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## avatest™帮您通过考试

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## 数学代写|拓扑学代写TOPOLOGY代考|Tight and overtwisted

In this section we are going to discuss a fundamental dichotomy of contact structures on 3-manifolds, introduced by Eliashberg [64], namely, the division of contact structures into tight and overtwisted ones. At first sight, the definition of theses types of contact structures looks slightly peculiar. We shall see later what motivated this definition, and why it has proved seminal for the development of 3 -dimensional contact topology.

Recall the definition of the standard overtwisted contact structure $\xi_{\text {ot }}$ on $\mathbb{R}^3$ given in Example 2.1.6. This was described by the equation (in cylindrical coordinates)
$$\cos r d z+r \sin r d \varphi=0$$
Let $\Delta$ be the $\operatorname{disc}{z=0, r \leq \pi} \subset \mathbb{R}^3$. The boundary $\partial \Delta$ of this disc is a Legendrian curve for $\xi_{\text {ot }}$, in fact, the disc $\Delta$ is tangent to $\xi_{\text {ot }}$ along its boundary. This means that the characteristic foliation $\Delta_{\xi_{\mathrm{ot}}}$ consists of all radial lines, with singular points at the origin and at all boundary points (Figure 4.9).

If the interior of $\Delta$ is pushed up slightly, the singular points at the boundary can be made to disappear. Only the singular point at the centre remains, and the characteristic foliation now looks as in Figure 4.10, with $\partial \Delta$ a closed leaf of the characteristic foliation $\Delta_{\xi}$. (We shall prove this presently by an explicit calculation in a related case.) We call $\Delta$ (in its perturbed or unperturbed form) the standard overtwisted disc.

For our discussion in the following chapter, it is useful to describe the properties of $\Delta$ in terms of the contact framing and the surface framing of Legendrian knots (Defns. 3.5.1 and 3.5.2).

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces in contact $3-$ manifolds

We now want to take a more systematic look at surfaces in contact $3-$ manifolds, with a view towards using them as a tool in the classification of contact structures. Obviously some of the material on hypersurfaces in contact manifolds of arbitrary dimension (Section 2.5.4) will be relevant here. I am going to reiterate some of the arguments from that section in the special 3-dimensional setting to spare the reader from having to leaf back and forth. Throughout I assume that $M$ is a 3 -manifold with oriented and cooriented contact structure $\xi=\operatorname{ker} \alpha$ (with $d \alpha$ defining the orientation of $\xi$ ), and $S \subset M$ an oriented surface embedded in $M$. Occasionally we allow $S$ to have boundary, but then there will be some control over the boundary, e.g. if it consists of Legendrian curves. All results can typically be proved for non-orientable surfaces by passing to a double cover.

As in Section 2.5.4 we identify a neighbourhood of $S$ in $M$ with $S \times \mathbb{R}$, and $S$ with $S \times{0}$, where we write $z \nmid$ for the $\mathbb{R}$-coordinate. We make this identification compatible with orientations: the orientation of $S$ followed by the natural orientation of $\mathbb{R}$ gives the orientation of $M$ (induced by $\xi$ ). We write the contact form $\alpha$ as
$$\alpha=\beta_z+u_z d z$$
where $\beta_z, z \in \mathbb{R}$, is a smooth family of 1 -forms on $S$, and $u_z: S \rightarrow \mathbb{R}$, a smooth family of functions. Then
$$d \alpha=d \beta_z-\dot{\beta}_z \wedge d z+d u_z \wedge d z$$
where the dot denotes the derivative with respect to $z$. Thus, the contact condition becomes
$$u_z d \beta_z+\beta_z \wedge\left(d u_z-\dot{\beta}_z\right)>0,$$
meaning that the $2-$ form on the left is a positive area form on $S$.

## 数学代写|拓扑学代写TOPOLOGY代考|Tight and overtwisted

$$\cos r d z+r \sin r d \varphi=0$$

## 数学代写|拓扑学代写TOPOLOGY代考|Surfaces in contact $3-$ manifolds

$$\alpha=\beta_z+u_z d z$$

$$d \alpha=d \beta_z-\dot{\beta}_z \wedge d z+d u_z \wedge d z$$

$$u_z d \beta_z+\beta_z \wedge\left(d u_z-\dot{\beta}_z\right)>0,$$

## MATLAB代写

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

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## avatest™帮您通过考试

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## 数学代写|拓扑学代写TOPOLOGY代考|Martinet’s construction

The first step towards the classification of overtwisted contact structures on 3 -manifolds is the following theorem of Martinet [175].

Theorem 4.1.1 (Martinet) Every closed, orientable 3-manifold $M$ admits a contact structure.

In view of the theorem of Lickorish and Wallace and the fact that $S^3$ admits a contact structure, Martinet’s theorem is a direct consequence of the following result.

Theorem 4.1.2 Let $\xi_0$ be a contact structure on a 3 -manifold $M_0$. Let $M$ be the manifold obtained from $M_0$ by a Dehn surgery along a knot $K$. Then $M$ admits a contact structure $\xi$ which coincides with $\xi_0$ outside the neighbourhood of $K$ where we perform surgery.

Proof By Theorem 3.3.1 we may assume that $K$ is positively transverse to $\xi_0$. Then, by the contact neighbourhood theorem (Example 2.5.16), we can find a tubular neighbourhood $\nu K$ of $K$ diffeomorphic to $S^1 \times D_{\delta_0}^2$, where $K$ is identified with $S^1 \times{\mathbf{0}}$ and $D_{\delta_0}^2$ denotes a disc of radius $\delta_0$, such that the contact structure $\xi_0$ is given as the kernel of $d \bar{\theta}+\bar{r}^2 d \bar{\varphi}$, with $\bar{\theta}$ denoting the $S^1$-coordinate and $(\bar{r}, \bar{\varphi})$ polar coordinates on $D_{\delta_0}^2$. Notice that this contact structure is rotationally symmetric about $S^1 \times{\mathbf{0}}$, so a transverse knot does not inherit any preferred framing from the contact structure. (We have seen in Definition 3.5.1 that the situation is markedly different for Legendrian knots.)

## 数学代写|拓扑学代写TOPOLOGY代考|2–plane fields on 3–manifolds

First we need the following well-known fact. It is unavoidable that at this point we use a little more algebraic or geometric topology than we have done so far. In order to ease the pain, I present three proofs, based on entirely different methods.

Theorem 4.2.1 Every closed, orientable 3-manifold $M$ is parallelisable, that is, the tangent bundle TM is trivial.

First Proof – obstruction theoretic The main point will be to show the vanishing of the second Stiefel-Whitney class $w_2(M)=w_2(T M) \in H^2\left(M ; \mathbb{Z}_2\right)$. Recall the following facts, which can be found in [35]; for the interpretation of Stiefel-Whitney classes as obstruction classes see also [188].
There are $\mathrm{Wu}$ classes $v_i \in H^i\left(M ; \mathbb{Z}_2\right)$ defined by
$$\left\langle\mathrm{Sq}^i(u),[M]\right\rangle=\left\langle v_i \cup u,[M]\right\rangle$$
for all $u \in H^{3-i}\left(M ; \mathbb{Z}_2\right)$, where Sq denotes the Steenrod squaring operations. Since $\mathrm{Sq}^i(u)=0$ for $i>3-i$, the only (potentially) non-zero Wu classes are $v_0=1$ and $v_1$. The $\mathrm{Wu}$ classes and the Stiefel-Whitney classes are related by $w_q=\sum_j \mathrm{Sq}^{q-j}\left(v_j\right)$. Hence $v_1=\mathrm{Sq}^0\left(v_1\right)=w_1$, which is the zero class, because $M$ is orientable. We conclude $w_2=0$.

Let $V_2\left(\mathbb{R}^3\right)=\mathrm{SO}(3) / \mathrm{SO}(1)=\mathrm{SO}(3)$ be the Stiefel manifold of oriented, orthonormal 2-frames in $\mathbb{R}^3$. This is connected, so there exists a section over the 1 -skeleton $\dagger$ of $M$ of the 2 -frame bundle $V_2(T M)$ associated with $T M$ (with a choice of Riemannian metric on $M$ understood $\ddagger$ ). The obstruction to extending this section over the 2 -skeleton is equal to $w_2$, which vanishes as we have just seen. The obstruction to extending the section over all of $M$ lies in $H^3\left(M ; \pi_2\left(V_2\left(\mathbb{R}^3\right)\right)\right)$, which is the zero group because of $\pi_2(\mathrm{SO}(3))=0$. (For that latter fact, recall that $\mathrm{SO}(3)$ is diffeomorphic to $\mathbb{R} P^3$, or appeal to the general result that the second homotopy group of any compact Lie group is trivial, see [37, Thm. V.7.1].)

## 数学代写|拓扑学代写TOPOLOGY代考|2–plane fields on 3–manifolds

$$\left\langle\mathrm{Sq}^i(u),[M]\right\rangle=\left\langle v_i \cup u,[M]\right\rangle$$

$(M, \xi)$中的横向结是一个嵌入$\gamma: S^1 \rightarrow M$，它无处不在地横向到$\xi$，即我们需要$\gamma^{\prime}(\theta) \notin \xi_{\gamma(\theta)}$对于所有$\theta \in S^1$。如果$\xi=\operatorname{ker} \alpha$是共取向的，就会说到一个正的或负的横向结，这取决于所有$\theta \in S^1$是$\alpha\left(\gamma^{\prime}(\theta)\right)>0$还是$\alpha\left(\gamma^{\prime}(\theta)\right)<0$。

## 数学代写|拓扑学代写TOPOLOGY代考|Front and Lagrangian projection

$$\alpha_{\mathrm{st}}=d z+x d y$$

$$\alpha_{\mathrm{st}}\left(\gamma^{\prime}\right)=z^{\prime}+x y^{\prime}$$

## MATLAB代写

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

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## avatest™帮您通过考试

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## 数学代写|拓扑学代写TOPOLOGY代考|Contact Hamiltonians

Let $X$ be a vector field on a contact manifold $(M, \xi=\operatorname{ker} \alpha)$. In Definition 1.5.7 and Lemma 1.5.8 we encountered special such vector fields called infinitesimal automorphisms of $\xi$ or $\alpha$, and we characterised such vector fields in terms of the Lie derivative. The next theorem relates infinitesimal automorphisms of $\xi$ to functions on $M$; the contact form plays an auxiliary role.

Theorem 2.3.1 With a fixed choice of contact form $\alpha$ there is a one-to-one correspondence between infinitesimal automorphisms $X$ of $\xi=\operatorname{ker} \alpha$ and smooth functions $H: M \rightarrow \mathbb{R}$. The correspondence is given by

• $X \longmapsto H_X=\alpha(X)$;
• $H \longmapsto X_H$, defined uniquely by $\alpha\left(X_H\right)=H$ and $i_{X_H} d \alpha=d H\left(R_\alpha\right) \alpha-d H$

The fact that $X_H$ is uniquely defined by the equations in the theorem follows as in the preceding section from the fact that $d \alpha$ is non-degenerate on $\xi$ and $R_\alpha \in \operatorname{ker}\left(d H\left(R_\alpha\right) \alpha-d H\right)$.

Proof Let $X$ be an infinitesimal automorphism of $\xi$. Set $H_X=\alpha(X)$ and write $d H_X+i_X d \alpha=\mathcal{L}X \alpha=\mu \alpha$ with $\mu: M \rightarrow \mathbb{R}$. Applying this last equation to $R\alpha$ yields $d H_X\left(R_\alpha\right)=\mu$. So $X$ satisfies the equations $\alpha(X)=H_X$ and $i_X d \alpha=d H_X\left(R_\alpha\right) \alpha-d H_X$. This means that we have $X_{H_X}=X$.

Conversely, given $H: M \rightarrow \mathbb{R}$ and with $X_H$ as defined in the theorem, we have
$$\mathcal{L}{X_H} \alpha=d\left(\alpha\left(X_H\right)\right)+i{X_H} d \alpha=d H\left(R_\alpha\right) \alpha,$$
so $X_H$ is an infinitesimal automorphism of $\xi$. Moreover, it is immediate from the definitions that $H_{X_H}=\alpha\left(X_H\right)=H$.

## 数学代写|拓扑学代写TOPOLOGY代考|Interlude: symplectic vector bundles

In Section 1.3 we saw how to construct an $\omega$-compatible complex structure on a symplectic vector space $(V, \omega)$, and we proved that the space $\mathcal{J}(\omega)$ of such complex structures is contractible. In the present interlude we extend this result from vector spaces to vector bundles. For further results on symplectic vector bundles see [177]. We also introduce the notion of an almost contact structure, which is the underlying bundle structure of a contact structure.

Definition 2.4.1 A symplectic vector bundle $(E, \omega)$ over a manifold $B$ is a (smooth) vector bundle $\pi: E \rightarrow B$ together with a symplectic linear form $\omega_b$ on each fibre $E_b=\pi^{-1}(b), b \in B$, with $\omega_b$ varying smoothly in $b$. Formally, this smoothness condition means that the map defined by $b \mapsto \omega_b$ is a smooth section of the bundle $\bigwedge^2 E^* \rightarrow B$, the second exterior power of the dual bundle of $E$.

Example 2.4.2 Given any vector bundle $E \rightarrow B$, there is a canonical symplectic bundle structure on the Whitney sum $E \oplus E^$, defined by $$\omega_b\left(X+\eta, X^{\prime}+\eta^{\prime}\right)=\eta\left(X^{\prime}\right)-\eta^{\prime}(X) \text { for } X, X^{\prime} \in E_b ; \eta, \eta^{\prime} \in E_b^ .$$

## 数学代写|拓扑学代写TOPOLOGY代考|Contact Hamiltonians

$X \longmapsto H_X=\alpha(X)$；

$H \longmapsto X_H$，由$\alpha\left(X_H\right)=H$和唯一定义 $i_{X_H} d \alpha=d H\left(R_\alpha\right) \alpha-d H$

$$\mathcal{L}{X_H} \alpha=d\left(\alpha\left(X_H\right)\right)+i{X_H} d \alpha=d H\left(R_\alpha\right) \alpha,$$

## 数学代写|拓扑学代写TOPOLOGY代考|Interlude: symplectic vector bundles

$$\begin{gathered} \delta_{\sharp}: \pi_1\left(X, x_1\right) \rightarrow \pi_1\left(X, x_2\right), \quad \delta_{\sharp}[\beta]=[i(\delta) * \beta * \delta], \ \gamma_{\sharp}: \pi_1\left(Y, f\left(x_1\right)\right) \rightarrow \pi_1\left(Y, f\left(x_2\right)\right), \quad \gamma_{\sharp}[\alpha]=[i(\gamma) * \alpha * \gamma] . \end{gathered}$$

$$\gamma_{\sharp} f_*[\beta]=\gamma_{\sharp}[f \beta]=[i(f \delta) * f \beta * f \delta]=[f(i(\delta) * \beta * \delta)]=f_* \delta_{\sharp}[\beta] .$$

## 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。

Posted on Categories:Topology, 拓扑学, 数学代写

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|拓扑学代写TOPOLOGY代考|Local Homeomorphisms and Sections

Definition 12.1 A continuous map $f: X \rightarrow Y$ is a local homeomorphism if for every $x \in X$ there exist open sets $A \subset X, B \subset Y$ such that $x \in A, f(A)=B$ and the restriction $f: A \rightarrow B$ is a homeomorphism.
Example 12.2 Every continuous, 1-1 and open map is a local homeomorphism.
Lemma 12.3 Every local homeomorphism $f: X \rightarrow Y$ is open, and the fibres $f^{-1}(y), y \in Y$, are discrete.

Proof We want to show that the image $f(V)$ of an open set $V \subset X$ is a neighbourhood of each of its points. That is to say, for every $y \in f(V)$ there exists $U \subset Y$ open such that $y \in U \subset f(V)$.

Let $x \in V$ be such that $f(x)=y$; by assumption there are open sets $A \subset X, B \subset$ $Y$ such that $x \in A, f(A)=B$ and the restriction $f: A \rightarrow B$ is a homeomorphism. In particular $y \in f(V \cap A)$, and $U=f(V \cap A)$ is open in $B$ so also open in $Y$.
For every $y \in Y$ and $x \in f^{-1}(y)$ there exists an open neighbourhood $x \in A$ for which the restriction $f: A \rightarrow Y$ is $1-1$. Hence $f^{-1}(y) \cap A={x}$, proving that the subspace topology on the fibres $f^{-1}(y)$ is discrete.

If $p: X \rightarrow Y$ is a map between sets, a function $s: Y \rightarrow X$ is called a section of $p$ if $p(s(y))=y$ for every $y \in Y$. A necessary condition for $p$ to have a section is that $p$ be onto; vice versa, the axiom of choice says exactly that any map admits sections.

In contrast-moving back to the topological world-continuous sections of continuous surjective maps do not exist, in general.

## 数学代写|拓扑学代写TOPOLOGY代考|Covering Spaces

Definition 12.5 Let $X$ be a connected space. A space $E$ together with a continuous map $p: E \rightarrow X$ is a covering space of $X$ if every point $x \in X$ is contained in an open set $V \subset X$ whose pre-image $p^{-1}(V)$ is the disjoint union of open sets $U_i$ with the property that $p: U_i \rightarrow V$ is a homeomorphism for every $i$.

The space $X$ is called the base (space) of the covering space, $E$ is the total space and $p$ is the covering map. The sets $p^{-1}(x), x \in X$, are called fibres of the covering space.

An open set $V \subset X$ is an admissible (open) set of the covering $p$ if it fulfils the condition of Definition 12.5. With other words $V \subset X$ is admissible if we can write $p^{-1}(V)=\cup_i U_i$, where:

1. every $U_i$ is open in $E$ and the restrictions $p: U_i \rightarrow V$ are homeomorphisms;
2. $U_i \cap U_j=\emptyset$ for every $i \neq j$.
Clearly, an open set contained in an admissible open set is still admissible. We will say that the covering space is trivial if the whole base $X$ is an admissible set.

If $p: E \rightarrow X$ is a covering space, from the definition every point $e \in E$ has an open neighbourhood homeomorphic to an open neighbourhood of $p(e)$. For later use we note that this implies that if $X$ is locally path connected, also $E$ is locally path connected.

Definition 12.6 A covering space $p: E \rightarrow X$ is connected if the total space $E$ is connected.

Example 12.7 Let $X$ be connected and $F$ a non-empty discrete space. The projection on the first factor $X \times F \rightarrow X$ is a trivial covering space, and it’s connected if and only if $F$ consists of one point.

## 数学代写|拓扑学代写TOPOLOGY代考|Local Homeomorphisms and Sections

$$C\left(\left(X, x_0\right),\left(Y, y_0\right)\right)=\left{f \in C(X, Y) \mid f\left(x_0\right)=y_0\right} .$$

## 数学代写|拓扑学代写TOPOLOGY代考|Path Homotopy

$$\Omega(X, a, b)={\alpha: I \rightarrow X \mid \alpha \text { continuous, } \alpha(0)=a, \alpha(1)=b}$$

$$\begin{gathered} *: \Omega(X, a, b) \times \Omega(X, b, c) \rightarrow \Omega(X, a, c), \quad \alpha * \beta(t)= \begin{cases}\alpha(2 t) & \text { if } 0 \leq t \leq \frac{1}{2}, \ \beta(2 t-1) & \text { if } \frac{1}{2} \leq t \leq 1 .\end{cases} \ i: \Omega(X, a, b) \rightarrow \Omega(X, b, a), \quad i(\alpha)(t)=\alpha(1-t) . \end{gathered}$$

$F(t, 0)=\alpha(t), F(t, 1)=\beta(t)$ 对于每个$t \in I$;

$F(0, s)=a, F(1, s)=b$ 对于每个$s \in I$。

$$F_s: I \rightarrow X, \quad F_s(t)=F(t, s),$$

## MATLAB代写

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

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## 数学代写|拓扑学代写TOPOLOGY代考|Noetherian Spaces

The upcoming result is very much used in algebraic geometry and commutative algebra.
Proposition 8.21 On an ordered set $(X, \leq)$ the following are equivalent:

1. every non-empty subset of $X$ contains maximal elements;
2. every countable ascending chain $\left{x_1 \leq x_2 \leq \cdots\right} \subset X$ stabilises (i.e. there’s an index $m \in \mathbb{N}$ such that $x_n=x_m$ for every $n \geq m$ ).

Proof For (1) $\Rightarrow(2)$ it’s enough to notice that any countable ascending chain $\left{x_1 \leq\right.$ $\left.x_2 \leq \cdots\right}$ contains a maximal element, say $x_m$, and therefore $x_n=x_m$ for every $n \geq m$

Let us prove (2) $\Rightarrow$ (1). By contradiction, suppose we have a non-empty $S \subset X$ with no maximal elements, i.e. ${y \in S \mid y>x} \neq \emptyset$ for every $x \in S$. By the axiom of choice there’s a function $f: S \rightarrow S$ such that $f(x)>x$ for every $x \in S$. Take $x_0 \in S$ : the ascending chain $\left{x_n=f^n\left(x_0\right) \mid n \in \mathbb{N}\right}$ does not stabilise.

Definition 8.22 A space is called Noetherian if every non-empty family of open sets has a maximal element for the inclusion.

By Proposition 8.21 a space is Noetherian if and only if every countable ascending chain stabilises.

Example 8.23 Let $\mathbb{K}$ be any field. The affine space $\mathbb{K}^n$ equipped with the Zariski topology (Example 3.11) is Noetherian. The proof is a simple consequence of Hilbert’s basis theorem, ${ }^1$ and as such we leave it to lecture courses on algebraic geometry and commutative algebra.

## 数学代写|拓扑学代写TOPOLOGY代考|A Long Exercise: Tietze’s Extension Theorem

Recall that a space is called normal when it is Hausdorff and disjoint closed sets have disjoint neighbourhoods. The exercises at the end of the section, solved in the given order, provide a proof of the following two results.

Lemma 8.29 (Urysohn’s lemma) Let $A, C$ be disjoint closed sets in a normal space $X$. There exists a continuous map $f: X \rightarrow[0,1]$ such that $f(x)=0$ when $x \in A$ and $f(x)=1$ when $x \in C$.

Theorem 8.30 (Tietze extension) Let $B$ be closed in a normal space $X, J \subset \mathbb{R} a$ convex subspace and $f: B \rightarrow J$ a continuous map. Then there exists a continuous map $g: X \rightarrow J$ such that $g(x)=f(x)$ for every $x \in B$.

For metric spaces Urysohn’s lemma is easy to prove: it suffices to recycle the argument of Proposition 7.31.

Let us remark that Lemma 8.29 is a special case of Theorem 8.30 when $J=[0,1]$ and $B=A \cup C$. The classical proof of Theorem 8.30 , for which we suggest consulting [Mu00, Du66], uses Urysohn’s lemma and the completeness of the space $B C(X, \mathbb{R})$ of continuous and bounded real functions on $X$.

## 数学代写|拓扑学代写TOPOLOGY代考|Noetherian Spaces

$X$的每个非空子集都包含最大元素;

## 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。