lab

数学代写|拓扑学代写Topology代考|Topological surgery

Basic references on surgery are the textbooks by Kosinski [153] and Ranicki [213]. Milnor’s beautiful article [183] remains one of the best introductions to surgery; the survey [216] by Rosenberg is also very useful. In the topological setting, it is usually not vital to have an explicit coordinate description for surgeries or the corresponding handle attachments. This situation changes, however, once we want to construct geometric structures on the surgered manifolds or the corresponding cobordisms. Such explicit coordinate descriptions are given in Milnor’s article, and, following Weinstein, we are going to adapt them to the contact geometric or symplectic setting, respectively. In the present section we first give the topological picture.
Let $D^{k+1}$ denote the unit disc in the Euclidean space $\mathbb{R}^{k+1}$, with boundary $\partial D^{k+1}=S^k$ the unit $k$-sphere. The product manifold $S^k \times S^{n-k-1}$ may be regarded as the boundary either of $S^k \times D^{n-k}$ or of $D^{k+1} \times S^{n-k-1}$.

Now let $M$ be an $n$-dimensional differential manifold, and suppose that we have an embedded $k$-sphere $S^k \subset M$ with trivial normal bundle. This implies that we can find an embedded copy of $S^k \times D^{n-k}$ in $M$.

Definition 6.1.1 Given $S^k \times D^{n-k} \subset M$, one can form a new manifold
$$
M^{\prime}:=\left(M \backslash S^k \times \operatorname{Int}\left(D^{n-k}\right)\right) \cup_{S^k \times S^{n-k-1}}\left(D^{k+1} \times S^{n-k-1}\right)
$$
by making the obvious identification along
$$
S^k \times S^{n-k-1}=\partial\left(M \backslash S^k \times \operatorname{Int}\left(D^{n-k}\right)\right)=\partial\left(D^{k+1} \times S^{n-k-1}\right) .
$$
(The fact that $M^{\prime}$ is indeed a manifold is a simple consequence of the collaring theorem for manifolds with boundary, see [38], [132] or [153].) The procedure of constructing $M^{\prime}$ from $M$ is called a surgery along $S^k \subset M$.
Observe that $M$ may be recovered from $M^{\prime}$ by a surgery along $S^{n-k-1}$.

数学代写|拓扑学代写Topology代考|Contact surgery and symplectic cobordisms

In the preceding section, the key point in the construction of the cobordism corresponding to a surgery was the following. Given $S^k \subset M$ with trivial normal bundle, it has a neighbourhood in $M$ diffeomorphic to a neighbourhood of the $k$-sphere given by ${\mathbf{y}=\mathbf{0}}$ in the lower boundary of $H$. This diffeomorphism extends to a diffeomorphism of the punctured neighbourhood $S^k \times\left(D^{n-k} \backslash{\mathbf{0}}\right)$ crossed with the interval $[-1,1]$ on the one side, and $H$ with the coordinate planes removed on the other, by mapping the obvious foliation by line segments in one space to the foliation by gradient flow lines $\gamma$ on the other. This allowed the gluing that gave us the desired cobordism.

We now want to show that one can perform surgery along an isotropic sphere with trivial conformal symplectic normal bundle in a given contact manifold such that the resulting manifold carries again a contact structure (which coincides with the old one outside the neighbourhood where surgery takes place). We shall call this a contact surgery. We are going to perform this surgery by constructing the corresponding symplectic cobordism. Thus, the task becomes obvious. First of all, we need a neighbourhood theorem for isotropic submanifolds in a contact manifold. This will enable us to identify (by a contactomorphism) a neighbourhood of an isotropic sphere in a given contact manifold with a neighbourhood of an isotropic sphere in a model handle. Such a neighbourhood theorem has already been proved (Thm. 2.5.8). As we shall see, the isotropic spheres in the model handle will have trivial conformal symplectic normal bundle, so the same condition will have to be imposed on any sphere along which we want to perform contact surgery.

Secondly, we need to work with a suitable flow, replacing the gradient flow in the topological picture, that will yield a symplectomorphism for the gluing construction. Observe that if $(M, \xi=\operatorname{ker} \alpha)$ is a contact manifold, then a natural symplectic structure $\omega$ on $[-1,1] \times M$ is obtained by symplectisation (see Example 1.4.7), that is, $\omega:=d\left(\mathrm{e}^t \alpha\right)$. Here the obvious foliation by line segments is given by the flow of the Liouville vector field $\partial_t$. Correspondingly, the gradient flow in our topological handle $H$ will have to be replaced by a Liouville flow on a symplectic handle. Moreover, the neighbourhood theorem for isotropic submanifolds will have to be strong enough so that the identification of flow lines in $[-1,1] \times M$ with those in $H$ actually yields a symplectomorphism.

For that last point, we shall need a version of Theorem 2.5 .8 for strict contact manifolds, see Definition 2.1.2. In the process, we also provide the argument hinted at in Remark 2.5.12. For the following definition recall Lemma 1.3.3 and Definition 2.5.3.

拓扑学代考

数学代写|拓扑学代写Topology代考|Topological surgery

外科的基本参考文献是Kosinski[153]和Ranicki[213]的教科书。米尔诺那篇漂亮的文章[183]仍然是对外科手术最好的介绍之一;Rosenberg的调查[216]也很有用。在拓扑设置中，通常不需要对手术或相应的手柄附件进行明确的坐标描述。然而，当我们想要在流形或相应的协坐标上构造几何结构时，这种情况就改变了。这种明确的坐标描述在Milnor的文章中给出，并且，根据Weinstein，我们将分别将它们适应于接触几何或辛设置。在本节中，我们首先给出拓扑图。
设$D^{k+1}$表示欧几里得空间$\mathbb{R}^{k+1}$中的单位圆盘，边界$\partial D^{k+1}=S^k$表示单位$k$球。积流形$S^k \times S^{n-k-1}$可以看作是$S^k \times D^{n-k}$或$D^{k+1} \times S^{n-k-1}$的边界。

现在设$M$是一个$n$维的微分流形，并假设我们有一个内嵌的$k$ -球$S^k \subset M$，它具有平凡的法向束。这意味着我们可以在$M$中找到一个嵌入的$S^k \times D^{n-k}$副本。

定义6.1.1给定$S^k \times D^{n-k} \subset M$，可以形成一个新的流形
$$
M^{\prime}:=\left(M \backslash S^k \times \operatorname{Int}\left(D^{n-k}\right)\right) \cup_{S^k \times S^{n-k-1}}\left(D^{k+1} \times S^{n-k-1}\right)
$$
通过进行明显的识别
$$
S^k \times S^{n-k-1}=\partial\left(M \backslash S^k \times \operatorname{Int}\left(D^{n-k}\right)\right)=\partial\left(D^{k+1} \times S^{n-k-1}\right) .
$$
($M^{\prime}$确实是流形的事实是有边界流形的项圈定理的一个简单结果，参见[38]，[132]或[153]。)从$M$构造$M^{\prime}$的过程称为沿$S^k \subset M$的手术。
观察$M$可以通过沿$S^{n-k-1}$的手术从$M^{\prime}$恢复。

数学代写|拓扑学代写Topology代考|Contact surgery and symplectic cobordisms

在上一节中，构建手术对应的协坐标的关键点如下。给定具有平凡法线束的$S^k \subset M$，它在$M$上有一个邻域，与${\mathbf{y}=\mathbf{0}}$在$H$下界上给出的$k$ -球的邻域是微分同构的。通过在一个空间中通过线段将明显的叶理映射到另一个空间中通过梯度流线$\gamma$的叶理，这种微同构扩展到与一边的区间$[-1,1]$相交的被刺破的邻域$S^k \times\left(D^{n-k} \backslash{\mathbf{0}}\right)$，以及与另一边的坐标平面相交的$H$。这样就可以粘合出我们想要的协同效果。

我们现在要证明，在给定的接触流形中，可以沿着具有平凡共形辛法向束的各向同性球体进行手术，使得所得到的流形再次携带一个接触结构(该结构与发生手术的邻域外的旧结构一致)。我们称之为接触手术。我们将通过构造相应的辛共来做这个手术。因此，任务变得显而易见。首先，我们需要接触流形中各向同性子流形的邻域定理。这将使我们能够(通过接触同构)在给定的接触流形中识别各向同性球体的邻域与模型柄中的各向同性球体的邻域。这样的邻域定理已经被证明了(公式2.5.8)。正如我们将看到的，模型柄中的各向同性球体将具有平凡的共形辛法向束，因此我们必须对任何想要进行接触手术的球体施加相同的条件。

其次，我们需要使用合适的流，取代拓扑图中的梯度流，这将为粘合构造产生辛形态。注意，如果$(M, \xi=\operatorname{ker} \alpha)$是一个接触流形，那么通过辛化(见例1.4.7)，在$[-1,1] \times M$上得到一个自然辛结构$\omega$，即$\omega:=d\left(\mathrm{e}^t \alpha\right)$。这里明显的线段分叶是由Liouville向量场的流动$\partial_t$给出的。相应地，我们的拓扑句柄$H$中的梯度流将不得不被辛句柄上的刘维尔流所取代。此外，各向同性子流形的邻域定理必须足够强大，才能使$[-1,1] \times M$中的流线与$H$中的流线的识别实际上产生一种复形态。

对于最后一点，我们需要定理2.5 .8的一个版本，用于严格接触流形，参见定义2.1.2。在此过程中，我们还提供了2.5.12中暗示的论证。对于下面的定义，请参考引理1.3.3和定义2.5.3。

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非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

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MATLAB代写

R语言代写	问卷设计与分析代写
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MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
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数学代写|拓扑学代写Topology代考|Math8750

Posted on 2023年8月9日2023年8月28日 by statistics-lab

数学代写|拓扑学代写Topology代考|Weak versus strong fillings

The following definitions of weak and strong symplectic fillings of contact manifolds are the obvious modifications of those in dimension 3. In this general setting, it is useful to consider an intermediate notion of domination, see [79], [176].

Definition 5.1.1 Let $M$ be a closed manifold of dimension $2 n-1$ with cooriented contact structure $\xi=\operatorname{ker} \alpha$. Notice that the (positive) conformal class of the symplectic bundle structure $\left.d \alpha\right|{\xi}$ is independent of the choice of contact form defining $\xi$ with given coorientation. We orient $\xi$ by $(d \alpha)^{n-1}$. (a) A compact symplectic manifold $(W, \omega)$ of dimension $2 n$ is called a weak (symplectic) filling of $(M, \xi)$ if $\partial W=M$ as oriented manifolds $\dagger$ and $\left.\omega^{n-1}\right|{\xi}>0$.
(b) We say that a compact symplectic manifold $(W, \omega)$ dominates $(M, \xi)$ if $\partial W=M$ as oriented manifolds and $\left.\omega\right|{\xi}$ is in the conformal class of $\left.d \alpha\right|{\xi}$.
(c) A compact symplectic manifold $(W, \omega)$ is called a strong (symplectic) filling of $(M, \xi)$ if $\partial W=M$ and there is a Liouville vector field $Y$ defined near $\partial W$, pointing outwards along $\partial W$, and satisfying $\xi=\operatorname{ker}\left(\left.i_Y \omega\right|_{T M}\right)$ (as cooriented contact structure). In this case we say that $(M, \xi)$ is the convex (or more precisely: $\omega$-convex) boundary of $(W, \omega)$.

Remark 5.1.2 In the terminology of Definition 1.4.5, the boundary of a strong filling is of contact type.

Example 5.1.3 By Examples 1.4.8 and 2.1.7, the disc $D^{2 n}$ with its standard symplectic form $\omega_{\mathrm{st}}$ is a strong filling of the sphere $S^{2 n-1}$ with its standard contact structure $\xi_{\text {st }}$.

数学代写|拓扑学代写Topology代考|Symplectic cobordisms

There is a natural correspondence between surgery and cobordisms, which we are going to recall in detail in Section 6.1. When one wants to perform surgery on contact manifolds, this gives rise to a corresponding notion of symplectic cobordisms. With the following discussion of such cobordisms we thus pave the way to the next chapter.

Definition 5.2.1 Let $\left(M_{ \pm}, \xi_{ \pm}\right)$be closed contact manifolds of dimension $2 n-1$ with cooriented contact structures, which induce the orientation of the respective manifold. A symplectic cobordism from $\left(M_{-}, \xi_{-}\right)$to $\left(M_{+}, \xi_{+}\right)$ is a compact $2 n$-dimensional symplectic manifold $(W, \omega)$, oriented by the volume form $\omega^n$, such that the following conditions hold.

The oriented boundary of $W$ equals $\partial W=M_{+} \sqcup \bar{M}{-}$, where $\bar{M}{-}$ stands for $M_{-}$with reversed orientation.
In a neighbourhood of $\partial W$, there is a Liouville vector field $Y$ for $\omega$, transverse to the boundary and pointing outwards along $M_{+}$, inwards along $M_{-}$.
The 1-form $\alpha:=i_Y \omega$ restricts to $T M_{ \pm}$as a contact form for $\xi_{ \pm}$.

We call $M_{+}$the convex boundary of the cobordism $W$ and $M_{-}$the concave boundary.

Example 5.2.2 Every closed contact manifold $(M, \xi=\operatorname{ker} \alpha)$ is symplectically cobordant to itself via the trivial symplectic cobordism
$$
\left([0,1] \times M, d\left(\mathrm{e}^t \alpha\right)\right),
$$
with $Y=\partial_t$.

拓扑学代考

数学代写|拓扑学代写Topology代考|Weak versus strong fillings

下面的接触流形弱辛填充和强辛填充的定义是对3维定义的明显修改。在这种一般情况下，考虑支配的中间概念是有用的，参见[79]，[176]。

定义5.1.1设$M$为维度为$2 n-1$的封闭流形，具有共向接触结构$\xi=\operatorname{ker} \alpha$。注意，辛束结构$\left.d \alpha\right|{\xi}$的(正)共形类与定义$\xi$的接触形式的选择无关。我们通过$(d \alpha)^{n-1}$定位$\xi$。(a)尺寸为$2 n$的紧辛流形$(W, \omega)$称为$(M, \xi)$的弱(辛)填充，如果$\partial W=M$是面向流形$\dagger$和$\left.\omega^{n-1}\right|{\xi}>0$。
(b)如果$\partial W=M$是有向流形，且$\left.\omega\right|{\xi}$在$\left.d \alpha\right|{\xi}$的保形类中，则紧辛流形$(W, \omega)$优于$(M, \xi)$。
(c)紧辛流形$(W, \omega)$称为$(M, \xi)$的强(辛)填充，如果$\partial W=M$，并且在$\partial W$附近定义一个Liouville向量场$Y$，沿着$\partial W$向外指向，并满足$\xi=\operatorname{ker}\left(\left.i_Y \omega\right|_{T M}\right)$(作为共向接触结构)。在这种情况下，我们说$(M, \xi)$是$(W, \omega)$的凸(或者更准确地说:$\omega$ -凸)边界。

注5.1.2在定义1.4.5的术语中，强填料的边界为接触型。

通过例1.4.8和2.1.7，具有标准辛形式$\omega_{\mathrm{st}}$的圆盘$D^{2 n}$是具有标准接触结构$\xi_{\text {st }}$的球体$S^{2 n-1}$的强填充。

数学代写|拓扑学代写Topology代考|Symplectic cobordisms

在外科手术和协方差之间有一种天然的对应关系，我们将在6.1节中详细回顾。当人们想要对接触流形进行手术时，这就产生了相应的辛协的概念。因此，通过下面对这种协调的讨论，我们为下一章铺平了道路。

5.2.1设$\left(M_{ \pm}, \xi_{ \pm}\right)$为尺寸为$2 n-1$的闭合接触流形，它们具有共取向的接触结构，由此推导出各自流形的取向。从$\left(M_{-}, \xi_{-}\right)$到$\left(M_{+}, \xi_{+}\right)$的辛协矩阵是一个紧致的$2 n$维辛流形$(W, \omega)$，由体积形式$\omega^n$定向，满足以下条件。

$W$的有向边界等于$\partial W=M_{+} \sqcup \bar{M}{-}$，其中$\bar{M}{-}$表示方向相反的$M_{-}$。

在$\partial W$的邻域中，$\omega$有一个Liouville向量场$Y$，横向于边界，沿$M_{+}$向外，沿$M_{-}$向内。

1-form $\alpha:=i_Y \omega$限制为$T M_{ \pm}$作为$\xi_{ \pm}$的联系表单。

我们称$M_{+}$为协坐标$W$的凸边界，称$M_{-}$为凹边界。

例5.2.2每一个闭合接触流形$(M, \xi=\operatorname{ker} \alpha)$都通过平凡的辛协度与自身辛协度
$$
\left([0,1] \times M, d\left(\mathrm{e}^t \alpha\right)\right),
$$
通过$Y=\partial_t$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|MAST90023

Posted on 2023年8月9日2023年8月28日 by statistics-lab

数学代写|拓扑学代写Topology代考|Construction near the 2–skeleton

The following result is the central tool of the present section. It tells us that a family of plane fields can be turned into a family of contact structures (relative to a subset where the contact condition was already satisfied), provided there is a field of directions along which one can twist the plane fields. This field of directions will be given by the intersection of the original plane field with some foliation. This will enable us to control the ‘norm’ (defined below) of the resulting contact structures and hence the characteristic foliations they induce on the $\partial B^i$ as described in the outline.

Proposition 4.7.15 Consider a bounded domain $U \subset \mathbb{R}^3$ and a closed subset $A \subset U$. Let $\mathfrak{P}$ be an oriented 2-dimensional foliation on the closure $\bar{U}$ of $U$. Let $\left(\xi_t\right){t \in[0,1]}$, be a continuous family of (cooriented) 2-plane distributions on $\bar{U}$ with the following properties: $(\xi 1) \xi_0$ and $\xi_1$ are contact structures; ( $\xi 2) \xi_t$ is a contact structure in a neighbourhood of $A$ for each $t \in[0,1]$; $(\mathfrak{P} 1) \xi_t$ is transverse to $\mathfrak{P}$ for each $t \in[0,1]$; $(\mathfrak{P} 2)$ for any leaf $\mathfrak{p}$ of $\mathfrak{P}$, the 1 -dimensional foliation $\mathfrak{p}{\xi_t}$, for any $t \in[0,1]$, consists of curves that hit $A$ at most once.

Then there exists a continuous family $\left(\xi_t^{\prime}\right)_{t \in[0,1]}$ of contact structures on $\bar{U}$ such that

$\xi_0^{\prime}=\xi_0$ and $\xi_1^{\prime}=\xi_1$ on $\bar{U}$;
$\left.\xi_t^{\prime}\right|_A=\left.\xi_t\right|_A$ for all $t \in[0,1]$.

数学代写|拓扑学代写Topology代考|Proof of the classification result

We are now ready to prove Theorem 4.7.2. As indicated previously, we only want to prove injectivity of $\left(i_{\Delta}\right)_{#}$ on the level of $\pi_0$. Recall the outline of the argument in Section 4.7.2. Thus, let $\xi_t \in \operatorname{Distr}(M, \Delta), t \in[0,1]$, be a continuous family of plane fields, with $\xi_0, \xi_1 \in \Xi^{\text {ot }}(M, \Delta)$. We need to find a family $\xi_t^{\prime \prime} \in \Xi^{\mathrm{ot}}(M, \Delta)$ with $\xi_0^{\prime \prime}=\xi_0, \xi_1^{\prime \prime}=\xi_1$.

First Step – Where we leave holes The plane fields $\xi_t$ all coincide at the centre of the disc $\Delta$, and $\xi_0, \xi_1$ coincide near $\Delta$ by Theorem 2.5.22. This allows us, by a first homotopy rel ${0,1}$ of the family $\left(\xi_t\right)$, to assume that all $\xi_t$ actually coincide near $\Delta$. Then we can find an embedded ball $B^0 \subset\left(M, \xi_t\right)$ contactomorphic to the ball $\left{(z / \delta)^2+r^2 \leq(\pi+\delta)^2\right}$ in $\left(\mathbb{R}^3, \xi_{\text {ot }}\right)$ for some small $\delta>0$. In particular, the characteristic foliation $\left(\partial B^0\right){\xi_t}$ is the foliation $\mathfrak{F}^0$ from Figure 4.35. We are going to write $B{\text {ot }}$ for this ball (for some fixed $\delta$ ) and call it the standard overtwisted ball.

Because of the extension character of the result discussed on page 217, we may proceed as follows:

consider the $\xi_t$ (for all $t$ simultaneously) on small subsets of $M$ contained in a Darboux chart for each $t$;
perturb them there to a family of plane fields satisfying the contact condition in the neighbourhood of the 2-skeleton of an auxiliary simplicial complex, relative to the subset of $M$ where this contact property had been achieved by a previous perturbation.

拓扑学代考

数学代写|拓扑学代写Topology代考|Construction near the 2–skeleton

下面的结果是本节的中心工具。它告诉我们，一组平面场可以转化为一组接触结构(相对于已经满足接触条件的子集)，只要有一个方向场，人们可以沿着这个方向扭转平面场。这个方向场将由原始平面场与一些叶理的交点给出。这将使我们能够控制由此产生的接触结构的“规范”(定义见下文)，从而在大纲中描述的$\partial B^i$上产生特征叶状结构。

命题4.7.15考虑一个有界域 $U \subset \mathbb{R}^3$ 一个封闭子集 $A \subset U$．让 $\mathfrak{P}$ 是闭合上有取向的二维叶理 $\bar{U}$ 的 $U$．让 $\left(\xi_t\right){t \in[0,1]}$，是上的(共向)2-平面分布的连续族 $\bar{U}$ 具有以下属性: $(\xi 1) \xi_0$ 和 $\xi_1$ 是接触结构;（ $\xi 2) \xi_t$ 邻域中的接触结构是 $A$ 对于每一个 $t \in[0,1]$； $(\mathfrak{P} 1) \xi_t$ 是横向的 $\mathfrak{P}$ 对于每一个 $t \in[0,1]$； $(\mathfrak{P} 2)$ 对于任何一片叶子 $\mathfrak{p}$ 的 $\mathfrak{P}$即一维叶理 $\mathfrak{p}{\xi_t}$对于任何人 $t \in[0,1]$，由命中的曲线组成 $A$ 最多一次。

那么在$\bar{U}$上存在一个连续族$\left(\xi_t^{\prime}\right)_{t \in[0,1]}$的接触结构，使得

$\xi_0^{\prime}=\xi_0$ 还有$\bar{U}$上的$\xi_1^{\prime}=\xi_1$;

$\left.\xi_t^{\prime}\right|_A=\left.\xi_t\right|_A$ 对于所有$t \in[0,1]$。

数学代写|拓扑学代写Topology代考|Proof of the classification result

现在我们准备证明定理4.7.2。如前所述，我们只想在$\pi_0$的水平上证明$\left(i_{\Delta}\right)_{#}$的注入性。回想一下4.7.2节中参数的大纲。因此，设$\xi_t \in \operatorname{Distr}(M, \Delta), t \in[0,1]$为平面场的连续族，其中$\xi_0, \xi_1 \in \Xi^{\text {ot }}(M, \Delta)$。我们需要找到一个家庭$\xi_t^{\prime \prime} \in \Xi^{\mathrm{ot}}(M, \Delta)$与$\xi_0^{\prime \prime}=\xi_0, \xi_1^{\prime \prime}=\xi_1$。

第一步-我们留下洞的地方平面场$\xi_t$都在圆盘的中心重合$\Delta$，并且根据定理2.5.22 $\xi_0, \xi_1$在$\Delta$附近重合。这允许我们，通过族$\left(\xi_t\right)$的第一个同伦关系${0,1}$，假设所有的$\xi_t$实际上都在$\Delta$附近重合。然后我们可以找到一个嵌入球$B^0 \subset\left(M, \xi_t\right)$接触形态的球$\left{(z / \delta)^2+r^2 \leq(\pi+\delta)^2\right}$在$\left(\mathbb{R}^3, \xi_{\text {ot }}\right)$为一些小$\delta>0$。其中，特征叶理$\left(\partial B^0\right){\xi_t}$为图4.35中的叶理$\mathfrak{F}^0$。我们将为这个球写$B{\text {ot }}$(对于一些固定的$\delta$)，并将其称为标准的过扭球。

由于第217页讨论的结果具有可拓性，我们可以这样进行:

考虑$\xi_t$(同时适用于所有$t$)在每个$t$的达布图中包含的$M$的小子集上;

将它们扰动到辅助简单复合体的2骨架附近满足接触条件的平面场族，相对于$M$的子集，其中这种接触性质已通过先前的扰动获得。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|AMAT540

Posted on 2023年7月25日2023年8月25日 by statistics-lab

数学代写|拓扑学代写Topology代考|The characteristic foliation

Recall that by a singular (and oriented) 1-dimensional foliation on the surface $S$ we mean an equivalence class of vector fields, where two vector fields $X, X^{\prime}$ are called equivalent if there is a smooth function $f: S \rightarrow \mathbb{R}^{+}$such that $X^{\prime}=f X$.

We now fix an area form $\Omega$ on $S$, compatible with the given orientation. This allows us to make the following definition, where we set $\beta:=\beta_0$.

Definition 4.6.1 The characteristic foliation $S_{\xi}$ on $S$ induced by the contact structure $\xi$ is the singular 1-dimensional foliation represented by the vector field $X$ that is defined by
$$
i_X \Omega=\beta .
$$
Notice that different choices of area form $\Omega$ and different choices of contact form $\alpha$ for the cooriented contact structure $\xi$ differ by a positive function, so the equivalence class of $X$ as a singular 1-dimensional foliation does indeed depend on $S$ and $\xi$ only. Furthermore, observe that by (4.4) and (4.1) the points $p \in S$ with $X_p=0$ are precisely those points of $S$ where the contact plane $\xi_p$ coincides with the tangent plane $T_p S$; if $X_p \neq \mathbf{0}$, then the line in $T_p S$ spanned by $X_p$ equals $T_p S \cap \xi_p$, since $\alpha_p\left(X_p\right)=\beta_p\left(X_p\right)=0$.

Definition 4.6.2 The divergence $\operatorname{div}{\Omega}(X)$ of a vector field $X$ on $S$ with respect to the area form $\Omega$ is defined by $\mathcal{L}_X \Omega=\operatorname{div}{\Omega}(X) \Omega$; using the Cartan formula we can rewrite this as
$$
\operatorname{div}_{\Omega}(X) \Omega=d\left(i_X \Omega\right) .
$$

数学代写|拓扑学代写Topology代考|Convex surfaces

In some of the examples of the previous section we saw how useful a statement such as Lemma 4.6.4 may be for writing down explicit contact forms. In the present section we collect a few more basic facts about vertically invariant contact structures on $S \times \mathbb{R}$ and a related concept, viz. convex surfaces. Here the emphasis will be on those results that we want to use in the next section for manipulating the characteristic foliation. The real power of convex surface theory will be explained in Section 4.8 below.

Definition 4.6.16 A convex surface $S$ in $(M, \xi)$ is an embedded surface with the property that there is a contact vector field (see Defn. 1.5.7) defined near and transverse to $S$.

Remark 4.6.17 This terminology, introduced by Giroux [104], is – as Giroux himself admits – a little unfortunate, since there is really no distinction between ‘convex’ and ‘concave’ in the present context. The notion ‘con’ has been suggested but seems unlikely to catch on, at least in the French literature.

Example 4.6.18 The vector field $Y:=x \partial_x+y \partial_y+2 z \partial_z$ is a contact vector field for the standard contact structure $\xi_2=\operatorname{ker} \alpha_2$ on $\mathbb{R}^3$, where $\alpha_2=d z+x d y-y d x$, since $\mathcal{L}_Y \alpha_2=2 \alpha_2$. This vector field is transverse to $S^2$, so the unit sphere is a convex surface in $\left(\mathbb{R}^3, \xi_2\right)$.

As before, our surface $S$ is always understood to be oriented, although most statements carry over to non-orientable surfaces.

Lemma 4.6.19 A closed surface $S \subset(M, \xi=\operatorname{ker} \alpha)$ is convex if and only if there is an embedding $\Psi: S \times \mathbb{R} \rightarrow M$ (with image inside an arbitrarily small prescribed neighbourhood of $S$ ) such that $p \mapsto \Psi(p, 0)$ defines the inclusion of $S$ in $M$, and $\Psi^* \alpha$ determines a vertically invariant (i.e. $\mathbb{R}$-invariant) contact structure on $S \times \mathbb{R}$.

拓扑学代考

数学代写|拓扑学代写Topology代考|The characteristic foliation

回想一下，在表面$S$上一个奇异的(定向的)一维叶理是指等价的向量场，其中两个向量场$X, X^{\prime}$被称为等价的，如果有一个平滑函数$f: S \rightarrow \mathbb{R}^{+}$使得$X^{\prime}=f X$。

现在我们在$S$上固定一个与给定方向兼容的区域表单$\Omega$。这允许我们创建以下定义，并在其中设置$\beta:=\beta_0$。

4.6.1接触结构$\xi$诱导的$S$上的特征叶理$S_{\xi}$是向量场$X$表示的一维奇异叶理，定义为
$$
i_X \Omega=\beta .
$$
注意，对于共取向接触结构$\xi$，不同选择的面积形式$\Omega$和不同选择的接触形式$\alpha$有一个正函数不同，因此，作为一维奇异叶状结构的$X$的等价类确实只依赖于$S$和$\xi$。此外，通过(4.4)和(4.1)可以观察到，$p \in S$与$X_p=0$的点正是$S$的接触平面$\xi_p$与切平面$T_p S$重合的点;如果是$X_p \neq \mathbf{0}$，那么$T_p S$中由$X_p$张成的直线等于$T_p S \cap \xi_p$，因为$\alpha_p\left(X_p\right)=\beta_p\left(X_p\right)=0$。

4.6.2在$S$上的矢量场$X$相对于面积形式$\Omega$的散度$\operatorname{div}{\Omega}(X)$定义为$\mathcal{L}X \Omega=\operatorname{div}{\Omega}(X) \Omega$;用卡坦公式我们可以把它写成
$$
\operatorname{div}{\Omega}(X)
\Omega=d\left(i_X \Omega\right) .
$$

数学代写|拓扑学代写Topology代考|Convex surfaces

在前一节的一些例子中，我们看到了引理4.6.4这样的语句对于写出显式的联系表单是多么有用。在本节中，我们收集了一些关于$S \times \mathbb{R}$上垂直不变接触结构的基本事实和一个相关的概念，即凸面。在这里，重点将放在那些结果上，我们希望在下一节中使用这些结果来操纵特征叶理。凸面理论的真正威力将在下面的第4.8节中解释。

定义4.6.16 $(M, \xi)$中的凸面$S$是一个嵌入面，其性质是在$S$附近和横向定义有一个接触向量场(见定义1.5.7)。

注释4.6.17这个术语是由吉鲁[104]引入的，正如吉鲁自己承认的那样，这有点不幸，因为在目前的上下文中，“凸”和“凹”之间确实没有区别。“con”这个概念已经被提出，但似乎不太可能流行起来，至少在法国文学中是这样。

向量场$Y:=x \partial_x+y \partial_y+2 z \partial_z$是$\mathbb{R}^3$上标准接触结构$\xi_2=\operatorname{ker} \alpha_2$的接触向量场，其中$\alpha_2=d z+x d y-y d x$，因为$\mathcal{L}_Y \alpha_2=2 \alpha_2$。这个向量场横向于$S^2$，所以单位球是$\left(\mathbb{R}^3, \xi_2\right)$的一个凸面。

像以前一样，我们的曲面$S$总是被理解为有取向的，尽管大多数语句适用于非定向曲面。

引理4.6.19一个封闭曲面 $S \subset(M, \xi=\operatorname{ker} \alpha)$ 凸当且仅当有嵌入 $\Psi: S \times \mathbb{R} \rightarrow M$ 的任意小的指定邻域内的图像 $S$ )这样 $p \mapsto \Psi(p, 0)$ 定义的包含 $S$ 在 $M$，和 $\Psi^* \alpha$ 确定垂直不变量(即。 $\mathbb{R}$(不变)接触结构 $S \times \mathbb{R}$．

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|拓扑学代写Topology代考|MATH452

Posted on 2023年7月25日2023年8月25日 by statistics-lab

数学代写|拓扑学代写Topology代考|Cobordism classes of links

We now want to relate elements in $H_1(M ; \mathbb{Z})$ to cobordism classes of links in $M$.

Theorem 4.2.4 Let $M$ be a closed, oriented 3-manifold. Any homology class $c \in H_1(M ; \mathbb{Z})$ is represented by an embedded, oriented link (of 1-spheres) $L_c$ in $M$. Two links $L_0, L_1$ represent the same class $\left[L_0\right]=\left[L_1\right]$ if and only if they are oriented cobordant in $M$, that is, there is an embedded, oriented surface $S$ in $[0,1] \times M$ with
$$
\partial S=L_1 \sqcup \bar{L}_0 \subset{1} \times M \sqcup{0} \times M,
$$
where $\sqcup$ denotes disjoint union and $\bar{L}$ the link $L$ with reversed orientation.
Proof The first part was already proved in Proposition 3.4.2. We retain the notation used there.
If two links $L_0, L_1$ are cobordant in $M$, then clearly
$$
\left[L_0\right]=\left[L_1\right] \in H_1([0,1] \times M ; \mathbb{Z}) \cong H_1(M ; \mathbb{Z})
$$

For the converse, suppose we are given two links $L_0, L_1 \subset M$ with $\left[L_0\right]=$ $\left[L_1\right]$. Choose arbitrary framings for these links and use this, as described above, to define smooth maps $f_0, f_1: M \rightarrow S^2$ with common regular value $p \in S^2$ such that $f_i^{-1}(p)=L_i, i=0,1$. Now identify $S^2$ with the standardly embedded $\mathbb{C} P^1 \subset \mathbb{C} P^2$. Let $P \subset \mathbb{C} P^2$ be a second copy of $\mathbb{C} P^1$, embedded in such a way that $[P]{\mathbb{C} P^2}=\left[\mathbb{C} P^1\right]{\mathbb{C} P^2}$ and $P$ intersects $\mathbb{C} P^1$ transversely in $p$ only. This is possible since $\mathbb{C} P^1 \subset \mathbb{C} P^2$ has self-intersection one. Then the maps $f_0, f_1$, regarded as maps into $\mathbb{C} P^2$, are transverse to $P$ and we have $f_i^{-1}(P)=L_i, i=0,1$. Hence
$$
f_i^* u_0=f_i^\left(P D[P]_{\mathbb{C} P^2}\right)=P D\left[f_i^{-1}(P)\right]_M=P D\left[L_i\right]_M $$ is the same for $i=0$ or 1 , and from the identification $$ \begin{aligned} {\left[M, \mathbb{C} P^2\right] } & \stackrel{\cong}{ } H^2(M, \mathbb{Z}) \ {[f] } & \longmapsto f^ u_0
\end{aligned}
$$
we conclude that $f_0$ and $f_1$ are homotopic as maps into $\mathbb{C} P^2$.

数学代写|拓扑学代写Topology代考|Framed cobordisms

We have seen that if $L_1, L_2 \subset M$ are links with $\left[L_1\right]=\left[L_2\right] \in H_1(M ; \mathbb{Z})$, then $L_1$ and $L_2$ are cobordant in $M . \dagger$ In general, however, a given framing on $L_1$ and $L_2$ does not extend over the cobordism. The following observation will be useful later on.

Write $\left(S^1, n\right)$ for a contractible loop in $M$ with framing $n \in \mathbb{Z}$ (by which we mean that $S^1$ and a second copy of $S^1$ obtained by pushing it away in the direction of one of the vectors in the frame have linking number $n)$. When writing $L=L^{\prime} \sqcup\left(S^1, n\right)$ it is understood that $\left(S^1, n\right)$ is not linked with any component of $L^{\prime}$.

Suppose we have two framed links $L_0, L_1 \subset M$ with $\left[L_0\right]=\left[L_1\right]$. Let $S \subset[0,1] \times M$ be an embedded surface with
$$
\partial S=L_1 \sqcup \bar{L}_0 \subset{1} \times M \sqcup{0} \times M .
$$
With $D^2$ a small disc embedded in $S$, the framing of $L_1$ and $L_2$ in $M$ extends to a framing of $S \backslash D^2$ in $[0,1] \times M$ (since $S \backslash D^2$ deformation retracts to a 1-dimensional complex containing $L_0$ and $L_1$, and over such a complex any orientable 2 -plane bundle is trivial). Now we embed a cylinder $[0,1] \times S^1$ into $[0,1] \times M$ such that
$$
\begin{gathered}
{[0,1] \times S^1 \cap{0} \times M=\emptyset,} \
{[0,1] \times S^1 \cap{1} \times M={1} \times S^1,}
\end{gathered}
$$
and
$$
[0,1] \times S^1 \cap\left(S \backslash \operatorname{Int}\left(D^2\right)\right)={0} \times S^1=\partial D^2,
$$
see Figure 4.2. This shows that $L_0$ is framed cobordant in $M$ to $L_1 \sqcup\left(S^1, n\right)$ for suitable $n \in \mathbb{Z}$.

拓扑学代考

数学代写|拓扑学代写Topology代考|Cobordism classes of links

现在，我们要将$H_1(M ; \mathbb{Z})$中的元素与$M$中链接的协同类关联起来。

定理4.2.4设$M$为一个封闭的定向三流形。任何同源类$c \in H_1(M ; \mathbb{Z})$都由$M$中嵌入的(1球的)定向链接$L_c$表示。两个链接$L_0, L_1$表示相同的类$\left[L_0\right]=\left[L_1\right]$，当且仅当它们在$M$中是定向协同的，也就是说，在$[0,1] \times M$中有一个嵌入的定向表面$S$
$$
\partial S=L_1 \sqcup \bar{L}_0 \subset{1} \times M \sqcup{0} \times M,
$$
其中$\sqcup$表示不连接连接，$\bar{L}$表示方向相反的连接$L$。
命题3.4.2已经证明了第一部分。我们保留这里使用的符号。
如果两个链接$L_0, L_1$在$M$中是协同的，那么很明显
$$
\left[L_0\right]=\left[L_1\right] \in H_1([0,1] \times M ; \mathbb{Z}) \cong H_1(M ; \mathbb{Z})
$$

相反，假设我们有两个链接$L_0, L_1 \subset M$和$\left[L_0\right]=$$\left[L_1\right]$。如上所述，为这些链接选择任意框架，并使用它来定义光滑映射$f_0, f_1: M \rightarrow S^2$，其具有共同的规则值$p \in S^2$，例如$f_i^{-1}(p)=L_i, i=0,1$。现在用标准嵌入的$\mathbb{C} P^1 \subset \mathbb{C} P^2$标识$S^2$。设$P \subset \mathbb{C} P^2$为$\mathbb{C} P^1$的第二个副本，以这样一种方式嵌入，即$[P]{\mathbb{C} P^2}=\left[\mathbb{C} P^1\right]{\mathbb{C} P^2}$和$P$仅在$p$中与$\mathbb{C} P^1$横向相交。这是可能的，因为$\mathbb{C} P^1 \subset \mathbb{C} P^2$有自交1。然后地图$f_0, f_1$，被看作是$\mathbb{C} P^2$的地图，横向到$P$，我们有$f_i^{-1}(P)=L_i, i=0,1$。因此
$$
f_i^* u_0=f_i^\left(P D[P]_{\mathbb{C} P^2}\right)=P D\left[f_i^{-1}(P)\right]_M=P D\left[L_i\right]_M $$与$i=0$或1相同，并且来自标识$$ \begin{aligned} {\left[M, \mathbb{C} P^2\right] } & \stackrel{\cong}{ } H^2(M, \mathbb{Z}) \ {[f] } & \longmapsto f^ u_0
\end{aligned}
$$
我们得出$f_0$和$f_1$是同伦映射到$\mathbb{C} P^2$的结论。

数学代写|拓扑学代写Topology代考|Framed cobordisms

我们已经看到，如果$L_1, L_2 \subset M$是与$\left[L_1\right]=\left[L_2\right] \in H_1(M ; \mathbb{Z})$的链接，那么$L_1$和$L_2$在$M . \dagger$中是协同的。但是，通常，在$L_1$和$L_2$上的给定框架不会扩展到协同上。下面的观察将在后面有用。

在$M$中使用帧$n \in \mathbb{Z}$为可收缩循环编写$\left(S^1, n\right)$(我们的意思是$S^1$和通过将其推向帧中的一个向量的方向而获得的$S^1$的第二个副本具有链接号$n)$)。在编写$L=L^{\prime} \sqcup\left(S^1, n\right)$时，请理解$\left(S^1, n\right)$不与$L^{\prime}$的任何组件链接。

假设我们有两个带框架的链接$L_0, L_1 \subset M$和$\left[L_0\right]=\left[L_1\right]$。设$S \subset[0,1] \times M$为嵌入面的
$$
\partial S=L_1 \sqcup \bar{L}_0 \subset{1} \times M \sqcup{0} \times M .
$$
$D^2$是一个嵌入在$S$中的小圆盘，$M$中的$L_1$和$L_2$的框架扩展到$[0,1] \times M$中的$S \backslash D^2$的框架(因为$S \backslash D^2$的变形缩回到包含$L_0$和$L_1$的一维复合体，在这样一个复合体上，任何可定向的2平面束都是微不足道的)。现在我们在$[0,1] \times M$中嵌入一个圆柱体$[0,1] \times S^1$，这样
$$
\begin{gathered}
{[0,1] \times S^1 \cap{0} \times M=\emptyset,} \
{[0,1] \times S^1 \cap{1} \times M={1} \times S^1,}
\end{gathered}
$$
和
$$
[0,1] \times S^1 \cap\left(S \backslash \operatorname{Int}\left(D^2\right)\right)={0} \times S^1=\partial D^2,
$$
见图4.2。这表明$L_0$是框架协作在$M$到$L_1 \sqcup\left(S^1, n\right)$为合适的$n \in \mathbb{Z}$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

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