## 数学代写|实分析作业代写Real analysis代考|Hypergeometric series

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|Hypergeometric series

$$1+\frac{\alpha \cdot \beta}{1 . \gamma} x+\frac{\alpha(\alpha+1) \beta(\beta+1)}{1.2 \cdot \gamma(\gamma+1)} x^2+\frac{\alpha(\alpha+1)(\alpha+2) \beta(\beta+1)(\beta+2)}{1.2 .3 \cdot \gamma(\gamma+1)(\gamma+2)} x^3+\cdots$$
where $\alpha, \beta, \gamma, x>0$.
Ignoring the first term, let $\sum_1^{\infty} u_n$ be the series.
Then $u_n=\frac{\alpha(\alpha+1) \cdots(\alpha+n-1) \beta(\beta+1) \cdots(\beta+n-1)}{1.2 \cdots \cdot n \gamma(\gamma+1) \cdots(\gamma+n-1)} x^n$ for $n \geq 1$.
$\frac{u_{n+1}}{u_n}=\frac{(\alpha+n)(\beta+n)}{(1+n)(\gamma+n)} x$ and $\lim {n \rightarrow \infty} \frac{u{n+1}}{u_n}=x$.
By D’Alembert’s ratio test, $\Sigma u_n$ is convergent if $01$.
When $x=1$,
\begin{aligned} & \frac{u_n}{u_{n+1}}=\frac{(n+1)(n+\gamma)}{(n+\alpha)(n+\beta)} \ & =1+\left(\frac{(\gamma+1-\alpha-\beta) n+(\gamma-\alpha \beta)}{n^2+(\alpha+\beta) n+\alpha \beta}\right) \ & =1+\left(\frac{\gamma+1-\alpha-\beta}{n}+\frac{\gamma-\alpha \beta}{n^2}\right)\left[1+\frac{\alpha+\beta}{n}+\frac{\alpha \beta}{n^2}\right]^{-1} \ & =1+\left(\frac{\gamma+1-\alpha-\beta}{n}+\frac{\gamma-\alpha \beta}{n^2}\right)\left[1-\frac{\alpha+\beta}{n}-\frac{\alpha \beta}{n^2}+\cdots\right] \ & =1+\frac{\gamma+1-\alpha-\beta}{n}+\frac{1}{n^2}[(\gamma-\alpha \beta)-(\alpha+\beta)(\gamma+1-\alpha-\beta)+\text { terms containing } \end{aligned}
$\frac{1}{n}$ and higher powers of $\frac{1}{n}$ ]
$=1+\frac{\gamma+1-\alpha-\beta}{n}+\frac{\phi(n)}{n^2}$, where $\lim _{n \rightarrow \infty} \phi(n)$ is finite and therefore ${\phi(n)}$ is bounded.
By Gauss’s test, when $x=1$,
$\Sigma u_n$ is convergent if $\gamma+1-\alpha-\beta>1$ and
$\Sigma u_n$ is divergent if $\gamma+1-\alpha-\beta \leq 1$.
Therefore the series is convergent if $01$. When $x=1$, the series is convergent if $\gamma>\alpha+\beta$ and divergent if $\gamma \leq \alpha+\beta$.

## 数学代写|实分析作业代写Real analysis代考|Series of arbitrary terms

Let $\Sigma u_n$ be a series of positive and negative real numbers.
Let $u_n^{\prime}=\left|u_n\right|$. Then $\Sigma u_n^{\prime}$ is a series of positive real numbers.
If $\Sigma u_n^{\prime}$ is convergent then $\Sigma u_n$ is said to be an absolutely convergent series.
Theorem 6.4.1. An absolutely convergent series is convergent.
Proof. Let $\Sigma u_n$ be a series of positive and negative real numbers and be absolutely convergent. Then $\Sigma\left|u_n\right|$ is a convergent series of positive terms.

Let us choose a positive $\epsilon$. Then there exists a natural number $m$ such that
||$u_{n+1}|+| u_{n+2}|+\cdots+| u_{n+p}||<\epsilon$ for all $n \geq m$ and for every natural number $p$.

That is, $\left|u_{n+1}\right|+\left|u_{n+2}\right|+\cdots+\left|u_{n+p}\right|<\epsilon$ for all $n \geq m$ and for every natural number $p$.
But $\left|u_{n+1}+u_{n+2}+\cdots+u_{n+p}\right| \leq\left|u_{n+1}\right|+\left|u_{n+2}\right|+\cdots+\left|u_{n+p}\right|$.
Therefore $\left|u_{n+1}+u_{n+2}+\cdots+u_{n+p}\right|<\epsilon$ for all $n \geq m$ and for every natural number. $p$.
By Cauchy’s principle of convergence, $\Sigma u_n$ is convergent.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Hypergeometric series

$$1+\frac{\alpha \cdot \beta}{1 . \gamma} x+\frac{\alpha(\alpha+1) \beta(\beta+1)}{1.2 \cdot \gamma(\gamma+1)} x^2+\frac{\alpha(\alpha+1)(\alpha+2) \beta(\beta+1)(\beta+2)}{1.2 .3 \cdot \gamma(\gamma+1)(\gamma+2)} x^3+\cdots$$

$\frac{u_{n+1}}{u_n}=\frac{(\alpha+n)(\beta+n)}{(1+n)(\gamma+n)} x$和$\lim {n \rightarrow \infty} \frac{u{n+1}}{u_n}=x$。

\begin{aligned} & \frac{u_n}{u_{n+1}}=\frac{(n+1)(n+\gamma)}{(n+\alpha)(n+\beta)} \ & =1+\left(\frac{(\gamma+1-\alpha-\beta) n+(\gamma-\alpha \beta)}{n^2+(\alpha+\beta) n+\alpha \beta}\right) \ & =1+\left(\frac{\gamma+1-\alpha-\beta}{n}+\frac{\gamma-\alpha \beta}{n^2}\right)\left[1+\frac{\alpha+\beta}{n}+\frac{\alpha \beta}{n^2}\right]^{-1} \ & =1+\left(\frac{\gamma+1-\alpha-\beta}{n}+\frac{\gamma-\alpha \beta}{n^2}\right)\left[1-\frac{\alpha+\beta}{n}-\frac{\alpha \beta}{n^2}+\cdots\right] \ & =1+\frac{\gamma+1-\alpha-\beta}{n}+\frac{1}{n^2}[(\gamma-\alpha \beta)-(\alpha+\beta)(\gamma+1-\alpha-\beta)+\text { terms containing } \end{aligned}
$\frac{1}{n}$和更高次幂$\frac{1}{n}$]
$=1+\frac{\gamma+1-\alpha-\beta}{n}+\frac{\phi(n)}{n^2}$，其中$\lim _{n \rightarrow \infty} \phi(n)$是有限的因此${\phi(n)}$是有界的。

$\Sigma u_n$是收敛的，如果$\gamma+1-\alpha-\beta>1$和
$\Sigma u_n$是发散的$\gamma+1-\alpha-\beta \leq 1$。

## 数学代写|实分析作业代写Real analysis代考|Series of arbitrary terms

|| $u_{n+1}|+| u_{n+2}|+\cdots+| u_{n+p}||<\epsilon$适用于所有$n \geq m$和所有自然数$p$。

## 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|Subsequential limit

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|Subsequential limit

Let $\left{u_n\right}$ be a real sequence. A real number $l$ is said to be a subsequential limit of the sequence $\left{u_n\right}$ if there exists a subsequence of $\left{u_n\right}$ that converges to $l$.

Theorem 5.12.1. A real number $l$ is a subsequential limit of a sequence $\left{u_n\right}$ if and only if every neighbourhood of $l$ contains infinitely many elements of the sequence $\left{u_n\right}$.

Proof. Let $l$ be a subsequential limit of the sequence $\left{u_n\right}$. Then there exists a subsequence $\left{u_{r_n}\right}$ such that $\lim {n \rightarrow \infty} u{r_n}=l$.

Let us choose a positive $\epsilon$. Then there exists a natural number $k$ such that $l-\epsilon<u_{r_n}<l+\epsilon$ for all $n \geq k$.
Therefore $l-\epsilon<u_n<l+\epsilon$ for infinitely many values of $n$.
Since $\epsilon$ is arbitrary, every neighbourhood of $l$ contains infinite number of elements of the sequence $\left{u_n\right}$.

Conversely, let the sequence $\left{u_n\right}$ be such that for each pre-assigned positive $\epsilon$ the $\epsilon$-neighbourhood of $l$ contains infinitely many elements of the sequence.

Let $\epsilon=1$. Then $l-1r_1\right)$ in $S_2$ such that $l-\frac{1}{2}<u_{r_2}<l+\frac{1}{2}$.

## 数学代写|实分析作业代写Real analysis代考|Characterisation of a compact set

Fheorem 5.13.1. Let $K$ be a non-empty subset of $\mathbb{R}$ : Then $K$ is compact if and only if every sequence in $K$ has a subsequence convergent to a point in $K$.
Proof. Let $K$ be a compact set. Let $\left{x_n\right}$ be a sequence in $K$.
Since $K$ is compact, $K$ is a closed and hounded set. Since $\left{x_n\right}$ is a sequence in $K$, it is a bounded sequence and by Bolzano-Weierstrass theorem it has a convergent subsequence, say $\left{x_{r_n}\right}$. Let $\lim {n \rightarrow \infty} x{r_n}=l$.
We prove that $l \in K$.
Let $l \notin K$. Then $l \in \mathbb{R}-K$. Since $K$ is a closed set, it follows that $\mathbb{R}-K$ is an open set and $l$ is an interior point of $\mathbb{R}-K$. So there exists a neighbourhood $N(l)$ of $l$ such that $N(l) \subset \mathbb{R}-K$.

Hence $N(l)$ contains no element of the sequence $\left{x_{r_n}\right}$ and therefore $l$ cannot be the limit of the sequence $\left{x_{r_n}\right}$, a contradiction.
Therefore $l \in K$.

Thus every sequence in $K$ has a subsequence convergent to a point in $K$.

Conversely, suppose that $K$ is a non-empty subset of $\mathbb{R}$ with the property that every sequence in $K$ has a subsequence convergent to a point in $K$. Let $T$ be an infinite subset of $K$.
Let $x_1 \in T, x_2 \in T-\left{x_1\right}, x_3 \in T-\left{x_1, x_2\right}, \ldots \ldots$
Continuing thus we obtain a sequence $\left{x_n\right}$ of distinct elements in $K$. By hypothesis there is a subsequence $\left{x_{r_n}\right}$ which converges to some point $x$ in $K$. Therefore $x$ is a limit point of the set $T$.

Thus $K$ is such that every infinite subset of $K$ has a limit point in $K$ and therefore $K$ is compact.
[Theorem 3.16.4] This completes the proof.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Characterisation of a compact set

forem 5.13.1。设$K$为$\mathbb{R}$的非空子集:则$K$是紧的，当且仅当$K$中的每个序列都有收敛于$K$中的一个点的子序列。

[定理3.16.4]这就完成了证明。

## 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|黎曼曲面代写Riemann surface代考|One-parameter semigroups

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface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。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups

We can now officially define the main object of study of this chapter, one-parameter semigroups on Riemann surfaces.

Definition 5.2.1. Let $X$ be a Riemann surface. A one-parameter semigroup of holomorphic maps (briefly, a one-parameter semigroup) on $X$ is a continuous semigroup homomorphism $\Phi$ from $\mathbb{R}^{+}$to $\operatorname{Hol}(X, X)$ endowed with the composition. A one-parameter group of holomorphic maps on $X$ is a continuous group homomorphism from $(\mathbb{R},+)$ to $\operatorname{Hol}(X, X)$. When $t \in \mathbb{R}^{+}$and $z \in X$, we shall often write $\Phi_t(z)$ or $\Phi(t, z)$ instead of $\Phi(t)(z)$. The trivial one-parameter semigroup is the trivial homomorphism $\Phi_t \equiv \operatorname{id}_X$ for all $t \in \mathbb{R}^{+}$. Finally, we shall say that a nontrivial one-parameter semigroup is periodic if there exists $t_0>0$ such that $\Phi_{t_0} \equiv \mathrm{id}_X$.

Remark 5.2.2. The definition of one-parameter semigroup as a continuous map $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$ has as an immediate consequence the fact that also the map, still denoted by $\Phi$, from $\mathbb{R}^{+} \times X$ to $X$ sending $(t, z)$ in $\Phi_t(z)$ is continuous.

Remark 5.2.3. If $\Phi_{t_0} \equiv \mathrm{id}X$, then $\Phi{k t_0} \equiv \mathrm{id}X$ for all $k \in \mathbb{N}$. Furthermore, if $t>t_0$, writing $t=s+k t_0$ with $k=\left\lfloor t / t_0\right\rfloor \in \mathbb{N}$ and $s \in\left[0, t_0\right)$ we see that $\Phi_t \equiv \Phi_s$, and hence $\Phi$ is completely determined by $\Phi{\left[0, t_0\right]}$.

Our first result shows that not every function can be imbedded in a one-parameter semigroup

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups on Riemann surfaces

The aim of this section is to thoroughly investigate one-parameter semigroups on Riemann surfaces different from the unit disk, postponing the study of one-parameter semigroups on $\mathbb{D}$ to the remaining sections of this chapter.
Proposition 5.3.1. Let $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$ be a one-parameter semigroup on a Riemann surface $X$ with non-Abelian fundamental group. Then $\Phi$ is trivial.
Proof. By Theorem 2.6.2, we should have $\Phi_t \equiv \mathrm{id}_X$ for small $t$, and hence for all $t$.
So, we are left with just a few cases to investigate; let us start with the Riemann sphere.

Proposition 5.3.2. Let $\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(\widehat{\mathbb{C}}, \widehat{\mathbb{C}})$ be a nontrivial one-parameter semigroup on the Riemann sphere $\widehat{\mathbb{C}}$. Then $\Phi$ extends to a one-parameter group, still denoted by $\Phi$, and there is $\gamma \in \operatorname{Aut}(\widehat{\mathbb{C}})$ such that either:
(i) $y^{-1} \circ \Phi_t \circ \gamma(z)=z+$ at for some $a \in \mathbb{C}^$, or (ii) $\gamma^{-1} \circ \Phi_t \circ \gamma(z)=e^{-b t} z$ for some $b \in \mathbb{C}^$.
In case (i), $\Phi$ has a unique fixed point with spectral value 0 and it is never periodic. In case (ii), $\Phi$ has two distinct fixed points with spectral value respectively $\pm b$; moreover, $\Phi$ is periodic if and only if $b \in \mathbb{R}^* i$ and then it has period $2 \pi /|b|$.

Proof. By Propositions 5.2.4 and 5.2.5, $\Phi$ extends to a one-parameter group, because the compactness of $\widehat{\mathbb{C}}$ implies that any injective holomorphic self-map of $\widehat{\mathbb{C}}$ is also surjective, and hence an automorphism.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups

5.2.1.定义设$X$为黎曼曲面。在$X$上的全纯映射的单参数半群(简称为单参数半群)是一个从$\mathbb{R}^{+}$到$\operatorname{Hol}(X, X)$的具有复合的连续半群同态$\Phi$。$X$上全纯映射的单参数群是从$(\mathbb{R},+)$到$\operatorname{Hol}(X, X)$的连续群同态。当$t \in \mathbb{R}^{+}$和$z \in X$时，我们经常写$\Phi_t(z)$或$\Phi(t, z)$而不是$\Phi(t)(z)$。平凡单参数半群是所有$t \in \mathbb{R}^{+}$的平凡同态$\Phi_t \equiv \operatorname{id}X$。最后，我们将说一个非平凡单参数半群是周期的，如果存在$t_0>0$使得$\Phi{t_0} \equiv \mathrm{id}_X$。

5.2.2.将单参数半群定义为连续映射$\Phi: \mathbb{R}^{+} \rightarrow \operatorname{Hol}(X, X)$的直接结果是，在$\Phi_t(z)$中发送$(t, z)$的从$\mathbb{R}^{+} \times X$到$X$的映射(仍然表示为$\Phi$)也是连续的。

5.2.3.如果是$\Phi_{t_0} \equiv \mathrm{id}X$，那么所有的$k \in \mathbb{N}$都是$\Phi{k t_0} \equiv \mathrm{id}X$。此外，如果$t>t_0$，用$k=\left\lfloor t / t_0\right\rfloor \in \mathbb{N}$和$s \in\left[0, t_0\right)$写$t=s+k t_0$，我们看到$\Phi_t \equiv \Phi_s$，因此$\Phi$完全由$\Phi{\left[0, t_0\right]}$决定。

## 数学代写|黎曼曲面代写Riemann surface代考|One-parameter semigroups on Riemann surfaces

(i) $y^{-1} \circ \Phi_t \circ \gamma(z)=z+$ at对于一些$a \in \mathbb{C}^$，或(ii) $\gamma^{-1} \circ \Phi_t \circ \gamma(z)=e^{-b t} z$对于一些$b \in \mathbb{C}^$。

## 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|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface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。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

In the last section, we saw that parabolic self-maps of $\mathbb{D}$ fall in two categories having different dynamical behavior: positive hyperbolic step and zero hyperbolic step. So it is interesting to have some procedure to decide to which category a given parabolic self-map belongs.

In this section, we collect a few results of this kind, assuming a bit of regularity at the Wolff point. The main technical step is the following.

Proposition 4.7.1. Let $F \in \operatorname{Hol}\left(\mathrm{H}^{+}, \mathbb{H}^{+}\right)$be of the form $F(w)=w+i \alpha+\eta(w)$ with $\alpha \in \mathbb{C}$ and
$$\lim _{w \rightarrow \infty} \eta(w)=0$$
Then:
(i) F is parabolic with Wolff point at infinity;
(ii) $\frac{1}{v} F^v\left(w_0\right) \rightarrow$ i $\alpha$ as $v \rightarrow+\infty$ for every $w_0 \in \mathbb{H}^{+}$;
(iii) $\operatorname{Re} \alpha \geq 0$;
(iv) for each $w_0 \in \mathbb{H}^{+}$, the sequence $\left{\operatorname{Im} F^v\right.$ ( $\left.\left.w_0\right)\right}$ is not decreasing;
(v) if $\alpha=0$, then $F$ has zero hyperbolic step;
(vi) $F$ has zero hyperbolic step if and only if $\operatorname{Im} F^v\left(w_0\right) \rightarrow+\infty$ for some (and hence all) $w_0 \in \mathbb{H}^{+}$;
(vii) if $\operatorname{Re} \alpha>0$, then $F$ has zero hyperbolic step;
(viii) if $\alpha \neq 0$, then the orbit $\left{F^v\left(w_0\right)\right}$ tends to $\infty$ nontangentially if and only if $\operatorname{Re} \alpha>0$.

## 数学代写|黎曼曲面代写Riemann surface代考|Boundary fixed points

Recall that a boundary fixed point of a $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ is a point $\sigma \in \partial \mathbb{D}$ such that $f(\sigma)=\sigma$, where $f(\sigma)$ is the nontangential limit of $f$ at $\sigma$ (see Definition 2.3.14). In Remark 2.3.15, we saw that if $\sigma$ is a boundary fixed point then we can define the derivative $f^{\prime}(\sigma)$ of $f$ at $\sigma$ by setting $f^{\prime}(\sigma)=\beta_f(\sigma) \in(0,+\infty]$; in particular, $f^{\prime}(\sigma)$ is the nontangential limit of $f^{\prime}$ at $\sigma$ when $\beta_f(\sigma)<+\infty$.

Definition 4.8.1. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be a holomorphic self-map of the unit disk. We say that $\sigma \in \partial \mathbb{D}$ is a boundary repelling fixed point if it is a boundary fixed point with $f^{\prime}(\sigma)>1$. Given $A>1$, we shall set
$$\operatorname{Fix}_A(f)=\left{\sigma \in \partial \mathbb{D} \mid f(\sigma)=\sigma \text { and } f^{\prime}(\sigma) \leq A\right}$$
Corollaries 2.3.16 and 2.5.5 say that if $f$ has a fixed point in $\mathbb{D}$, then all boundary fixed points are repelling, and that if $f$ has no fixed points in $\mathbb{D}$ then exactly one boundary fixed point is not repelling, the Wolff point of $f$. Furthermore, we have $f^{\prime}\left(\sigma_1\right) f^{\prime}\left(\sigma_2\right) \geq 1$ for all pairs of boundary fixed points (Theorem 2.3.13 contains a more precise estimate for boundary contact points).

In this section, we shall prove a precise quantitative generalization of these facts that we shall use in the next section to study the backward dynamics of a holomorphic self-map of $\mathbb{D}$.

We shall need two lemmas. The first one concerns Blaschke products (see Definition 1.5.5).

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Parabolic type and boundary smoothness

$$\lim _{w \rightarrow \infty} \eta(w)=0$$

(i) F在无穷远处具有Wolff点的抛物线;
(ii)对于每一个$w_0 \in \mathbb{H}^{+}$, $\frac{1}{v} F^v\left(w_0\right) \rightarrow$ I $\alpha$为$v \rightarrow+\infty$;
(iii) $\operatorname{Re} \alpha \geq 0$;
(iv)对于每个$w_0 \in \mathbb{H}^{+}$，顺序$\left{\operatorname{Im} F^v\right.$ ($\left.\left.w_0\right)\right}$)不递减;
(v)如果$\alpha=0$，则$F$的双曲步长为零;
(vi) $F$有零双曲阶跃当且仅当$\operatorname{Im} F^v\left(w_0\right) \rightarrow+\infty$对于一些(因此全部)$w_0 \in \mathbb{H}^{+}$;
(vii)若$\operatorname{Re} \alpha>0$，则$F$的双曲步长为零;
(viii)如果$\alpha \neq 0$，则轨道$\left{F^v\left(w_0\right)\right}$非切向$\infty$当且仅当$\operatorname{Re} \alpha>0$。

## 数学代写|黎曼曲面代写Riemann surface代考|Boundary fixed points

4.8.1.定义设$f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$为单位盘的全纯自映射。如果$\sigma \in \partial \mathbb{D}$与$f^{\prime}(\sigma)>1$为边界不动点，则称其为边界排斥不动点。给定$A>1$，我们将设置
$$\operatorname{Fix}_A(f)=\left{\sigma \in \partial \mathbb{D} \mid f(\sigma)=\sigma \text { and } f^{\prime}(\sigma) \leq A\right}$$

## 有限元方法代写

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

## MATLAB代写

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

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic dynamics

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼曲面Riemann surface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。我们的专家在代写黎曼曲面Riemann surface代写方面经验极为丰富，各种代写黎曼曲面Riemann surface相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼曲面代写Riemann surface代考|Elliptic dynamics

Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D}) \backslash\left{\mathrm{id}{\mathbb{D}}\right}$ with Wolff point $\tau_f \in \overline{\mathbb{D}}$. If $\tau_f \in \mathbb{D}$, the Schwarz-Pick lemma implies that $\left|f^{\prime}\left(\tau_f\right)\right| \leq 1$, with equality if and only if $f$ is an elliptic automorphism. On the other hand, if $\tau_f \in \partial \mathbb{D}$ then Corollary 2.5.5 implies that $0{\mathbb{D}}\right}$ with Wolff point $\tau_f \in \overline{\mathbb{D}}$. We say that $f$ is:

• elliptic if $\tau_f \in \mathbb{D}$
• hyperbolic if $\tau_f \in \partial \mathbb{D}$ and $0<f^{\prime}\left(\tau_f\right)<1$;
• parabolic if $\tau_f \in \partial \mathbb{D}$ and $f^{\prime}\left(\tau_f\right)=1$.
Moreover, if $f$ is elliptic we shall say that it is attracting if $0<\left|f^{\prime}\left(\tau_f\right)\right|<1$ and that it is superattracting if $f^{\prime}\left(\tau_f\right)=0$.

We begin studying attracting elliptic functions, which is the easiest case. We shall see that the dynamics is modeled on the dynamics of the linear map $F(z)=f^{\prime}\left(\tau_f\right) z$; in particular, we shall obtain a model (in the sense of Definition 3.5.2) of the form $(\mathbb{C}, \psi, F)$ and we shall show that the orbits approach the Wolff point in a way comparable to the way the orbits of $F$ approach the origin. This is the content of the Kœnigs theorem.

## 数学代写|黎曼曲面代写Riemann surface代考|Superattracting dynamics

The superattracting elliptic case has slightly different features, mainly because the function $f$ is never injective in a neighborhood of its Wolff point, and thus it cannot have a model in the sense of Theorem 3.5.10. However, we shall still be able to change variables so that in the new coordinates $f$ will be expressed in a simple form; but in general it will not be possible to extend the coordinate map to the whole of $\mathbb{D}$. To express our results, we need a couple of definitions.
Definition 4.2.1. Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ and let
$$f(z)=a_0+a_1\left(z-z_0\right)+a_2\left(z-z_0\right)^2+\cdots$$
be the power series expansion of $f$ at a point $z_0 \in \mathbb{D}$. The multiplicity $m_f^1\left(z_0\right)$ of $f$ at $z_0$ is given by $m_f^1\left(z_0\right)=\min \left{k \mid a_k \neq 0\right}$. More generally, given $v \geq 1$ the $v$-multiplicity $m_f^v\left(z_0\right)$ of $f$ at $z_0$ is the multiplicity of $f^v$ at $z_0$, i. e., $m_f^v\left(z_0\right)=m_{f^v}^1\left(z_0\right)$.

Clearly, we have $f(0)=0$ if and only if $m_f^1(0) \geq 1$ and 0 is superattracting if and only if $m_f^1(0) \geq 2$.

We shall now prove the superattracting version of Theorem 4.1.2, the Böttcher theorem.

Theorem 4.2.2 (Böttcher, 1904). Let $f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$ be superattracting elliptic. Let $\tau_f \in \mathbb{D}$ be its Wolff point, and $m \geq 2$ the multiplicity of $f-\tau_f$ at $\tau_f$. Then:
(i) there exists a simply connected $f$-absorbing domain $A \subset \mathbb{D}$ containing $\tau_f$ and $a$ never vanishing holomorphic function $\psi \in \operatorname{Hol}(A, \mathbb{C})$ with $\psi\left(\tau_f\right)=1$ such that the function $\varphi(z)=z \psi(z)$ is the unique solution of the functional equation
$$\varphi \circ f(z)=\varphi(z)^m$$
satisfying $\varphi\left(\tau_f\right)=0$ and $\varphi^{\prime}\left(\tau_f\right)=1$;
(ii) for every $z \in A \backslash\left{\tau_f\right}$, we have
$$\lim _{v \rightarrow+\infty}\left[\frac{f^{v+1}(z)-\tau_f}{f^v(z)-\tau_f}\right]^{1 / m^v}=\varphi(z)^{m-1} .$$
Proof. As we have seen in the proof of Theorem 4.1.2, recalling in particular (4.4), without loss of generality we can assume that $\tau_f=0$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|Superattracting dynamics

4.2.1.定义让$f \in \operatorname{Hol}(\mathbb{D}, \mathbb{D})$和让
$$f(z)=a_0+a_1\left(z-z_0\right)+a_2\left(z-z_0\right)^2+\cdots$$

(1)存在一个含有$\tau_f$和$a$不灭全纯函数$\psi \in \operatorname{Hol}(A, \mathbb{C})$与$\psi\left(\tau_f\right)=1$的单连通$f$吸收域$A \subset \mathbb{D}$，使得函数$\varphi(z)=z \psi(z)$是泛函方程的唯一解
$$\varphi \circ f(z)=\varphi(z)^m$$

(ii)对于每一个$z \in A \backslash\left{\tau_f\right}$，我们有
$$\lim _{v \rightarrow+\infty}\left[\frac{f^{v+1}(z)-\tau_f}{f^v(z)-\tau_f}\right]^{1 / m^v}=\varphi(z)^{m-1} .$$

## 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|最优化理论作业代写optimization theory代考|Minimum-Time Control of Time-Invariant Linear Systems

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

## 数学代写|最优化理论作业代写optimization theory代考|Minimum-Time Control of Time-Invariant Linear Systems

Armed with our knowledge about the form of time-optimal controls, for the remainder of this section we shall consider the following important class of problems: A linear, stationary system of order $n$ having $m$ controls is described by the state equation
$$\dot{\mathbf{x}}(t)=\mathbf{A} \mathbf{x}(t)+\mathbf{B u}(t)$$

where $\mathbf{A}$ and $\mathbf{B}$ are constant $n \times n$ and $n \times m$ matrices, respectively. The components of the control vector are constrained by
$$\left|u_i(t)\right| \leq 1, \quad i=1,2, \ldots, m$$
Assuming that the system is completely controllable and normal (no singular intervals exist), find a control, if one exists, which transfers the system from an arbitrary initial state $\mathbf{x}_0$ at time $t=0$ to the final state $\mathbf{x}\left(t_f\right)=\mathbf{0}$ in minimum time. We shall refer to this problem as the stationary, linear regulator, minimum-time problem.

From Eq. (5.4-20) we know that the optimal control, if it exists, is bangbang. Let us now state without proof some important theorems due to Pontryagin et al. [P-1] which apply to stationary, linear regulator, minimumtime problems.
THEOREM 5.4-1 (EXISTENCE)
If all of the eigenvalues of $\mathbf{A}$ have nonpositive real parts, then an optimal control exists that transfers any initial state $\mathbf{x}_0$ to the origin.
THEOREM 5.4-2 (UNIQUENESS)
If an extremal control exists, then it is unique. $\dagger$

## 数学代写|最优化理论作业代写optimization theory代考|MINIMUM CONTROL-EFFORT PROBLEMS

In the preceding section we considered problems in which the objective was to transfer a system from an arbitrary initial state to a specific target set as quickly as possible. Let us now consider problems in which control effort required, rather than elapsed time, is the criterion of optimality. Such problems arise frequently in aerospace applications, where often there are limited control resources available for achieving desired objectives.

The class of problems we will discuss is the following: Find a control $\mathbf{u}^*(t)$ satisfying constraints of the form
$$M_{i-} \leq u_i(t) \leq M_{i+}, \quad i=1,2, \ldots, m$$
which transfers a system described by
$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$
from an arbitrary initial state $\mathbf{x}_0$ to a specified target set $S(t)$ with a minimum expenditure of control effort.

As measures of control effort we shall consider the two performance indices
$$J_1(\mathbf{u})=\int_{t_0}^{t s}\left[\sum_i^m \beta_i\left|u_i(t)\right|\right] d t$$
and
$$J_2(\mathbf{u})=\int_{t_0}^{t t}\left[\sum_{i=1}^m r_i u_i^2(t)\right] d t,$$

## 数学代写|最优化理论作业代写optimization theory代考|Minimum-Time Control of Time-Invariant Linear Systems

$$\dot{\mathbf{x}}(t)=\mathbf{A} \mathbf{x}(t)+\mathbf{B u}(t)$$

$$\left|u_i(t)\right| \leq 1, \quad i=1,2, \ldots, m$$

## 数学代写|最优化理论作业代写optimization theory代考|MINIMUM CONTROL-EFFORT PROBLEMS

$$M_{i-} \leq u_i(t) \leq M_{i+}, \quad i=1,2, \ldots, m$$

$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$

$$J_1(\mathbf{u})=\int_{t_0}^{t s}\left[\sum_i^m \beta_i\left|u_i(t)\right|\right] d t$$

$$J_2(\mathbf{u})=\int_{t_0}^{t t}\left[\sum_{i=1}^m r_i u_i^2(t)\right] d t,$$

## 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|最优化理论作业代写optimization theory代考|NECESSARY CONDITIONS FOR OPTIMAL CONTROL

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

## 数学代写|最优化理论作业代写optimization theory代考|NECESSARY CONDITIONS FOR OPTIMAL CONTROL

Let us now employ the techniques introduced in Chapter 4 to determine necessary conditions for optimal control. As stated in Chapter 1, the problem is to find an admissible control $\mathbf{u}^$ that causes the system $$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$ to follow an admissible trajectory $\mathbf{x}^$ that minimizes the performance measure
$$J(\mathbf{u})=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_s} g(\mathbf{x}(t), \mathbf{u}(t), t) d t$$
We shall initially assume that the admissible state and control regions are not bounded, and that the initial conditions $\mathbf{x}\left(t_0\right)=\mathbf{x}_0$ and the initial time $t_0$ are specified. As usual, $\mathbf{x}$ is the $n \times 1$ state vector and $\mathbf{u}$ is the $m \times 1$ vector of control inputs.

In the terminology of Chapter 4 , we have a problem involving $n+m$ functions which must satisfy the $n$ differential equation constraints (5.1-1). The $m$ control inputs are the independent functions.

The only difference between Eq. (5.1-2) and the functionals considered in Chapter 4 is the term involving the final states and final time. However, assuming that $h$ is a differentiable function, we can write
$$h\left(\mathbf{x}\left(t_f\right), t_f\right)=\int_{t_0}^{t_s} \frac{d}{d t}[h(\mathbf{x}(t), t)] d t+h\left(\mathbf{x}\left(t_0\right), t_0\right),$$
so that the performance measure can be expressed as
$$J(\mathbf{u})=\int_{t_0}^{t s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$
Since $\mathbf{x}\left(t_0\right)$ and $t_0$ are fixed, the minimization does not affect the $h\left(\mathbf{x}\left(t_0\right), t_0\right)$ term, so we need consider only the functional
$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t .$$

## 数学代写|最优化理论作业代写optimization theory代考|Boundary Conditions

In a particular problem either $g$ or $h$ may be missing; in this case, we simply strike out the terms involving the missing function. To determine the boundary conditions is a matter of making the appropriate substitutions in Eq. (5.1-18). In all cases it will be assumed that we have the $n$ equations $\mathbf{x}^*\left(t_0\right)=\mathbf{x}_0$.

Problems with Fixed Final Time. If the final time $t_f$ is specified, $\mathbf{x}\left(t_f\right)$ may be specified, free, or required to lie on some surface in the state space.

CASE I. Final state specified. Since $\mathbf{x}\left(t_f\right)$ and $t_f$ are specified, we substitute $\delta \mathbf{x}_f=0$ and $\delta t_f=0$ in (5.1-18). The required $n$ equations are
$$\mathbf{x}^\left(t_f\right)=\mathbf{x}_f$$ CASE II. Final state free. We substitute $\delta t_f=0$ in Eq. (5.1-18); since $\delta \mathbf{x}_f$ is arbitrary, the $n$ equations $$\frac{\partial h}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right)-\mathbf{p}^*\left(t_f\right)=0 \dagger$$
must be satisfied.

CASE III. Final state lying on the surface defined by $\mathbf{m}(\mathbf{x}(t))=\mathbf{0}$. Since this is a new situation, let us consider an introductory example. Suppose that the final state of a second-order system is required to lie on the circle
$$m(\mathbf{x}(t))=\left[x_1(t)-3\right]^2+\left[x_2(t)-4\right]^2-4=0$$
shown in Fig. 5-1. Notice that admissible changes in $\mathbf{x}\left(t_f\right)$ are (to first-order) tangent to the circle at the point $\left(x^\left(t_f\right), t_f\right)$. The tangent line is normal to the gradient vector $$\frac{\partial m}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right)=\left[\begin{array}{l} 2\left[x_1^\left(t_f\right)-3\right] \ 2\left[x_2^\left(t_f\right)-4\right] \end{array}\right]$$
at the point ( $\left.\mathbf{x}^\left(t_f\right), t_f\right)$. Thus, $\delta \mathbf{x}\left(t_f\right)$ must be normal to the gradient (5.1-22), so that $$\left[\frac{\partial m}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right)\right]^T \delta \mathbf{x}\left(t_f\right)=2\left[x_1^\left(t_f\right)-3\right] \delta x_1\left(t_f\right)+2\left[x_2^\left(t_f\right)-4\right] \delta x_2\left(t_f\right)=0 .$$

## 数学代写|最优化理论作业代写optimization theory代考|NECESSARY CONDITIONS FOR OPTIMAL CONTROL

$$J(\mathbf{u})=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_s} g(\mathbf{x}(t), \mathbf{u}(t), t) d t$$

Eq.(5.1-2)和第4章中考虑的泛函之间的唯一区别是涉及最终状态和最终时间的术语。然而，假设$h$是一个可微函数，我们可以写
$$h\left(\mathbf{x}\left(t_f\right), t_f\right)=\int_{t_0}^{t_s} \frac{d}{d t}[h(\mathbf{x}(t), t)] d t+h\left(\mathbf{x}\left(t_0\right), t_0\right),$$

$$J(\mathbf{u})=\int_{t_0}^{t s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$

$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t .$$

## 数学代写|最优化理论作业代写optimization theory代考|Boundary Conditions

$$\mathbf{x}^\left(t_f\right)=\mathbf{x}_f$$案例二。最终状态自由。我们将$\delta t_f=0$代入式(5.1-18);因为$\delta \mathbf{x}_f$是任意的，所以$n$等于$$\frac{\partial h}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right)-\mathbf{p}^*\left(t_f\right)=0 \dagger$$

$$m(\mathbf{x}(t))=\left[x_1(t)-3\right]^2+\left[x_2(t)-4\right]^2-4=0$$

## 有限元方法代写

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

## MATLAB代写

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

## 数学代写|最优化理论作业代写optimization theory代考|Maxima and Minima of Functionals

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

## 数学代写|最优化理论作业代写optimization theory代考|Maxima and Minima of Functionals

Let us now review the definition of an extreme value of a function.

A function $f$ with domain $\mathscr{D}$ has a relative extremum at the point $\mathbf{q}^$ if there is an $\epsilon>0$ such that for all points $q$ in $\mathscr{D}$ that satisfy $\left|\mathbf{q}-\mathbf{q}^\right|<\epsilon$ the increment of $f$ has the same sign. If
$$\Delta f=f(\mathbf{q})-f\left(\mathbf{q}^\right) \geq 0$$ $f\left(\mathbf{q}^\right)$ is a relative minimum; if
$$\Delta f=f(q)-f\left(\mathbf{q}^\right) \leq 0$$ $f\left(\mathbf{q}^\right)$ is a relative maximum.
If (4.1-45) is satisfied for arbitrarily large $\epsilon$, then $f\left(\mathbf{q}^\right)$ is a global, or absolute, minimum. Similarly, if (4.1-46) holds for arbitrarily large $\epsilon$, then $f\left(\mathbf{q}^\right)$ is a global, or absolute, maximum.

Recall the procedure for locating extrema of functions. Generally, one attempts to find points where the differential vanishes-a necessary condition for an extremum at an interior point of $\mathscr{D}$. Assuming that there are such points and that they can be determined, then one can examine the behavior of the function in the vicinity of these points.

## 数学代写|最优化理论作业代写optimization theory代考|The Fundamental Theorem of the Calculus of Variations

The fundamental theorem used in finding extreme values of functions is the necessary condition that the differential vanish at an extreme point (except extrema at the boundaries of closed regions). In variational problems, the analogous theorem is that the variation must be zero on an extremal curve, provided that there are no bounds imposed on the curves. We next state this theorem and give the proof.

Let $\mathrm{x}$ be a vector function of $t$ in the class $\Omega$, and $J(\mathbf{x})$ be a differentiable functional of $\mathbf{x}$. Assume that the functions in $\Omega$ are not constrained by any boundaries.
The fundamental theorem of the calculus of variations is
If $\mathbf{x}^$ is an extremal, the variation of $J$ must vanish on $\mathbf{x}^$; that is,
$$\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right)=0 \text { for all admissible } \delta \mathbf{x} . \dagger$$ Proof by contradiction: Assume that $\mathbf{x}^$ is an extremal and that $\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right) \neq 0$. Let us show that these assumptions imply that the increment $\Delta J$ can be made to change sign in an arbitrarily small neighborhood of $\mathbf{x}^$.
The increment is
\begin{aligned} \Delta J\left(\mathbf{x}^, \delta \mathbf{x}\right) & =J\left(\mathbf{x}^+\delta \mathbf{x}\right)-J\left(\mathbf{x}^\right) \ & =\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right)+g\left(\mathbf{x}^, \delta \mathbf{x}\right) \cdot|\delta \mathbf{x}|, \end{aligned} where $g\left(\mathbf{x}^, \delta \mathbf{x}\right) \rightarrow 0$ as $|\delta \mathbf{x}| \rightarrow 0$; thus, there is a neighborhood, $|\delta \mathbf{x}|<\epsilon$, where $g\left(\mathbf{x}^*, \delta \mathbf{x}\right) \cdot|\delta \mathbf{x}|$ is small enough so that $\delta J$ dominates the expression for $\Delta J$.

## 数学代写|最优化理论作业代写optimization theory代考|Maxima and Minima of Functionals

$$\Delta f=f(\mathbf{q})-f\left(\mathbf{q}^\right) \geq 0$$$f\left(\mathbf{q}^\right)$是相对最小值;如果
$$\Delta f=f(q)-f\left(\mathbf{q}^\right) \leq 0$$$f\left(\mathbf{q}^\right)$是相对最大值。

## 数学代写|最优化理论作业代写optimization theory代考|The Fundamental Theorem of the Calculus of Variations

$$\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right)=0 \text { for all admissible } \delta \mathbf{x} . \dagger$$反证法:假设$\mathbf{x}^$是一个极值，$\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right) \neq 0$。让我们来证明，这些假设意味着，增量$\Delta J$可以在$\mathbf{x}^$的任意小邻域内改变符号。

\begin{aligned} \Delta J\left(\mathbf{x}^, \delta \mathbf{x}\right) & =J\left(\mathbf{x}^+\delta \mathbf{x}\right)-J\left(\mathbf{x}^\right) \ & =\delta J\left(\mathbf{x}^, \delta \mathbf{x}\right)+g\left(\mathbf{x}^, \delta \mathbf{x}\right) \cdot|\delta \mathbf{x}|, \end{aligned}，其中$g\left(\mathbf{x}^, \delta \mathbf{x}\right) \rightarrow 0$表示$|\delta \mathbf{x}| \rightarrow 0$;因此，存在一个邻域$|\delta \mathbf{x}|<\epsilon$，其中$g\left(\mathbf{x}^*, \delta \mathbf{x}\right) \cdot|\delta \mathbf{x}|$足够小，因此$\delta J$支配了$\Delta J$的表达式。

## 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。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Many of the ethical issues in international business are rooted in differences in political systems, laws, economic development, and culture across countries. What is considered normal practice in one nation may be considered unethical in another. Managers in a multinational firm need to be particularly sensitive to these differences. In the international business setting, the most common ethical issues involve employment practices, human rights, environmental regulations, corruption, and the moral obligation of multinational corporations.
EMPLOYMENT PRACTICES
When work conditions in a host nation are clearly inferior to those in a multinational’s home nation, which standards should be applied? Those of the home nation, those of the host nation, or something in between? While few would suggest that pay and work conditions should be the same across nations, how much divergence is acceptable? For example, while 12-hour workdays, extremely low pay, and a failure to protect workers against toxic chemicals may be common in some less developed nations, does this mean that it is okay for a multinational company to tolerate such working conditions in its subsidiaries or to condone it by using local subcontractors?

## 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。