Posted on Categories:Riemann surface, 数学代写, 黎曼曲面

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|黎曼曲面代写Riemann surface代考|Definition and Simplest Properties

Consider a $g$-dimensional complex torus $\mathbb{C}^g / \Lambda$ where $\Lambda$ is a lattice of full rank:
$$\Lambda=A N+B M, \quad A, B \in g l(g, \mathbb{C}), N, M \in \mathbb{Z}^g,$$
and the $2 g$ columns of $A, B$ are $\mathbb{R}$-linearly independent. Non-constant meromorphic functions on $\mathbb{C}^g / \Lambda$ exist only (see, for example, [Sie71]) if the complex torus is an Abelian torus, i.e., if by an appropriate linear choice of coordinates on $\mathbb{C}^g$ the lattice (1.80) can be reduced to a special form: $A$ is a diagonal matrix of the form
$$A=2 \pi \mathrm{i} \operatorname{diag}\left(a_1=1, \ldots, a_g\right), \quad a_k \in \mathbb{N}, a_k \text { divides } a_{k+1},$$
and $B$ is a symmetric matrix with negative real part. An Abelian torus with $a_1=\ldots=a_g=1$ is called principally polarized. Jacobi varieties of Riemann surfaces are principally polarized Abelian tori. Meromorphic functions on Abelian tori are constructed in terms of theta functions, which are defined by their Fourier series.

Definition 31. Let $B$ be a symmetric $g \times g$ matrix with negative real part. The theta function is defined by the following series
$$\theta(z)=\sum_{m \in \mathbb{Z}^g} \exp \left{\frac{1}{2}(B m, m)+(z, m)\right}, \quad z \in \mathbb{C} .$$
Here
$$(B m, m)=\sum_{i j} B_{i j} m_i m_j, \quad(z, m)=\sum_j z_j m_j .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Theta Functions of Riemann Surfaces

From now on we consider an Abelian torus which is a Jacobi variety, $\mathbb{C} / \Lambda=$ $J a c(\mathcal{R})$. By combining the theta function with the Abel map, one obtains the following useful map on a Riemann surface:
$$\Theta(P):=\theta\left(\mathcal{A}{P_0}(P)-d\right), \quad \mathcal{A}{P_0}(P)=\int_{P_0}^P \omega .$$
Here we incorporated the base point $P_0 \in \mathcal{R}$ in the notation of the Abel map, and the parameter $d \in \mathbb{C}^g$ is arbitrary. The periodicity properties of the theta function (1.81) imply the following

Proposition 8. $\Theta(P)$ is an entire function on the universal covering $\tilde{\mathcal{R}}$ of $\mathcal{R}$. Under analytical continuation $\mathcal{M}{a_k}, \mathcal{M}{b_k}$ along a-and b-cycles on the Riemann surface, it is transformed as follows:
$$\begin{gathered} \mathcal{M}{a_k} \Theta(P)=\Theta(P), \ \mathcal{M}{b_k} \Theta(P)=\exp \left{-\frac{1}{2} B_{k k}-\int_{P_0}^P \omega_k+d_k\right} \Theta(P) . \end{gathered}$$
The zero divisor $(\Theta)$ of $\Theta(P)$ on $\mathcal{R}$ is well defined.
Theorem 22. The theta function $\Theta(P)$ either vanishes identically on $\mathcal{R}$ or has exactly $g$ zeros (counting multiplicities):
$$\operatorname{deg}(\Theta)=g .$$
Suppose $\Theta \not \equiv 0$. As in Sect. 1.4 consider the simply connected model $F_g$ of the Riemann surface. The differential $\mathrm{d} \log \Theta$ is well defined on $F_g$ and the number of zeros of $\Theta$ is equal to
$$\operatorname{deg}(\Theta)=\frac{1}{2 \pi \mathrm{i}} \int_{\partial F_g} \mathrm{~d} \log \Theta(P) .$$

## 数学代写|黎曼曲面代写Riemann surface代考|Definition and Simplest Properties

$$\Lambda=A N+B M, \quad A, B \in g l(g, \mathbb{C}), N, M \in \mathbb{Z}^g,$$

$$A=2 \pi \mathrm{i} \operatorname{diag}\left(a_1=1, \ldots, a_g\right), \quad a_k \in \mathbb{N}, a_k \text { divides } a_{k+1},$$
$B$是一个实部为负的对称矩阵。带$a_1=\ldots=a_g=1$的阿贝尔环面称为主极化环面。黎曼曲面的雅可比变体主要是极化的阿贝尔环面。阿贝尔环面上的亚纯函数是由函数构成的，而函数是由它们的傅立叶级数定义的。

$$\theta(z)=\sum_{m \in \mathbb{Z}^g} \exp \left{\frac{1}{2}(B m, m)+(z, m)\right}, \quad z \in \mathbb{C} .$$

$$(B m, m)=\sum_{i j} B_{i j} m_i m_j, \quad(z, m)=\sum_j z_j m_j .$$

## 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:Riemann surface, 数学代写, 黎曼曲面

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Hurwitz formula – Applications

6.1 Example: $f: \mathbb{P}^1 \rightarrow \mathbb{P}^1$ a polynomial of degree $d$. We have $g(R)=g(S)=0$, so RiemannHurwitz gives
$$-1=-d+\frac{1}{2} b$$
or $b=2(d-1)$.
To see why that is, we pull the following theorem out of our algebraic hat.
6.2 Theorem: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a polynomial mapping of degree $d$; then the total branching index over the finite branch points is exactly $(d-1)$.

This means that the equation $f(x)=y$ has multiple roots for precisely $(d-1)$ values of $y$, counting $y k$ times when the equation $f(x)-y$ has only $d-k$ distinct roots.
6.3 Remark: This is quite clear in examples such as $f(x)=x^n$, or for any polynomial of degree 2. In general, $f$ has multiple roots iff a certain expression – the discriminant of $f$ – vanishes; and $\operatorname{disc}(f(x)-\alpha)$ is a polynomial in $\alpha$ of degree $(d-1)$. For generic $f, f-\alpha$ will have a double root for precisely $(d-1)$ values of $\alpha$. Of course, reversing our arguments here will deduce thm. (6.2) from the Riemann-Hurwitz formula.

But we need another $(d-1)$ to agree with Riemann-Hurwitz. This of course comes from the point at $\infty$.

## 数学代写|黎曼曲面代写Riemann surface代考|Proof of Riemann-Hurwitz

To prove the Riemann-Hurwitz formula, we need to introduce a new notion – that of a triangulation of a surface – and give a rigorous definition of the Euler characteristic. To be completely rigorous, though, a mild digression on topological technology is required.
7.1 Definition: A topological space is second countable, or has a countable base, if it contains a countable family of open subsets $U_n$, such that every open set is a union of some of the $U_n$.
7.2 Example: A countable base for $\mathbb{R}$ is the collection of open intervals with rational endpoints. The exact same argument for this also proves the following.
7.3 Proposition: A metric space has a countable base iff it contains a dense countable subset.
7.4 Remark: Such metric spaces are often called separable.
Another easy observation is:
7.5 Proposition: A toplogical surface has a countable base iff it can be covered by countably many discs.

In particular, compact surfaces have a countable base. Every connected surface you can easily imagine has a countable base, but Prüfer has given an example of a connected surface admitting none. Such examples are necessarily quite pathological; it is common to exclude them by building the ‘countable base’ requirement into the definition of a surface.
7.6 Remark: It turns out that one does not exclude any interesting Riemann surfaces by insisting on the countable base condition. That is, it can be proved that every connected Riemann surface has a countable base, even if the condition was not included in the definition to begin with. (The proof is not obvious; see, for example, Springer, Introduction to Riemann Surfaces; you’ll also find Prüfer’s example there.)

The relevance of this topological techno-digression is the following theorem; ‘triangulable’ means pretty much what you’d think, but is defined precisely below.
7.7 Proposition: A connected surface is triangulable iff it admits a countable base. In particular, every Riemann surface is triangulable. (This was given a direct proof by Radò (1925).)

## 数学代写|黎曼曲面代写Riemann surface代考|The Riemann-Hurwitz formula – Applications

6.1示例:$f: \mathbb{P}^1 \rightarrow \mathbb{P}^1$次多项式$d$。我们有$g(R)=g(S)=0$, riemanhurwitz给出
$$-1=-d+\frac{1}{2} b$$

6.2定理:设$f: \mathbb{C} \rightarrow \mathbb{C}$为次$d$的多项式映射;那么有限个分支点上的总分支索引正好是$(d-1)$。

## 数学代写|黎曼曲面代写Riemann surface代考|Proof of Riemann-Hurwitz

7.1定义:一个拓扑空间是次可数的，或者有一个可数基，如果它包含一个可数的开子集族$U_n$，使得每个开集都是若干个$U_n$的并集。
7.2示例:$\mathbb{R}$的可数基数是具有有理端点的开区间的集合。同样的论证也证明了以下几点。
7.3命题:度量空间如果包含密集可数子集，则具有可数基。
7.4注:这种度量空间通常称为可分离空间。

7.6注:结果表明，坚持可数基条件并不会排除任何有趣的黎曼曲面。也就是说，可以证明每一个连通的黎曼曲面都有一个可数基，即使这个条件一开始没有包含在定义中。(证据并不明显;例如，参见Springer的《黎曼曲面导论》;你也可以在这里找到普莱尔的例子。)

## 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:Riemann surface, 数学代写, 黎曼曲面

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|黎曼曲面代写Riemann surface代考|Meromorphic functions and maps to P1

We now study the special case of holomorphic maps with target space $\mathbb{P}^1$. It turns out that we recover the more familiar notion of meromorphic function.
4.1 Definition: A function $f: U \rightarrow \mathbb{C} \cup{\infty}(U \subseteq \mathbb{C}$ open) is called meromorphic if it is holomorphic at every point where it has a finite value, whereas, near every point $z_0$ with $f\left(z_0\right)=\infty, f(z)=\phi(z) /\left(z-z_0\right)^n$ for some holomorphic function $\phi$, defined and non-zero around $z_0$. The positive number $n$ is the order of the pole at $z_0$.
4.2 Remark: Equivalently, we ask that, locally, $f=\phi / \psi$ with $\phi$ and $\psi$ holomorphic. We can always arrange that $\phi\left(z_0\right)$ or $\psi\left(z_0\right)$ are non-zero, by dividing out any $\left(z-z_0\right)$ power, and we define $a / 0=\infty$ for any $a \neq 0$.
4.3 Theorem: A meromorphic function on $U$ is the same as a holomorphic map $U \rightarrow \mathbb{P}^1$ which is not identically $\infty$.

Proof: Let $f$ be meromorphic. Clearly it defines a continuous map to $\mathbb{P}^1$, because $f(z) \rightarrow \infty$ near a pole. Clearly, also, it is holomorphic away from its poles. Holomorphicity near a pole $z_0$ means: for every function $g$, defined and holomorphic near $\infty \in \mathbb{P}^1, g \circ f$ is holomorphic near $z_0$. But $g$ is holomorphic at $\infty$ iff the function $h$ defined by
$$h(z)= \begin{cases}g(1 / z) & \text { if } z \neq 0 \ g(\infty) & \text { if } z=0\end{cases}$$
is holomorphic near 0. But then, $g \circ f=h(1 / f)=h\left(\left(z-z_0\right)^n / \phi(z)\right)$ which is holomorphic, being the composition of holomorphic functions. (Recall $\phi(z) \neq 0$ near $z_0$.)

Conversely, let $f: U \rightarrow \mathbb{P}^1$ be a holomorphic map. By definition, using the function $w \mapsto w$ defined on $\mathbb{C} \subset \mathbb{P}^1$, the composite function $f:\left(f^{-1}(\mathbb{C})=U \backslash f^{-1}(\infty)\right) \rightarrow \mathbb{C}$ is holomorphic; so we must only check the behaviour near the infinite value. For that, we use the function $w \mapsto 1 / w$ holomorphic on $\mathbb{P}^1 \backslash{0}$ and conclude that $1 / f$ is holomorphic on $U$, away from the zeroes of $f$. But then $f$ is meromorphic.

## 数学代写|黎曼曲面代写Riemann surface代考|Algebra with meromorphic functions

There is a slight difference between meromorphic functions and maps to $\mathbb{P}^1$; it stems from the condition that $f$ should not be identically $\infty$, to be called meromorphic. This has a significant consequence, as far as algebra is concerned:
4.7 Proposition: The meromorphic functions on a connected Riemann surface form a field, called the field of fractions of the Riemann surface.

Recall that a field is a set with associative and commutative operations, addition and multiplication, such that multiplication is distributive for addition; and, moreover, the ratio $a / b$ of any two elements, with $b$ not equal two zero, is defined and has the familiar property $a / b * b=a$.
Proof: This is clear from the local definition (4.1).
Remark: From another point of view, this may seem curious. Recall from calculus that certain arithmetic operations involving $\infty$ and 0 cannot be consistently defined: $\infty-\infty, \infty / \infty, 0 / 0$ and $\infty \cdot 0$ cannot be assigned meanings consistent with the usual arithmetic laws. For meromorphic functions, we assign a meaning to this undefined expressions by taking the limit of the nearby values of the function; the local expression (4.1) of a meromorphic function ensures that the limit exists.

## 数学代写|黎曼曲面代写Riemann surface代考|Meromorphic functions and maps to P1

4.1定义:函数 $f: U \rightarrow \mathbb{C} \cup{\infty}(U \subseteq \mathbb{C}$ 如果开(Open)在其有有限值的每一点上都是全纯的，则称其为亚纯的，而在每一点附近则称其为全纯的 $z_0$ 有 $f\left(z_0\right)=\infty, f(z)=\phi(z) /\left(z-z_0\right)^n$ 对于某个全纯函数 $\phi$，且周围是非零的 $z_0$． 正数 $n$ 杆子的顺序是多少 $z_0$．
4.2备注:同样，我们要求，在局部， $f=\phi / \psi$ 有 $\phi$ 和 $\psi$ 全纯的我们总是可以安排的 $\phi\left(z_0\right)$ 或 $\psi\left(z_0\right)$ 都是非零的，除以任何 $\left(z-z_0\right)$ 权力，我们定义 $a / 0=\infty$ 对于任何 $a \neq 0$．

$$h(z)= \begin{cases}g(1 / z) & \text { if } z \neq 0 \ g(\infty) & \text { if } z=0\end{cases}$$

## 数学代写|黎曼曲面代写Riemann surface代考|Algebra with meromorphic functions

4.7命题:连通黎曼曲面上的亚纯函数形成一个场，称为黎曼曲面的分数场。

## 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:Riemann surface, 数学代写, 黎曼曲面

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|黎曼曲面代写Riemann surface代考|What are Riemann surfaces?

Problem: Natural algebraic expressions have ‘ambiguities’ in their solutions; that is, they define multi-valued rather than single-valued functions.

In the real case, there is usually an obvious way to fix this ambiguity, by selecting one branch of the function. For example, consider $f(x)=\sqrt{x}$. For real $x$, this is only defined for $x \geq 0$, where we conventionally select the positive square root (Fig.1.1).

We get a continuous function $[0, \infty) \rightarrow \mathbb{R}$, analytic everywhere except at 0 . Clearly, there is a problem at 0 , where the function is not differentiable; so this is the best we can do.

In the complex story, we take ‘ $w=\sqrt{z}$ ‘ to mean $w^2=z$; but, to get a single-valued function of $z$, we must make a choice, and a continuous choice requires a cut in the domain.

A standard way to do that is to define ‘ $\sqrt{z}$ ‘ $: \mathbb{C} \backslash \mathbb{R}^{-} \rightarrow \mathbb{C}$ to be the square root with positive real part. There is a unique such, for $z$ away from the negative real axis. This function is continuous and in fact complex-analytic, or holomorphic, away from the negative real axis.

A different choice for $\sqrt{z}$ is the square root with positive imaginary part. This is uniquely defined away from the positive real axis, and determines a complex-analytic function on $\mathbb{C} \backslash \mathbb{R}^{+}$.
In formulae: $z=r e^{i \theta} \Longrightarrow \sqrt{z}=\sqrt{r} e^{i \theta / 2}$, but in the first case we take $-\pi<\theta<\pi$, and, in the second, $0<\theta<2 \pi$.

Either way, there is no continuous extension of the function over the missing half-line: when $z$ approaches a point on the half-line from opposite sides, the limits of the chosen values of $\sqrt{z}$ differ by a sign. A restatement of this familiar problem is: starting at a point $z_0 \neq 0$ in the plane, any choice of $\sqrt{z_0}$, followed continuously around the origin once, will lead to the opposite choice of $\sqrt{z_0}$ upon return; $z_0$ needs to travel around the origin twice, before $\sqrt{z_0}$ travels once.

## 数学代写|黎曼曲面代写Riemann surface代考|An interesting example

Let us conclude the lecture with an example of a Riemann surface with an interesting shape, which cannot be identified by projection (or in any other way) with the $z$-plane or the $w$-plane.
Start with the function $w=\sqrt{\left(z^2-1\right)\left(z^2-k^2\right)}$ where $k \in \mathbb{C}, k \neq \pm 1$, whose graph is the Riemann surface
$$T=\left{(z, w) \in \mathbb{C}^2 \mid w^2=\left(z^2-1\right)\left(z^2-k^2\right)\right}$$
There are two values for $w$ for every value of $z$, other than $z= \pm 1$ and $z= \pm k$, in which cases $w=0$. A real snapshot of the graph (when $k \in \mathbb{R}$ ) is indicated in Fig. (1.3), where the dotted lines indicate that the values are imaginary.

Near $z=1, z=1+\epsilon$ and the function is expressible as
$$w=\sqrt{\epsilon(2+\epsilon)(1+\epsilon+k)(1+\epsilon-k)}=\sqrt{\epsilon} \sqrt{2+\epsilon} \sqrt{(1+k)+\epsilon} \sqrt{(1-k)+\epsilon} .$$
A choice of sign for $\sqrt{2(1+k)(1-k)}$ leads to a holomorphic function $\sqrt{2+\epsilon} \sqrt{(1+k)+\epsilon} \sqrt{(1-k)+\epsilon}$ for small $\epsilon$, so $w=\sqrt{\epsilon} \times$ (a holomorphic function of $\epsilon$ ), and the qualitative behaviour of the function near $w=1$ is like that of $\sqrt{\epsilon}=\sqrt{z-1}$.

Similarly, $w$ behaves like the square root near $-1, \pm k$. The important thing is that there is no continuous single-valued choice of $w$ near these points: any choice of $w$, followed continuously round any of the four points, leads to the opposite choice upon return.

Defining a continuous branch for the function necessitates some cuts. The simplest way is to remove the open line segments joining 1 with $k$ and -1 with $-k$. On the complement of these segments, we can make a continuous choice of $w$, which gives an analytic function (for $z \neq \pm 1, \pm k$ ). The other ‘branch’ of the graph is obtained by a global change of sign.

## 数学代写|黎曼曲面代写Riemann surface代考|What are Riemann surfaces?

$\sqrt{z}$的另一个选项是虚部为正的平方根。这是唯一的定义远离正实轴，并确定在$\mathbb{C} \backslash \mathbb{R}^{+}$上的复解析函数。

## 数学代写|黎曼曲面代写Riemann surface代考|An interesting example

$$T=\left{(z, w) \in \mathbb{C}^2 \mid w^2=\left(z^2-1\right)\left(z^2-k^2\right)\right}$$

$$w=\sqrt{\epsilon(2+\epsilon)(1+\epsilon+k)(1+\epsilon-k)}=\sqrt{\epsilon} \sqrt{2+\epsilon} \sqrt{(1+k)+\epsilon} \sqrt{(1-k)+\epsilon} .$$

## 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:Riemann surface, 数学代写, 黎曼曲面

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|黎曼曲面代写Riemann surface代考|Continuous dynamics on Riemann surfaces

In this chapter, we shall look at dynamics from another point of view. Let $X$ be a Riemann surface; then $\operatorname{Hol}(X, X)$, endowed, as usual, with the compact-open topology, is a topological semigroup with identity, i. e., the operation given by the composition $(f, g) \mapsto f \circ g$ is continuous, associative, and has an identity. From this point of view, a semigroup homomorphism $\Phi: \mathbb{N} \rightarrow \operatorname{Hol}(X, X)$ is the same thing as the sequence of iterates of the single function $\Phi(1)$. In other words, in the previous chapters we have actually studied semigroup homomorphisms of $\mathbb{N}$ into $\operatorname{Hol}(X, X)$.

From this point of view, a natural generalization of the sequence of iterates is a one-parameter semigroup, i. e., a continuous semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow$ $\operatorname{Hol}(X, X)$. In this chapter, we shall thoroughly study these objects, aiming toward a complete classification. This will be possible because on Riemann surfaces with nonAbelian fundamental group every one-parameter semigroup $\Phi$ is trivial, i. e., $\Phi_t=\mathrm{id}_X$ for all $t \geq 0$. Furthermore, the one-parameter semigroups on other Riemann surfaces different from the disk can be classified (Section 5.3); so the main problem is the description of one-parameter semigroups on $\mathbb{D}$.

We shall actually provide several different descriptions of one-parameter semigroups on $\mathbb{D}$, useful in different contexts. We shall show how to relate one-parameter semigroups to Cauchy problems and ordinary differential equations, proving that a semigroup is completely determined by a holomorphic function $F: \mathbb{D} \rightarrow \mathbb{C}$, its infinitesimal generator. We shall give both a differential characterization and a completely explicit description of infinitesimal generators. Finally, we shall show how to replace $\mathbb{D}$ by another simply connected domain (in essentially a unique way) so to express a generic one-parameter semigroup in a particularly simple form; in a sense we shall transfer the analytic intricacies of one-parameter semigroups in a geometrically simple domain as $\mathbb{D}$ to the geometrical intricacies of a domain of definition for analytically very simple one-parameter semigroups, expressed in terms of affine maps.

## 数学代写|黎曼曲面代写Riemann surface代考|Algebraic semigroup homomorphisms

In this section, we collect some well-known facts about algebraic semigroups homomorphism of $\mathbb{R}^{+}$into other groups or semigroups that we shall need later. In this section, as operation on $\mathbb{R}^{+}$we shall always consider the sum, that makes $\mathbb{R}^{+}$in a semigroup but of course not a group. Moreover, we shall put $\mathbb{R}^{+}=(0,+\infty)$, so that $\left(\mathbb{R}^{+}, \cdot\right)$ is a topological group.

Definition 5.1.1. Let $G$ be a semigroup with identity element $e$. A function $\Phi: \mathbb{R}^{+} \rightarrow G$ is a semigroup homomorphism if $\Phi(0)=e$ and $\Phi(t+s)=\Phi(t) \circ \Phi(s)$ for all $t, s \geq 0$, where – denotes the operation in $G$. In the sequel, we shall often write $\Phi_t$ instead of $\Phi(t)$.
Lemma 5.1.2. Let $G$ be a group. Then:
(i) every semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow G$ can be extended in a unique way to a group homomorphism $\tilde{\Phi}: \mathbb{R} \rightarrow G$; in particular, if $G$ is a topological group and $\Phi$ is continuous, then $\Phi$ is continuous too;
(ii) if $G$ is finite, then every semigroup homomorphism $\Phi: \mathbb{R}^{+} \rightarrow G$ is trivial.
Proof. (i) The (unique) extension is obviously given by
$$\tilde{\Phi}(t)= \begin{cases}\Phi(t) & \text { if } t \geq 0 ; \ {[\Phi(-t)]^{-1}} & \text { if } t \leq 0,\end{cases}$$
where $[\cdot]^{-1}$ denotes the inverse operator in $G$. The continuity of $\tilde{\Phi}$ follows immediately from the continuity of the group operations.

## 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。