## 数学代写|拓扑学代写Topology代考|MAST90023

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|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

$\xi_0^{\prime}=\xi_0$ 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$t \in[0,1]$。

## 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|Darboux’s theorem

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

## 数学代写|拓扑学代写Topology代考|Darboux’s theorem

Theorem 2.5.1 (Darboux’s theorem) Let $\alpha$ be a contact form on the $(2 n+1)$-dimensional manifold $M$ and $p$ a point on $M$. Then there are coordinates $x_1, \ldots, x_n, y_1, \ldots, y_n, z$ on a neighbourhood $U \subset M$ of $p$ such that $p=(0, \ldots, 0)$ and
$$\left.\alpha\right|U=d z+\sum{j=1}^n x_j d y_j .$$
Remark 2.5.2 Observe that the $\operatorname{map}(\mathbf{x}, \mathbf{y}, z) \mapsto\left(\varepsilon \mathbf{x}, \varepsilon \mathbf{y}, \varepsilon^2 z\right)$ is a contactomorphism of the standard contact structure $\xi_{\text {st }}$ on $\mathbb{R}^{2 n+1}$ for any $\varepsilon \in \mathbb{R}^{+}$. Therefore it is an immediate consequence of the Darboux theorem that there is a contact embedding of the closed unit ball $B_{\mathrm{st}}$ in $\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$ into $(M, \xi=\operatorname{ker} \alpha)$ which sends the origin to $p$. Here ‘contact embedding of $B_{\mathrm{st}}$ ‘ simply means a contactomorphism of a small open neighbourhood of $B_{\mathrm{st}}$ in $\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$ onto its image in $(M, \xi)$; later we shall encounter a more general concept of contact embeddings.

In fact, by Proposition 2.1.8 and Example 2.1.10 there is a contactomorphism of $\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$ with a relatively compact subset of itself, and hence by scaling with a subset of $B_{\text {st }}$. So we can also construct a contactomorphism between $\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$ and a neighbourhood of $p$ in $(M, \xi)$.

Proof of Theorem 2.5.1 We may assume without loss of generality that $M=\mathbb{R}^{2 n+1}$ and $p=\mathbf{0}$ is the origin of $\mathbb{R}^{2 n+1}$. Choose linear coordinates
$$x_1, \ldots, x_n, y_1, \ldots y_n, z$$
on $\mathbb{R}^{2 n+1}$ such that
$$\text { on } T_0 \mathbb{R}^{2 n+1}:\left{\begin{array}{l} \alpha\left(\partial_z\right)=1, \quad i_{\partial_z} d \alpha=0, \ \partial_{x_j}, \partial_{y_j} \in \operatorname{ker} \alpha(j=1, \ldots, n), d \alpha=\sum_{j=1}^n d x_j \wedge d y_j . \end{array}\right.$$

## 数学代写|拓扑学代写Topology代考|Isotropic submanifolds

Let $L \subset(M, \xi=\operatorname{ker} \alpha)$ be an isotropic submanifold in a contact manifold with cooriented contact structure. Write $\left.(T L)^{\perp} \subset \xi\right|_L$ for the sub-bundle of $\left.\xi\right|L$ that is symplectically orthogonal to $T L$ with respect to the symplectic bundle structure $\left.d \alpha\right|{\xi}$. As we have seen in the preceding symplectic interlude, the conformal class of this symplectic bundle structure only depends on the contact structure $\xi$, not on the choice of contact form $\alpha$ defining $\xi$. So the bundle $(T L)^{\perp}$ is determined by $\xi$.

The fact that $L$ is isotropic implies $T L \subset(T L)^{\perp}$. Lemma 1.3 .3 allows us to make the following definition, see [241].
Definition 2.5.3 The quotient bundle
$$\operatorname{CSN}_M(L):=(T L)^{\perp} / T L$$
with the conformal symplectic structure induced by $d \alpha$ is called the conformal symplectic normal bundle of $L$ in $M$.
So the normal bundle $N L:=\left(\left.T M\right|_L\right) / T L$ of $L$ in $M$ can be split as
$$N L \cong\left(\left.T M\right|_L\right) /\left(\left.\xi\right|_L\right) \oplus\left(\left.\xi\right|_L\right) /(T L)^{\perp} \oplus \operatorname{CSN}_M(L) .$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Darboux’s theorem

$$\left.\alpha\right|U=d z+\sum{j=1}^n x_j d y_j .$$
2.5.2注意:对于任何$\varepsilon \in \mathbb{R}^{+}$, $\operatorname{map}(\mathbf{x}, \mathbf{y}, z) \mapsto\left(\varepsilon \mathbf{x}, \varepsilon \mathbf{y}, \varepsilon^2 z\right)$都是$\mathbb{R}^{2 n+1}$上的标准触点结构$\xi_{\text {st }}$的触点形态。因此，达布定理的直接结果是，$\left(\mathbb{R}^{2 n+1}, \xi_{\mathrm{st}}\right)$中的闭合单位球$B_{\mathrm{st}}$在$(M, \xi=\operatorname{ker} \alpha)$中有一个接触嵌入，它将原点发送到$p$。在这里，“$B_{\mathrm{st}}$的接触嵌入”仅仅意味着将$\left(\mathbb{R}^{2 n+1}, \xi_{\text {st }}\right)$中的一个小的开放邻域$B_{\mathrm{st}}$与其在$(M, \xi)$中的图像的接触形态;稍后我们将遇到更一般的接触嵌入概念。

$$x_1, \ldots, x_n, y_1, \ldots y_n, z$$

$$\text { on } T_0 \mathbb{R}^{2 n+1}:\left{\begin{array}{l} \alpha\left(\partial_z\right)=1, \quad i_{\partial_z} d \alpha=0, \ \partial_{x_j}, \partial_{y_j} \in \operatorname{ker} \alpha(j=1, \ldots, n), d \alpha=\sum_{j=1}^n d x_j \wedge d y_j . \end{array}\right.$$

## 数学代写|拓扑学代写Topology代考|Isotropic submanifolds

$L$是各向同性的这一事实意味着$T L \subset(T L)^{\perp}$。引理1.3 .3允许我们做出如下定义，参见[241]。
2.5.3商束
$$\operatorname{CSN}_M(L):=(T L)^{\perp} / T L$$

$$N L \cong\left(\left.T M\right|_L\right) /\left(\left.\xi\right|_L\right) \oplus\left(\left.\xi\right|_L\right) /(T L)^{\perp} \oplus \operatorname{CSN}_M(L) .$$

## 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|Cerf ’s theorem

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

## 数学代写|拓扑学代写Topology代考|Cerf ’s theorem

Write $\operatorname{Diff}(M)$ for the group of orientation-preserving diffeomorphisms of an orientable differential manifold $M$ (the group multiplication being given by composition of diffeomorphisms). Let $D^n$ be the $n$-dimensional unit disc in $\mathbb{R}^n$, and $S^{n-1}=\partial D^n$ its boundary, the standard $(n-1)$-dimensional unit sphere. Since diffeomorphisms of a manifold with boundary preserve that boundary, we have a natural restriction homomorphism
$$\begin{array}{ccc} \rho_n: \quad \operatorname{Diff}\left(D^n\right) & \longrightarrow \operatorname{Diff}\left(S^{n-1}\right) \ f & \longmapsto & \left.f\right|_{S^{n-1}} . \end{array}$$
The group $\Gamma_n$ is defined as
$$\Gamma_n=\operatorname{Diff}\left(S^{n-1}\right) / \operatorname{im} \rho_n .$$
In order to show that this is indeed a group, we need to prove that $\operatorname{im} \rho_n$ is a normal subgroup in $\operatorname{Diff}\left(S^{n-1}\right)$.

We begin with two lemmata. Write $\operatorname{Diff}_0\left(S^{n-1}\right)$ for the group of diffeomorphisms of $S^{n-1}$ that are isotopic to the identity.

## 数学代写|拓扑学代写Topology代考|Property P for knots

We begin by recalling a few facts about Dehn surgery on knots in 3 -manifolds, mostly to set up notation. For a textbook reference on this topic see [209] or [215].

Let $K$ be a knot in the 3 -sphere $S^3$ (or, more generally, in some oriented 3 manifold $M$ ), by which we mean a smoothly embedded copy of the circle $S^1$. Write $\nu K$ for a (closed) tubular neighbourhood of $K$. The neighbourhood $\nu K$ is diffeomorphic to a solid torus $S^1 \times D^2$, since this is the only orientable $D^2$-bundle over $S^1$. Let $C$ be the closure of the complement $S^3 \backslash \nu K$ of $\nu K$ in $S^3$. (Part of) the Mayer-Vietoris sequence $\dagger$ for $S^3=\nu K \cup C$ with $\nu K \cap C=T^2$ reads
\begin{aligned} & H_2\left(S^3\right) \rightarrow H_1\left(T^2\right) \quad \rightarrow \quad H_1(\nu K) \oplus H_1(C) \rightarrow H_1\left(S^3\right) \ & 0 \quad \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \quad \oplus \quad H_1(C) \rightarrow 0 \quad 0 . \ & \end{aligned}
We conclude that $H_1(C) \cong \mathbb{Z}$. We also see that on $T^2=\partial(\nu K)$ there are two distinguished curves, unique up to isotopy.
(1) The meridian $\mu$, defined as a simple closed curve that generates the kernel of the homomorphism $H_1\left(T^2\right) \rightarrow H_1(\nu K)$.
(2) The preferred longitude $\lambda$, a simple closed curve that generates the kernel of the homomorphism $H_1\left(T^2\right) \rightarrow H_1(C)$.

We assume that $S^3$ is equipped with its standard orientation as the boundary of $D^4 \cdot \ddagger$ We give $T^2=\partial(\nu K)$ the boundary orientation. We also assume $K$ to be oriented. Then $\lambda$ can be oriented by requiring it to be isotopic to $K$ in $\nu K$ as oriented curve; the orientation we choose for $\mu$ is the one that turns $\mu, \lambda$ into a positive basis for that homology group. (Occasionally we allow ourselves to denote a simple closed curve on $T^2$ by the same symbol as the class it represents in $H_1\left(T^2\right)$, since that class determines the curve up to isotopy.) This is illustrated in Figure 1.5, with the standard (right-hand) orientation of ambient 3-space.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Cerf ’s theorem

$$\begin{array}{ccc} \rho_n: \quad \operatorname{Diff}\left(D^n\right) & \longrightarrow \operatorname{Diff}\left(S^{n-1}\right) \ f & \longmapsto & \left.f\right|_{S^{n-1}} . \end{array}$$

$$\Gamma_n=\operatorname{Diff}\left(S^{n-1}\right) / \operatorname{im} \rho_n .$$

## 数学代写|拓扑学代写Topology代考|Property P for knots

\begin{aligned} & H_2\left(S^3\right) \rightarrow H_1\left(T^2\right) \quad \rightarrow \quad H_1(\nu K) \oplus H_1(C) \rightarrow H_1\left(S^3\right) \ & 0 \quad \rightarrow \mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \quad \oplus \quad H_1(C) \rightarrow 0 \quad 0 . \ & \end{aligned}

(1)子午线$\mu$，定义为生成同态核$H_1\left(T^2\right) \rightarrow H_1(\nu K)$的简单封闭曲线。
(2)首选经度$\lambda$，一条生成同态核$H_1\left(T^2\right) \rightarrow H_1(C)$的简单封闭曲线。

## 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|Universal Coverings

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|Universal Coverings

Definition 13.29 A covering $u: \widetilde{X} \rightarrow X$ is said to be universal if the total space $\widetilde{X}$ is connected and simply connected.

We saw in Example 13.22 that universal coverings are regular. In particular, if $u: \widetilde{X} \rightarrow X$ is universal then $\operatorname{Aut}(\widetilde{X}, u)$ acts freely and transitively on fibres, and $\pi_1(X) \simeq \operatorname{Aut}(\widetilde{X}, u)$ are isomorphic.

Proposition 13.30 (Universal property of universal coverings) Let $u: \widetilde{X} \rightarrow X$ be a universal covering. For every covering $p: E \rightarrow X$ and any points $\tilde{x} \in \tilde{X}, e \in E$ such that $u(\tilde{x})=p(e)$, there exists a unique covering morphism $\phi: \widetilde{X} \rightarrow E$ such that $\phi(\tilde{x})=e$. In particular, all universal coverings of a space $X$ are isomorphic to one another.

Proof Since $0=u_* \pi_1(\tilde{X}, \tilde{x}) \subset p_* \pi_1(E, e), \phi$ exists by virtue of Theorem 13.18. Additionally, if $p: E \rightarrow X$ is universal then the previous arguments show that there’s a covering morphism $\psi: E \rightarrow \widetilde{X}$ with $\psi(e)=\tilde{x}$. But $\widetilde{X}$ and $E$ are connected by definition, so the lift’s uniqueness forces $\phi \psi$ and $\psi \phi$ to be identity maps.

This proves the uniqueness of universal coverings. The remaining part of the section is devoted entirely to the issue of existence. We begin with a simple necessary condition.

## 数学代写|拓扑学代写Topology代考|Coverings with Given Monodromy

Consider a covering $p: E \rightarrow X$, a point $x \in X$ and the monodromy action
$$p^{-1}(x) \times \pi_1(X, x) \rightarrow p^{-1}(x) .$$
It’s not hard to show that $E$ is connected if and only if the monodromy action is transitive. In fact if $E$ is connected, for every pair $a, b \in p^{-1}(x)$ we can find a path $\alpha \in \Omega(E, a, b)$ and hence $b=a \cdot[p \alpha]$. Conversely, if the monodromy is transitive, the fibre $p^{-1}(x)$ is contained in a path component. Given any point $a \in E$ we choose a path $\alpha: I \rightarrow X$ such that $\alpha(0)=p(a), \alpha(1)=x$. The lift $\alpha_a: I \rightarrow E$ joins $a$ to some point in $p^{-1}(x)$, so that $a$ belongs in the same connected component where $p^{-1}(x)$ lies.

We saw already, in Theorem 13.1, that the stabiliser of any $e \in p^{-1}(x)$, i.e. the subgroup
$$\operatorname{Stab}(e)=\left{a \in \pi_1(X, x) \mid e \cdot a=e\right},$$
coincides with $p_* \pi_1(E, e)$. In particular the covering $p: E \rightarrow X$ is universal if and only if the monodromy is free and transitive. Moreover, the covering is regular if and only if the monodromy acts transitively and all stabilisers are normal subgroups.
Theorem 13.35 Let $X$ be connected, locally path connected and semi-locally simply connected. For every non-empty set $T$ and every right action
$$T \times \pi_1(X, x) \stackrel{\bullet}{\longrightarrow} T$$
there exists a covering $p: E \rightarrow X$ and a bijection $\phi: T \rightarrow p^{-1}(x)$ such that $\phi(t \bullet a)=\phi(t) \cdot a$, for every $t \in T$ and $a \in \pi_1(X, x)$. The pair $(p, \phi)$ is unique up to isomorphism.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Coverings with Given Monodromy

$$p^{-1}(x) \times \pi_1(X, x) \rightarrow p^{-1}(x) .$$

$$\operatorname{Stab}(e)=\left{a \in \pi_1(X, x) \mid e \cdot a=e\right},$$

$$T \times \pi_1(X, x) \stackrel{\bullet}{\longrightarrow} T$$

## 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|实分析作业代写Real analysis代考|Operator Norm

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

## 数学代写|实分析作业代写Real analysis代考|Operator Norm

This section works with linear functions from $n$-dimensional column-vector space to $m$-dimensional column-vector space. It will have applications within this chapter both when the scalars are real and when the scalars are complex. To be neutral let us therefore write $\mathbb{F}$ for $\mathbb{R}$ or $\mathbb{C}$. Material on the correspondence between linear functions and matrices may be found in Section A7 of the appendix.
Specifically let $L\left(\mathbb{F}^n, \mathbb{F}^m\right)$ be the vector space of all linear functions from $\mathbb{F}^n$ into $\mathbb{F}^m$. This space corresponds to the vector space of $m$-by- $n$ matrices with entries in $\mathbb{F}$, as follows: In the notation in Section $A 7$ of the appendix, we let $\left(e_1, \ldots, e_n\right)$ be the standard ordered basis of $\mathbb{F}^n$, and $\left(u_1, \ldots, u_m\right)$ the standard ordered basis of $\mathbb{F}^m$. We define a dot product in $\mathbb{F}^m$ by
$$\left(a_1, \ldots, a_m\right) \cdot\left(b_1, \ldots, b_m\right)=a_1 b_1+\cdots+a_m b_m$$
with no complex conjugations involved. The correspondence of a linear function $T$ in $L\left(\mathbb{F}^n, \mathbb{F}^m\right)$ to a matrix $A$ with entries in $\mathbb{F}$ is then given by $A_{i j}=T\left(e_j\right) \cdot u_i$.
Let $|\cdot|$ denote the Euclidean norm on $\mathbb{F}^n$ or $\mathbb{F}^m$, given as in Section II.1 by the square root of the sum of the absolute values squared of the entries. The Euclidean norm makes $\mathbb{F}^n$ and $\mathbb{F}^m$ into metric spaces, the distance between two points being the Euclidean norm of the difference.

Proposition 3.1. If $T$ is a member of the space $L\left(\mathbb{F}^n, \mathbb{F}^m\right)$ of linear functions from $\mathbb{F}^n$ to $\mathbb{F}^m$, then there exists a finite $M$ such that $|T(x)| \leq M|x|$ for all $x$ in $\mathbb{F}^n$. Consequently $T$ is uniformly continuous on $\mathbb{F}^n$.

PROOF. Each $x$ in $\mathbb{F}^n$ has $x=\sum_{j=1}^n\left(x \cdot e_j\right) e_j$, and linearity gives $T(x)=$ $\sum_{j=1}^n\left(x \cdot e_j\right) T\left(e_j\right)$. Thus
$$|T(x)|=\left|\sum_{j=1}^n\left(x \cdot e_j\right) T\left(e_j\right)\right| \leq \sum_{j=1}^n\left|T\left(e_j\right)\right|\left|x \cdot e_j\right| .$$
The expression $x \cdot e_j$ is just the $j^{\text {th }}$ entry of $x$, and hence $\left|x \cdot e_j\right| \leq|x|$. Therefore $|T(x)| \leq\left(\sum_{j=1}^n\left|T\left(e_j\right)\right|\right)|x|$, and the first conclusion has been proved with $M=\sum_{j=1}^n\left|T\left(e_j\right)\right|$. Replacing $x$ by $x-y$ gives
$$|T(x)-T(y)|=|T(x-y)| \leq M|x-y|,$$
and uniform continuity of $T$ follows with $\delta=\epsilon / M$.

## 数学代写|实分析作业代写Real analysis代考|Nonlinear Functions and Differentiation

We begin a discussion of more general functions between Euclidean spaces by defining the multivariable derivative for such a function and giving conditions for its existence. Let $E$ be an open set in $\mathbb{R}^n$, and let $f: E \rightarrow \mathbb{R}^m$ be a function. We can write $f(x)=\left(\begin{array}{c}f_1(x) \ \vdots \ f_m(x)\end{array}\right)$, where $f_i(x)=f(x) \cdot u_i$. Then $f(x)=\sum_{i=1}^m f_i(x) u_i$. The functions $f_i: E \rightarrow \mathbb{R}$ are called the components of $f$. The associated partial derivatives are given by
$$\frac{\partial f_i}{\partial x_j}(x)=\left.\frac{d}{d t} f_i\left(x+t e_j\right)\right|{t=0}$$ We say that $f$ is differentiable at $x$ in $E$ if there is some $T$ in $L\left(\mathbb{R}^n, \mathbb{R}^m\right)$ with $$\lim {h \rightarrow 0} \frac{|f(x+h)-f(x)-T(h)|}{|h|}=0 .$$
The linear function $T$ is unique if it exists. In fact, if $T_1$ and $T_2$ both serve as $T$ in this limit relation, then we write
$$T_2(h)-T_1(h)=\left(f(x+h)-f(x)-T_1(h)\right)-\left(f(x+h)-f(x)-T_2(h)\right)$$
and find that
\begin{aligned} \frac{\left|T_1(h)-T_2(h)\right|}{|h|} & \leq \frac{\left|f(x+h)-f(x)-T_1(h)\right|}{|h|}+\frac{\left|f(x+h)-f(x)-T_2(h)\right|}{|h|} \ & \longrightarrow 0 . \end{aligned}

If $T_1 \neq T_2$, choose some $v \in \mathbb{R}^n$ with $|v|=1$ and $T_1(v) \neq T_2(v)$. As a nonzero real parameter $t$ tends to 0 , we must have
\begin{aligned} & \left|T_1(v)-T_2(v)\right| \ & \quad=|t v|^{-1}\left|\left(f(x+t v)-f(x)-T_1(t v)\right)-\left(f(x+t v)-f(x)-T_2(t v)\right)\right| \ & \longrightarrow 0 . \end{aligned}

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Operator Norm

$$\left(a_1, \ldots, a_m\right) \cdot\left(b_1, \ldots, b_m\right)=a_1 b_1+\cdots+a_m b_m$$

(i) $d\left(z_1, z_2\right) \geq 0$当且仅当$z_1=z_2$，
(ii) $d\left(z_1, z_2\right)=d\left(z_2, z_1\right)$;
(iii) $d\left(z_1, z_2\right) \leq d\left(z_1, z_3\right)+d\left(z_3, z_2\right)$。

$\left{z_n\right}$序列从$\mathbb{C}$收敛到$z$有两种可能的解释:要么$\left{\operatorname{Re} z_n\right}$收敛到$\operatorname{Re} z$，要么$\left{\operatorname{Im} z_n\right}$收敛到$\operatorname{Im} z$，要么$d\left(z_n, z\right)$收敛到$\mathbb{R}$。这些解释是一样的，因为
$$\max {\operatorname{Re} w, \operatorname{Im} w} \leq|w| \leq \sqrt{2} \max {\operatorname{Re} w, \operatorname{Im} w}$$

## 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|Quotients by Properly Discontinuous Actions

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|Quotients by Properly Discontinuous Actions

Definition 12.15 Let $G$ be a subgroup of the group $\operatorname{Homeo}(E)$ of homeomorphisms of a space $E$. The group $G$ is said to act properly discontinuously if every point $e \in E$ has a neighbourhood $U$ such that $g(U) \cap U=\emptyset$ for any $g \in G$ different from the identity.

isomorphic to $\mathbb{Z}$, that acts properly discontinuously.
Example 12.17 The subgroup in Homeo $\left(\mathbb{R}^2-{0}\right)$ generated by the multiplication by a number $\lambda>1$ acts in a properly discontinuous fashion. acting properly discontinuously. If $E / G$ is connected, then the quotient map $p: E \rightarrow$ $E / G$ is a covering map.

Proof Fix $e \in E$ and choose an open set $U \subset E$ such that $e \in U$ and $g(U) \cap U=\emptyset$ for every $g$ different from the identity.
Proposition 5.15 implies that $p: E \rightarrow E / G$ is an open map, and
$$p^{-1}(p(U))=\cup{g(U) \mid g \in G}$$
So we just need to prove that, for any $g \in G$, the open sets $g(U)$ are disjoint and that $p: g(U) \rightarrow p(U)$ is a homeomorphism.

Since $g(U) \cap h(U)=h\left(h^{-1} g(U) \cap U\right)$, it follows $g(U) \cap h(U)=\emptyset$ for every $g \neq$ $h$. The quotient map $p: U \rightarrow p(U)$ is open and bijective hence a homeomorphism. The map $p: g(U) \rightarrow p(U)$ is the composite of the homeomorphisms $g^{-1}: g(U) \rightarrow$ $U$ with $p: U \rightarrow p(U)$.

## 数学代写|拓扑学代写Topology代考|Lifting Homotopies

Definition 12.23 Let $f: Y \rightarrow X$ be a continuous map and $p: E \rightarrow X$ a covering space. A continuous mapping $g: Y \rightarrow E$ is called a lift of $f$ when the diagram commutes, i.e. $f=p g$.
Lemma 12.24 For any covering space $p: E \rightarrow X$ the diagonal $\Delta \subset E \times E$ is open and closed in the fibred product
$$E \times_X E={(u, v) \in E \times E \mid p(u)=p(v)} .$$
Proof Take $(e, e) \in \Delta$ and choose an open set $U \subset E$ such that $e \in U$ and the restriction $p: U \rightarrow X$ is $1-1$. Then
$$(U \times U) \cap\left(E \times_X E\right)=U \times_X U$$
is an open neighbourhood of $(e, e)$ in the fibred product. On the other hand
$$(U \times U) \cap\left(E \times_X E\right)={(u, v) \in U \times U \mid p(u)=p(v)} \subset \Delta,$$
proving that $\Delta$ is a neighbourhood of any of its points, inside the fibred product.
Conversely, if $\left(e_1, e_2\right) \in E \times_X E-\Delta$ we pick an admissible open set $V$ containing $p\left(e_1\right)=p\left(e_2\right)$. Since $e_1 \neq e_2$, there exist disjoint open sets $U_1, U_2 \subset p^{-1}(V)$ such that $e_1 \in U_1, e_2 \in U_2$. Therefore
$$\left(e_1, e_2\right) \in\left(U_1 \times U_2\right) \cap\left(E \times_X E\right) \subset E \times_X E-\Delta,$$
so that the diagonal is closed in the fibred product.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Quotients by Properly Discontinuous Actions

$$\pi_1(f): \pi_1(X, a) \rightarrow \pi_1(Y, f(a)), \quad \pi_1(f)([\alpha])=[f \alpha]$$

$$F: I^2 \rightarrow X, \quad F(t, s)=R(\beta(t), s)$$

## 有限元方法代写

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

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

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

## 数学代写|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

Definition 10.1 A space is locally connected if every point has local basis of connected neighbourhoods.

From Lemma 4.28 the connected components of a locally connected space are open. While general connected spaces may not be locally connected (Exercise 10.1), open sets in $\mathbb{R}^n$ are locally connected, and the product of two locally connected spaces is locally connected.

Definition 10.2 Let $X$ be a topological space. Denote by $\pi_0(X)=X / \sim$ the quotient space under the relation $\sim$ that identifies points connected by a path in $X$.

To be more precise, for any two points $x, y \in X$ one defines the set of paths from $x$ to $y$ :
$$\Omega(X, x, y)={\alpha:[0,1] \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=x, \alpha(1)=y}$$
and then
$$\pi_0(X)=X / \sim, \quad \text { where } \quad x \sim y \Longleftrightarrow \Omega(X, x, y) \neq \emptyset$$
We have to make sure $\sim$ is an equivalence relation.
Reflexivity. To prove $x \sim x$ we consider the constant path
$$1_x:[0,1] \rightarrow X, \quad 1_x(t)=x \text { for every } t \in[0,1] .$$
Symmetry. For every $x, y \in X$ we have the path-reverting operator
$$i: \Omega(X, x, y) \rightarrow \Omega(X, y, x), \quad i(\alpha)(t)=\alpha(1-t),$$
that is clearly invertible. In particular $\Omega(X, x, y)$ is empty precisely when $\Omega(X, y, x)$ is empty.
Transitivity. We just consider the product of paths (or composite)
$$*: \Omega(X, x, y) \times \Omega(X, y, z) \rightarrow \Omega(X, x, z), \quad(\alpha, \beta) \mapsto \alpha * \beta,$$
where
$$\alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq 1 / 2 \ \beta(2 t-1) & \text { if } 1 / 2 \leq t \leq 1\end{cases}$$

## 数学代写|拓扑学代写Topology代考|Homotopy

Definition 10.8 Two continuous maps $f_0, f_1: X \rightarrow Y$ are said to be homotopic if there is a continuous function
$$F: X \times[0,1] \rightarrow Y$$
such that $F(x, 0)=f_0(x)$ and $F(x, 1)=f_1(x)$ for every $x \in X$. Such an $F$ is called a homotopy between $f_0$ and $f_1$.

To help one ‘visualise’ the meaning of the above definition let’s write $f_t(x)=$ $F(x, t)$ for every $(x, t) \in X \times[0,1]$. Then for any $t \in[0,1]$ the map
$$f_t: X \rightarrow Y$$
is continuous. When $t=0$ we recover $f_0$, which deforms in a continuous way, as $t$ varies, until it becomes $f_1$ for $t=1$.

Example 10.9 Let $Y \subset \mathbb{R}^n$ be a convex subspace. For any topological space $X$, two continuous maps $f_0, f_1: X \rightarrow Y$ are homotopic: it suffices to define the homotopy as
$$F: X \times[0,1] \rightarrow Y, \quad F(x, t)=(1-t) f_0(x)+t f_1(x)$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Locally Connected Spaces and the Functor π0

$$\Omega(X, x, y)={\alpha:[0,1] \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=x, \alpha(1)=y}$$

$$\pi_0(X)=X / \sim, \quad \text { where } \quad x \sim y \Longleftrightarrow \Omega(X, x, y) \neq \emptyset$$

$$1_x:[0,1] \rightarrow X, \quad 1_x(t)=x \text { for every } t \in[0,1] .$$

$$i: \Omega(X, x, y) \rightarrow \Omega(X, y, x), \quad i(\alpha)(t)=\alpha(1-t),$$

$$*: \Omega(X, x, y) \times \Omega(X, y, z) \rightarrow \Omega(X, x, z), \quad(\alpha, \beta) \mapsto \alpha * \beta,$$

$$\alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq 1 / 2 \ \beta(2 t-1) & \text { if } 1 / 2 \leq t \leq 1\end{cases}$$

## 数学代写|拓扑学代写Topology代考|Homotopy

$$F: X \times[0,1] \rightarrow Y$$

## 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。